EMOIRS

of the

American Mathematical Society

Volume 229

•

Number 1074 (first of 5 numbers)

•

May 2014

Global and Local Regularity of

Fourier Integral Operators on

Weighted and Unweighted Spaces

David Dos Santos Ferreira

Wolfgang Staubach

ISSN 0065-9266 (print)

ISSN 1947-6221 (online)

American Mathematical Society

EMOIRS

of the

American Mathematical Society

Volume 229

•

Number 1074 (first of 5 numbers)

•

May 2014

Global and Local Regularity of

Fourier Integral Operators on

Weighted and Unweighted Spaces

David Dos Santos Ferreira

Wolfgang Staubach

ISSN 0065-9266 (print)

ISSN 1947-6221 (online)

American Mathematical Society

Providence, Rhode Island

Library of Congress Cataloging-in-Publication Data

Ferreira, David Dos Santos, 1975-

Global and local regularity of Fourier integral operators on weighted and unweighted spaces /

David Dos Santos Ferreira, Wolfgang Staubach.

pages cm. – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; volume 229,

number 1074)

“May 2014, volume 229, number 1074 (ﬁrst of 5 numbers).”
Includes bibliographical references.
ISBN 978-0-8218-9119-3 (alk. paper)
1. Fourier integral operators.

2. Mathematical analysis.

I. Staubach, Wolfgang, 1970-

II. American Mathematical Society.

III. Title.

QA403.5.F47

2014

515

.723–dc23

2013051215

DOI: http://dx.doi.org/10.1090/memo/1074

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Contents

Introduction

vii

Chapter 1.

Prolegomena

1.1.

Deﬁnitions, notations and preliminaries

1.2.

Tools in proving L

boundedness

1.3.

Links between nonsmoothness and global boundedness

Chapter 2.

Global Boundedness of Fourier Integral Operators

2.1.

Global L

boundedness of rough Fourier integral operators

2.2.

Local and global L

boundedness of Fourier integral operators

2.3.

Global L

∞

boundedness of rough Fourier integral operators

2.4.

Global L

-L

and L

-L

boundedness of Fourier integral operators

Chapter 3.

Global and Local Weighted L

Boundedness of Fourier Integral

Operators

3.1.

Tools in proving weighted boundedness

3.2.

Counterexamples in the context of weighted boundedness

3.3.

Invariant formulation in the local boundedness

3.4.

Weighted local and global L

boundedness of Fourier integral operators 48

Chapter 4.

Applications in Harmonic Analysis and Partial Diﬀerential

Equations

4.1.

Estimates in weighted Triebel-Lizorkin spaces

4.2.

Commutators with BMO functions

4.3.

Applications to hyperbolic partial diﬀerential equations

Bibliography

iii

Abstract

We investigate the global continuity on L

spaces with p

∈ [1, ∞] of Fourier

integral operators with smooth and rough amplitudes and/or phase functions sub-
ject to certain necessary non-degeneracy conditions. In this context we also prove
the optimal global L

boundedness result for Fourier integral operators with non-

degenerate phase functions and the most general smooth H¨

ormander class ampli-

tudes i.e. those in S

,δ

with , δ

∈ [0, 1]. We also prove the very ﬁrst results con-

cerning the continuity of smooth and rough Fourier integral operators on weighted
L

spaces, L

with 1 < p <

∞ and w ∈ A

, (i.e. the Muckenhoupt weights)

for operators with rough and smooth amplitudes and phase functions satisfying a
suitable rank condition. These results are shown to be optimal for operators with
amplitudes in classical H¨

ormander classes and can also be given a geometrically in-

variant formulation. The weighted results are in turn applied to prove, for the ﬁrst
time, weighted and unweighted estimates for the commutators of Fourier integral
operators with functions of bounded mean oscillation BMO, estimates on weighted
Triebel-Lizorkin spaces, and ﬁnally global unweighted and local weighted estimates
for the solutions of the Cauchy problem for m-th and second order hyperbolic par-
tial diﬀerential equations on R

. The global estimates in this context, when the

Sobolev spaces are L

based, are the best possible.

Received by the editor Nov 17 2011, and, in revised form, May 31, 2012.
Article electronically published on September 24, 2013.
DOI: http://dx.doi.org/10.1090/memo/1074
2010 Mathematics Subject Classiﬁcation. Primary 35S30, 42B99.
Key words and phrases. Fourier integral operators, Weighted estimates, BMO commutators.
During the preparation of this manuscript the ﬁrst author was partially supported by ANR

grant Equa-disp.

During the preparation of this manuscript the second author was partially supported by the

EPSRC First Grant Scheme, reference number EP/H051368/1.

2013 American Mathematical Society

Introduction

A Fourier integral operator is an operator that can be written locally in the

form

a,ϕ

u(x) = (2π)

−n

iϕ(x,ξ)

a(x, ξ)

u(ξ) dξ,

(0.1)

where a(x, ξ) is the amplitude and ϕ(x, ξ) is the phase function. Historically, a sys-
tematic study of smooth Fourier integral operators with amplitudes in C

∞

) with

|∂

∂

a(x, ξ)

| ≤ C

αβ

(1 +

|ξ|)

−|α|+δ|β|

(i.e. a(x, ξ)

∈ S

,δ

), and phase

functions in C

∞

×R

\0) homogenous of degree 1 in the frequency variable ξ and

with non-vanishing determinant of the mixed Hessian matrix (i.e. non-degenerate
phase functions), was initiated in the classical paper of L. H¨

ormander [H3]. Fur-

thermore, G. Eskin [Esk] (in the case a(x, ξ)

∈ S

1,0

) and H¨

ormander [H3] (in the

case a(x, ξ)

∈ S

−

1
2

≤ 1) showed the local L

boundedness of Fourier inte-

gral operators with non-degenerate phase functions. Later on, H¨

ormander’s local

result was extended by R. Beals [RBE] and A. Greenleaf and G. Uhlmann [GU]

to the case of amplitudes in S

1
2

After the pioneering investigations by M. Beals [Be], the optimal results con-

cerning local continuity properties of smooth Fourier integral operators (with non-
degenerate and homogeneous phase functions) in L

for 1

≤ p ≤ ∞, were obtained

in the seminal paper of A. Seeger, C. D. Sogge and E.M. Stein [SSS]. This also
paved the way for further investigations by G. Mockenhaupt, Seeger and Sogge in
[MSS1] and [MSS2], see also [So] and [So1]. In these investigations the bounded-
ness, from L

comp

to L

p
loc

and from L

comp

to L

q
loc

of smooth Fourier integral operators

with non-degenerate phase functions have been established, and furthermore it was
shown that the maximal operators associated with certain Fourier integral opera-
tors (and in particular constant and variable coeﬃcient hypersurface averages) are
L

bounded.

In the context of H¨

ormander type amplitudes and non-degenerate homogeneous

phase functions which are most frequently used in applications in partial diﬀerential
equations, it has been comparatively small amount of activity concerning global L

boundedness of Fourier integral operators. Among these, we would like to men-
tion the global L

boundedness of Fourier integral operators with homogeneous

phases in C

∞

× R

\ 0) and amplitudes in the H¨ormander class S

0,0

, due to

D. Fujiwara [Fuji]; the global L

boundedness of operators with inhomogeneous

phases in C

∞

× R

) and amplitudes in S

0,0

, due to K. Asada and D. Fujiwara

[AF]; the global L

boundedness of operators with smooth amplitudes in the so

called SG classes, due to E. Cordero, F. Nicola and L. Rodino in [CNR1]; the

vii

viii

INTRODUCTION

boundedness of operators with amplitudes in S

1,0

on the space of compactly sup-

ported distributions whose Fourier transform is in L

) (i.e. the

spaces)

due to Cordero, Nicola and Rodino in [CNR2] and Nicola’s reﬁnement of this in-
vestigation in [Nic] (where the roles of the smooth spatial factorizations and aﬃne
ﬁbrations have been emphasised); and ﬁnally, S. Coriasco and M. Ruzhansky’s
global L

boundedness of Fourier integral operators [CR], with smooth amplitudes

in a suitable subclass of the H¨

ormander class S

1,0

, where certain decay of the am-

plitudes in the spatial variables are assumed. We should also mention that before
the appearance of the paper [CR], M. Ruzhansky and M. Sugimoto had already
proved in [Ruz 2] certain weighted L

boundedness (with some power weights) as

well as the global unweighted L

boundedness of Fourier integrals operators with

phases in C

∞

×R

) that are not necessarily homogeneous in the frequency vari-

ables, and amplitudes that are in the class S

0,0

. In all the aforementioned results,

one has assumed certain bounds on the derivatives of the phase functions and also
a stronger non-degeneracy condition than the one required in the local L

estimates.

In this paper, we shall take all these results as our point of departure and make

a systematic study of the global L

boundedness of Fourier integral operators with

amplitudes in S

,δ

with and δ in [0, 1], which cover all the possible ranges of

’s and δ’s. Furthermore we initiate the study of weighted norm inequalities for
Fourier integral operators with weights in the A

class of Muckenhoupt and use our

global unweighted L

results to prove a sharp weighted L

boundedness theorem

for Fourier integral operators. The weighted results in turn will be used to establish
the validity of certain vector-valued inequalities and more importantly to prove the
weighted and unweighted boundedness of commutators of Fourier integral opera-
tors with functions of bounded mean oscillation BMO. Thus, all the results of this
paper are connected and each chapter uses the results of the previous ones. This
has been reﬂected in the structure of the paper and the presentation of the results.

As mentioned earlier, in [SSS] the sharp local L

boundedness of the Fourier

integral operators was established under the assumption of the non-degeneracy of
the phase function. Furthermore, in the same context, it was shown in [Nic] that
if the rank of the spatial Hessian ∂

ϕ(x, ξ) is bounded from above, then for an

appropriate order of the amplitude which depends on that upper bound (and turns
out to be sharp), the corresponding Fourier integral operator is bounded on

Concerning the speciﬁc conditions that are put in this paper on the phase functions,
it has been known at least since the appearance of the papers [Fuji], [AF], [Ruz 2]
and [CR], that one has to assume stronger conditions, than mere non-degeneracy,
on the phase function in order to obtain global L

boundedness results. It turns out

that the assumption on the phase function, referred to in this paper as the strong
non-degeneracy condition, which requires a nonzero lower bound on the modulus
of the determinant of the mixed Hessian of the phase, is actually necessary for the
validity of global regularity of Fourier integral operators, see section 1.2.5. Further-
more, we also introduce the class Φ

of homogeneous (of degree 1) phase functions

with a speciﬁc control over the derivatives of orders greater than or equal to k,
and assume our phases to be strongly non-degenerate and belong to Φ

for some

k. At ﬁrst glance, these conditions might seem restrictive, but fortunately they are

INTRODUCTION

general enough to accommodate the phase functions arising in the study of hyper-
bolic partial diﬀerential equations and will still apply to the most generic phases in
practical applications.

Concerning our choice of amplitudes, there are some features that set our in-

vestigations apart from those made previously, for example partly motivated by
the investigation of C.E. Kenig and W. Staubach [KS], of the L

boundedness of

the so called pseudo-pseudodiﬀerential operators, we consider the global and local
L

boundedness of Fourier integral operators when the amplitude a(x, ξ) belongs

to the class L

∞

, wherein a(x, ξ) behaves in the spatial variable x like an L

∞

function, and in the frequency variable ξ, the behaviour is that of a symbol in the
H¨

ormander class S

It is worth mentioning that the conditions deﬁning classes Φ

, L

∞

and the

assumption of strong non-degeneracy make the global results obtained here natural
extensions of the local boundedness results of Seeger, Sogge and Stein’s in [SSS].
Apart from the obvious local to global generalizations, this is because on one hand,
our methods can handle the singularity of the phase function in the frequency vari-
ables at the origin and therefore the usual assumption that ξ

= 0 in the support of

the amplitude becomes obsolete. On the other hand, we do not require any regular-
ity (and therefore no decay of derivatives) in the spatial variables of the amplitudes.
Therefore, our amplitudes are close to, and in fact are spatially non-smooth ver-
sions of those in the Seeger-Sogge-Stein’s paper [SSS]. Indeed, in [SSS] the authors
although dealing with spatially smooth amplitudes, assume neither any decay in
the spatial variables nor the vanishing of the amplitude in a neighbourhood of the
singularity of the phase function.

There are several steps involved in the proof of the results of the paper and

there are discussions about various conditions that we impose on the operators as
well as some motivations for the study of rough operators. Moreover, giving exam-
ples and counterexamples when necessary, we have strived to give motivations for
our assumptions in the statements of the theorems. Here we will not mention all
the results that have been proven in this paper, instead we chose to highlight those
that are the main outcomes of our investigations.

In Chapter 1, we set up the notations and give the deﬁnitions of the classes of

amplitudes, phase functions and weights that will be used throughout the paper.
We also include the tools that we need in proving our global boundedness results, in
this chapter. We close the chapter with a discussion about the connections between
rough amplitudes and global boundedness of Fourier integral operators.

Chapter 2 is devoted to the investigation of the global boundedness of Fourier

integral operators with smooth or rough phases, and smooth or rough amplitudes.
To achieve our global boundedness results, we split the operators in low and high
frequency parts and show the boundedness of each and one of them separately. In
proving the L

boundedness of the low frequency portion, see Theorem 1.18, we

utilise Lemma 1.17 which yields a favourable kernel estimate for the low frequency

INTRODUCTION

operator, thereafter we use the phase reduction of Lemma 1.2.3 to bring the oper-
ator to a canonical form, and ﬁnally we use the L

boundedness of the non-smooth

substation in Corollary 1.16 to conclude the proof. Thus, we are able to establish
the global L

boundedness of the frequency-localised portion of the operator, for

all p’s in [1,

∞] simultaneously.

The global boundedness of the high frequency portion of the Fourier integral

operator needs to be investigated in three steps. First we show the L

−L

bounded-

ness then we proceed to the L

−L

boundedness and ﬁnally we prove the L

∞

−L

∞

boundedness.

In order to show the L

boundedness of Theorem 2.1, we use a semi-classical

reduction from Subsection 1.2.1 and Lemma 1.8, which will be used throughout
the paper. Thereafter we use the semiclassical version of the Seeger-Sogge-Stein
decomposition which was introduced in a microlocal form in [SSS].

For our global L

boundedness result, we also consider amplitudes with < δ

which appear in the study of Fourier integral operators on nilpotent Lie groups and
also in scattering theory. In Theorem 2.7, we show a global L

boundedness result

for operators with smooth H¨

ormander class amplitudes in S

min(0,

(

−δ))

,δ

∈ [0, 1],

∈ [0, 1). Also, in Theorem 2.8 we prove the L

boundedness of operators with am-

plitudes belonging to S

, with m <

(

−1). In both of these theorems, the phase

functions are assumed to satisfy the strong non-degeneracy condition and both of
these results are sharp. It should be noted that the previous local results in this
context, e.g. those due to H¨

ormander were based on the calculus of Fourier integral

operators, which in our global setting and for the most general ranges of parameters
, δ treated here, is not available. The L

boundedness theorems here can be viewed

as extensions of the celebrated Calder´

on-Vaillancourt theorem [CV] to the case of

Fourier integral operators. Indeed, the phase function of a pseudodiﬀerential oper-
ator, which is

x, ξ is in class Φ

and satisﬁes the strong non-degeneracy condition

and therefore our L

boundedness result completes the L

boundedness theory of

smooth Fourier integral operators with homogeneous non-degenerate phases.
Finally, in Theorem 2.16 we prove the global L

∞

boundedness of Fourier integral

operators, where in the proof we follow almost the same line of argument as in
the proof of the L

boundedness case, but to obtain the result which we desire,

we make a more detailed analysis of the kernel estimates which bring us beyond
the result implied by the mere utilisation of the Seeger-Sogge-Stein decomposition.
Furthermore, in this case, no non-degeneracy assumption on the phase is required.
Our results above are summarised in the following global L

boundedness theorem,

see Theorem 2.17:

A. Global L

boundedness of smooth Fourier integral operators. Let T be

a Fourier integral operator given by (0.1) with amplitude a

∈ S

,δ

and a strongly

non-degenerate phase function ϕ(x, ξ)

∈ Φ

. Setting λ := min(0, n(

− δ)), suppose

that either of the following conditions hold:

INTRODUCTION

(a) 1

≤ p ≤ 2, 0 ≤ ≤ 1, 0 ≤ δ ≤ 1, and

m < n(

− 1)

− 1

−

+ λ

−

;

(b) 2

≤ p ≤ ∞, 0 ≤ ≤ 1, 0 ≤ δ ≤ 1, and

m < n(

− 1)

−

+ (n

− 1)

−

;

≤ ≤ 1, 0 ≤ δ < 1, and

m =

Then there exists a constant C > 0 such that

T u

≤ Cu

. For Fourier inte-

gral operators with rough amplitudes we show in Theorem 2.18 the following:

B. Global L

boundedness of rough Fourier integral operators. Let T be a

Fourier integral operator given by (0.1) with amplitude a

∈ L

∞

, 0

≤ ≤ 1 and

a strongly non-degenerate phase function ϕ(x, ξ)

∈ Φ

. Suppose that either of the

following conditions hold:

(a) 1

≤ p ≤ 2 and

m <

(

− 1) +

− 1

−

;

(b) 2

≤ p ≤ ∞ and

m <

(

− 1) + (n − 1)

−

Then there exists a constant C > 0 such that

T u

≤ Cu

. We also extend

both of the results above, i.e. the L

−L

regularity of smooth and rough operators,

to the L

− L

regularity, in Theorem 2.20.

After proving the global regularity of Fourier integral operators with smooth phase
functions, we turn to the problem of local and global boundedness of operators
which are merely bounded in the spatial variables in both their phases and am-
plitudes. A motivation for this investigation stems from the study of maximal
estimates involving Fourier integral operators, where a standard stopping time ar-
gument (here referred to as linearisation) reduces the problem to a Fourier integral
operator with a non-smooth phase and sometimes also a non-smooth amplitude.
For instance, estimates for the maximal spherical average operator

Au(x) = sup

∈[0,1]

−1

u(x + tω) dσ(ω)

which is directly related to the rough Fourier integral operator

T u(x) = (2π)

−n

a(x, ξ)e

it(x)

|ξ|+ix,ξ

u(ξ) dξ

where t(x) is a measurable function in x, with values in [0, 1] and a(x, ξ)

∈ L

∞

−

−1

Here, the phase function of the Fourier integral operator is ϕ(x, ξ) = it(x)

|ξ|+ix, ξ

which is merely an L

∞

function in the spatial variables x, but is smooth outside

xii

INTRODUCTION

the origin in the frequency variables ξ. As we shall see later, according to Deﬁnition
1.5, this phase belongs to the class L

∞

In our investigation of local or global L

boundedness of the rough Fourier integral

operators above for p

= 2, the results obtained are similar to those of the local

results for amplitudes in the class S

1,0

obtained in [MSS1], [MSS2] and [So1] for

(2.49). However, we consider more general classes of amplitudes (i.e. the S

,δ

class)

and also require only measurability and boundedness in the spatial variables (i.e.
the L

∞

class). The main results in this context are the L

boundedness results

which apart from the case of Fourier integral operators in dimension n = 1, yield a
problem of considerable diﬃculty in case one desires to prove a L

regularity result

under the sole assumption of rough non-degeneracy, see Deﬁnition 1.6.

Using the geometric conditions (imposed on the phase functions) which are the

rough analogues of the non-degeneracy and corank conditions for smooth phases
(the rough corank condition 2.2.2), we are able to prove a local L

boundedness

result with a certain loss of derivatives depending on the rough corank of the phase.
More explicitly we prove in Theorem 2.10:

C. Local L

boundedness of Fourier integral operators with rough am-

plitudes and phases. Let T be a Fourier integral operator given by (0.1) with
amplitude a

∈ L

∞

and phase function ϕ

∈ L

∞

. Suppose that the phase

satisﬁes the rough corank condition 2.2.2, then T can be extended as a bounded
operator from L

comp

to L

2
loc

provided m <

−

n+k

−1

−k)(−1)

Despite the lack of sharpness in the above theorem, the proof is rather techni-
cal. However, in case n = 1 this theorem can be improved to yield a local L

boundedness result with m <

−1

, and if the assumptions on the phase function

are also strengthen with a Lipschitz condition on the ξ derivatives of order 2 and
higher of the phase, then the above theorem holds with a loss m <

−

k
2

−k)(−1)

In Chapter 3 we turn to the problem of weighted norm inequalities for Fourier

integral operators. To our knowledge this question has never been investigated
previously in the context of Muckenhoupt’s A

weights which are the most natural

class of weights in harmonic analysis. Here we start this investigation by establish-
ing sharp boundedness results for a fairly wide class of Fourier integral operators,
somehow modeled on the parametrices of hyperbolic partial diﬀerential equations.
One notable feature of our investigation is that we also prove the results for Fourier
integral operators whose phase functions and amplitudes are only bounded and
measurable in the spatial variables and exhibit suitable symbol type behavior in
the frequency variables.
As before, we begin by discussing the weighted estimates for the low frequency
portion of the Fourier integral operators which can be handled by Lemma 1.17. As
a matter of fact, the weighted L

boundedness of low frequency parts of Fourier

integral operators is merely an analytic issue involving the right decay rates of the
phase function and does not involve any rank condition on the phase. The situation
in the high frequency case is entirely diﬀerent. Here, there is also a signiﬁcant dis-
tinction between the weighted and unweighted case, in the sense that, if one desires
to prove sharp weighted estimates, then a rank condition on the phase function
is absolutely crucial. This fact has been discussed in detail in Section 3.2, where

INTRODUCTION

xiii

one ﬁnds various examples, including one related to the wave equation, and coun-
terexamples which will lead us to the correct condition on the phase. Then we will
proceed with the local high frequency and global high frequency boundedness esti-
mates. As a rule, in the investigation of boundedness of Fourier integral operators,
the local estimates require somewhat milder conditions on the phase functions com-
pared to the global estimates and our case study of the weighted norm inequalities
here is no exception to this rule. Furthermore, we are able to formulate the local
weighted boundedness results in an invariant way involving the canonical relation
of the Fourier integral operator in question. Our main results in this context are
contained in Theorem 3.11:

D. Weighted L

boundedness of Fourier integral operators. Let a(x, ξ)

∈

∞

−

n+1

+n(

−1)

and

∈ [0, 1]. Suppose that either

(a) a(x, ξ) is compactly supported in the x variable and the phase function

ϕ(x, ξ)

∈ C

∞

× R

\ 0), is positively homogeneous of degree 1 in ξ

and satisﬁes, det ∂

xξ

ϕ(x, ξ)

= 0 as well as rank ∂

ξξ

ϕ(x, ξ) = n

− 1; or

(b) ϕ(x, ξ)

− x, ξ ∈ L

∞

, ϕ satisﬁes the rough non-degeneracy condition

as well as

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0.

Then the operator T

a,ϕ

is bounded on L

for p

∈ (1, ∞) and all w ∈ A

. Further-

more, for = 1 this result is sharp.
Here, it is worth mentioning that in the non-endpoint case, i.e. if a(x, ξ)

∈ L

∞

with m <

−

n+1

+ n(

− 1), we can prove a result that requires no non-degeneracy

assumption on the phase function. The proof of these statements are long and
technical and use several steps involving careful kernel estimates, uniform pointwise
estimates on certain oscillatory integrals, unweighted local and global L

bounded-

ness, interpolation, and extrapolation.

In Chapter 4 we are motivated by the fact that weighted norm inequalities with

weights can be used as an eﬃcient tool in proving vector valued inequalities and

also boundedness of commutators of operators with functions of bounded mean os-
cillation BMO. Therefore, we start the chapter by showing boundedness of certain
Fourier integral operators in weighted Triebel-Lizorkin spaces (see (4.4)). This is
based on a vector valued inequality for Fourier integral operators.
But more importantly we prove for the ﬁrst time, in Theorems 4.8 and 4.9, the
boundedness and weighted boundedness of BMO commutators of Fourier integral
operators, namely

E. L

boundedness of BMO commutators of Fourier integral operators.

Suppose either

(a) T

∈ I

,comp

×R

;

C) with

1
2

≤ ≤ 1 and m < (−n)|

1
p

−

1
2

|, satisﬁes

all the conditions of Theorem 3.12 or;

(b) T

a,ϕ

with a

∈ S

,δ

, 0

≤ ≤ 1, 0 ≤ δ ≤ 1, λ = min(0, n(−δ)) and ϕ(x, ξ)

is a strongly non-degenerate phase function with ϕ(x, ξ)

− x, ξ ∈ Φ

where in the range 1 < p

≤ 2,

m < n(

− 1)

− 1

−

+ λ

−

;

xiv

INTRODUCTION

and in the range 2

≤ p < ∞

m < n(

− 1)

−

+ (n

− 1)

−

;

a,ϕ

with a

∈ L

∞

, 0

≤ ≤ 1 and ϕ is a strongly non-degenerate

phase function with ϕ(x, ξ)

− x, ξ ∈ Φ

, where in the range 1 < p

≤ 2,

m <

(

− 1) +

− 1

−

and for the range 2

≤ p < ∞

m <

(

− 1) + (n − 1)

−

Then for b

∈ BMO, the commutators [b, T ] and [b, T

a,ϕ

] are bounded on L

with

1 < p <

∞. Here we like to mention that once again, the global L

bounded in

Theorem A above is used in the proof of the L

boundedness of the BMO commu-

tators. Finally, the weighted norm inequalities with weights in all A

classes have

the advantage of implying weighted boundedness of repeated commutators, namely
one has

F. Weighted L

boundedness of k-th BMO commutators of Fourier in-

tegral operators. Let a(x, ξ)

∈ L

∞

−

n+1

+n(

−1)

and

∈ [0, 1]. Suppose that

either

(a) a(x, ξ) is compactly supported in the x variable and the phase function

ϕ(x, ξ)

∈ C

∞

× R

\ 0), is positively homogeneous of degree 1 in ξ

and satisﬁes, det ∂

xξ

ϕ(x, ξ)

= 0 as well as rank ∂

ξξ

ϕ(x, ξ) = n

− 1; or

(b) ϕ(x, ξ)

− x, ξ ∈ L

∞

, ϕ satisﬁes the rough non-degeneracy condition

as well as

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0.

Then, for b

∈ BMO and k a positive integer, the k-th commutator deﬁned by

a,b,k

u(x) := T

(b(x)

− b(·))

(x)

is bounded on L

for each w

∈ A

and p

∈ (1, ∞).

These BMO estimates have no predecessors in the literature and are useful in
connection to the study of hyperbolic partial diﬀerential equations with rough co-
eﬃcients.
In the last section of Chapter 4, we also brieﬂy discuss global unweighted and local
weighted estimates for the solutions of the Cauchy problem for m-th and second
order hyperbolic partial diﬀerential equations.

Acknowledgements. Part of this work was undertaken while one of the authors
was visiting the department of Mathematics of the Heriot-Watt University. The ﬁrst
author wishes to express his gratitude for the hospitality of Heriot-Watt University.

CHAPTER 1

Prolegomena

In this chapter, we gather some results which will be useful in the study of

boundedness of Fourier integral operators. We also illustrate some of the connec-
tions between global boundedness results for operators with smooth phases and
amplitudes and local boundedness results for operators with rough phases and am-
plitudes, thus justifying a joint study of those operators.

1.1. Deﬁnitions, notations and preliminaries

1.1.1. Phases and amplitudes. In our investigation of the regularity prop-

erties of Fourier integral operators, we will be concerned with both smooth and
non-smooth amplitudes and phase functions. Below, we shall recall some basic def-
initions and ﬁx some notations which will be used throughout the paper. Also, in
the sequel we use the notation

ξ for (1 + |ξ|

)

1
2

. The following deﬁnition which

is due to H¨

ormander [H0], yields one of the most widely used classes of smooth

symbols/amplitudes.

Definition

1.1. Let m

∈ R, 0 ≤ δ ≤ 1, 0 ≤ ≤ 1. A function a(x, ξ) ∈

∞

× R

) belongs to the class S

,δ

, if for all multi-indices α, β it satisﬁes

sup

∈R

−m+|α|−δ|β|

|∂

∂

a(x, ξ)

| < +∞.

We shall also deal with the class L

∞

of rough symbols/amplitudes intro-

duced by Kenig and Staubach in [KS].

Definition

1.2. Let m

∈ R and 0 ≤ ≤ 1. A function a(x, ξ) which is

smooth in the frequency variable ξ and bounded measurable in the spatial variable
x, belongs to the symbol class L

∞

, if for all multi-indices α it satisﬁes

sup

∈R

−m+|α|

∂

· , ξ)

∞

)

< +

∞.

We also need to describe the type of phase functions that we will deal with. To

this end, the class Φ

deﬁned below, will play a signiﬁcant role in our investigations.

Definition

1.3. A real valued function ϕ(x, ξ) belongs to the class Φ

, if

ϕ(x, ξ)

∈ C

∞

× R

\ 0), is positively homogeneous of degree 1 in the frequency

variable ξ, and satisﬁes the following condition: For any pair of multi-indices α and
β, satisfying

|α| + |β| ≥ k, there exists a positive constant C

α,β

such that

sup

(x, ξ)

∈R

×R

|ξ|

−1+|α|

|∂

∂

ϕ(x, ξ)

| ≤ C

α,β

In connection to the problem of local boundedness of Fourier integral operators,

one considers phase functions ϕ(x, ξ) that are positively homogeneous of degree 1

1. PROLEGOMENA

in the frequency variable ξ for which det[∂

ϕ(x, ξ)]

= 0. The latter is referred

to as the non-degeneracy condition. However, for the purpose of proving global
regularity results, we require a stronger condition than the aforementioned weak
non-degeneracy condition.

Definition

1.4. (The strong non-degeneracy condition). A real valued phase

∈ C

× R

\ 0) satisﬁes the strong non-degeneracy condition, if there exists

a positive constant c such that

det

∂

ϕ(x, ξ)

∂x

∂ξ

≥ c,

for all (x, ξ)

∈ R

× R

\ 0.

The phases in class Φ

satisfying the strong non-degeneracy condition arise

naturally in the study of hyperbolic partial diﬀerential equations, indeed a phase
function closely related to that of the wave operator, namely ϕ(x, ξ) =

|ξ| + x, ξ

belongs to the class Φ

and is strongly non-degenerate.

We also introduce the non-smooth version of the class Φ

which will be used

throughout the paper.

Definition

1.5. A real valued function ϕ(x, ξ) belongs to the phase class

∞

, if it is homogeneous of degree 1 and smooth on R

\ 0 in the frequency

variable ξ, bounded measurable in the spatial variable x, and if for all multi-indices
|α| ≥ k it satisﬁes

sup

∈R

|ξ|

−1+|α|

∂

ϕ(

· , ξ)

∞

)

< +

∞.

We observe that if t(x)

∈ L

∞

then the phase function ϕ(x, ξ) = t(x)

|ξ| + x, ξ

belongs to the class L

∞

, hence phase functions originating from the linearisation

of the maximal functions associated with averages on surfaces, can be considered
as members of the L

∞

class. We will also need a rough analogue of the non-

degeneracy condition, which we deﬁne below.

Definition

1.6. (The rough non-degeneracy condition). A real valued phase ϕ

satisﬁes the rough non-degeneracy condition, if it is C

on R

\ 0 in the frequency

variable ξ, bounded measurable in the spatial variable x, and there exists a constant
c > 0 (depending only on the dimension) such that for all x, y

∈ R

and ξ

∈ R

\ 0

(1.1)

|∂

ϕ(x, ξ)

− ∂

ϕ(y, ξ)

| ≥ c|x − y|.

1.1.2. Basic notions of weighted inequalities. Our main reference for the

material in this section are [G] and [S]. Given u

∈ L

p
loc

, the L

maximal function

(u) is deﬁned by

(1.2)

(u)(x) = sup

|B|

|u(y)|

1
p

where the supremum is taken over balls B in R

containing x. Clearly then, the

Hardy-Littlewood maximal function is given by

M (u) := M

(u).

1.2. TOOLS IN PROVING L

BOUNDEDNESS

An immediate consequence of H¨

older’s inequality is that M (u)(x)

≤ M

(u)(x) for

≥ 1. We shall use the notation

|B|

|u(y)| dy

for the average of the function u over B. One can then deﬁne the class of Mucken-
houpt A

weights as follows.

Definition

1.7. Let w

∈ L

1
loc

be a positive function. One says that w

∈ A

there exists a constant C > 0 such that

(1.3)

M w(x)

≤ Cw(x),

for almost all

∈ R

One says that w

∈ A

for p

∈ (1, ∞) if

(1.4)

sup

B balls in R

−

−1

∞.

The A

constants of a weight w

∈ A

are deﬁned by

(1.5)

[w]

sup

B balls in R

−1

∞

(B)

and

(1.6)

[w]

sup

B balls in R

−

−1

Example

1. The function

|x|

is in A

if and only if

−n < α ≤ 0 and is in

with 1 < p <

∞ iﬀ −n < α < n(p − 1). Also u(x) = log

|x|

when

|x| <

1
e

and

u(x) = 1 otherwise, is an A

weight.

1.1.3. Additional conventions. As is common practice, we will denote con-

stants which can be determined by known parameters in a given situation, but
whose value is not crucial to the problem at hand, by C. Such parameters in this
paper would be, for example, m, , p, n, [w]

, and the constants C

in Deﬁnition

1.2. The value of C may diﬀer from line to line, but in each instance could be
estimated if necessary. We sometimes write a

b as shorthand for a ≤ Cb. Our

goal is to prove estimates of the form

T u

≤ Cu

∈ S (R

)

when a

∈ L

∞

, ϕ

∈ L

∞

and m <

−σ ≤ 0 or equivalently

T u

≤ Cu

s,p

∈ S (R

)

when a

∈ L

∞

and s > σ and H

s,p

{u ∈ S

; (I

− Δ)

s
2

∈ L

}. We will use

indiﬀerently one or the other equivalent formulation and we will refer to σ as the
loss of derivatives in the L

boundedness of T .

1.2. Tools in proving L

boundedness

1.2.1. Semi-classical reduction and decomposition of the operators.

It is convenient to work with semi-classical estimates: let A be the annulus

A =

∈ R

;

1
2

≤ |ξ| ≤ 2

and χ

∈ C

∞

(A) be a cutoﬀ function, we will prove estimates on the following

semi-classical Fourier integral operator

u = (2πh)

−n

ϕ(x,ξ)

χ(ξ)a(x, ξ/h)

u(ξ/h) dξ

1. PROLEGOMENA

with h

∈ (0, 1]. We will also need to investigate the low frequency component of

the operator

u = (2π)

−n

iϕ(x,ξ)

(ξ)a(x, ξ)

u(ξ) dξ

where χ

∈ C

∞

(B(0, 2)). The following lemma shows how semi-classical estimates

translate into classical ones. We choose to state the result in the realm of weighted
L

spaces with weights in the Muckenhoupt’s A

class. This extent of generality

will be needed when we deal with the weighted boundedness of Fourier integral
operators.

Lemma

1.8. Let a

∈ L

∞

and ϕ

∈ L

∞

, suppose that for all h

∈ (0, 1] and

∈ A

, there exist constants C

, C

> 0 (only depending on the A

constants of

w) such that the following estimates hold

≤ C

−m−s

∈ S (R

This implies the bound

T u

≤ C

∈ S (R

)

provided m <

−s.

Proof.

We start by taking a dyadic partition of unity

(ξ) +

∞

j=1

(ξ) = 1,

where χ

∈ C

∞

(B(0, 2)), χ

(ξ) = χ(2

−j

ξ) when j

≥ 1 with χ ∈ C

∞

(A) and we

decompose the operator T as

T = T χ

(D) +

∞

j=1

T χ

(D).

(1.7)

The ﬁrst term in (1.7) is bounded from L

to itself by assumption. After a change

of variables, we have

T χ

(D)u = (2π)

−n

ϕ(x,ξ)

χ(ξ)a(x, 2

ξ)

u(2

ξ) dξ

therefore using the semi-classical estimate with h = 2

−j

we obtain

T χ

(D)u

≤ C

(m+s)j

This ﬁnally gives

T u

≤ C

+ C

∞

j=1

(m+s)j

since the series is convergent when m <

−s. This completes the proof of our

lemma.

1.2. TOOLS IN PROVING L

BOUNDEDNESS

1.2.2. Seeger-Sogge-Stein decomposition. To get useful estimates for the

symbol and the phase function, one imposes a second microlocalization on the
former semi-classical operator in such a way that the annulus A is partitioned
into truncated cones of thickness roughly

√

h. Roughly h

−(n−1)/2

such pieces are

needed to cover the annulus A. For each h

∈ (0, 1] we ﬁx a collection of unit vectors

{ξ

}

≤ν≤J

which satisfy:

(i)

|ξ

− ξ

| ≥ h

−

1
2

, if ν

= μ,

(ii) If ξ

∈ S

−1

, then there exists a ξ

so that

|ξ − ξ

| ≤ h

1
2

Let Γ

denote the cone in the ξ space with aperture

√

h whose central direction is

, i.e.

(1.8)

∈ R

;

|ξ|

− ξ

≤

√

One can construct an associated partition of unity given by functions ψ

, each

homogeneous of degree 0 in ξ and supported in Γ

with

(1.9)

ν=1

(ξ) = 1,

for all ξ

= 0

and

sup

∈R

|∂

(ξ)

| ≤ C

−

|α|

(1.10)

We decompose the operator T

ν=1

(D) =

ν=1

(1.11)

where the kernel of the operator T

is given by

(x, y) = (2πh)

−n

ϕ(x,ξ)

−

y,ξ

χ(ξ)ψ

(ξ)a(x, ξ/h) dξ

(1.12)

= (2πh)

−n

∇

ϕ(x,ξ

)

−y,ξ

(x, ξ, h) dξ

with amplitude b

(x, ξ, h) = e

∇

ϕ(x,ξ)

−∇

ϕ(x,ξ

),ξ

χ(ξ)ψ

(ξ)a(x, ξ/h). We choose

our coordinates on R

= Rξ

⊕ ξ

⊥

in the following way

ξ = ξ

+ ξ

⊥ ξ

Also it is worth noticing that the symbol χ(ξ)a(x, ξ/h) satisﬁes the following bound

(1.13)

sup

∂

χ(ξ) a(

·, ξ/h)

∞

≤ C

−m−|α|(1−)

Lemma

1.9. Let a

∈ L

∞

and ϕ(x, ξ)

∈ L

∞

. Then the symbol

(x, ξ, h) = e

∇

ϕ(x,ξ)

−∇

ϕ(x,ξ

),ξ

(ξ)χ(ξ)a(x, ξ/h)

satisﬁes the estimates

sup

∂

(

·, ξ, h)

∞

≤ C

−m−|α|(1−)−

|α|

1. PROLEGOMENA

Proof.

We ﬁrst observe that the bounds (1.10) may be improved to

sup

∈A

∂

(ξ)

≤

−

|α|

(1.14)

This can be seen by induction on

|α|; by Euler’s identity, we have

∂

−|ξ|

−1

− ξ

∇∂

+ |α|∂

from which we deduce

|∂

∂

| ≤

|ξ|

− ξ

|∇∂

| + |α||∂

1
2

−

|α|
2

+ h

−

|α|

This ends the induction. Similarly we have

sup

∈A∩Γ

∂

∇

ϕ(

·,ξ)−∇

ϕ(

·,ξ

),ξ

∞

−

|α|

(1.15)

To prove this bound, we proceed by induction on

|α|, we have

∇

∂

∇

ϕ(x,ξ)

−∇

ϕ(x,ξ

),ξ

∂

∇

ϕ(x, ξ)

− ∇

ϕ(x, ξ

)

∇

ϕ(x,ξ)

−∇

ϕ(x,ξ

),ξ

and by the Leibniz rule, it suﬃces to verify that for

|β| ≤ 1

sup

∈A∩Γ

∂

ϕ(

·, ξ) − ∂

ϕ(

·, ξ

)

∞

−|β|

sup

∈A∩Γ

∂

ϕ(

·, ξ) − ∂

ϕ(

·, ξ

)

∞

−

|β|

where for the case β = 0 one simply uses the mean value theorem on

∇

ϕ(x, ξ)

−

∇

ϕ(x, ξ

), which due the condition ϕ

∈ L

∞

yields the desired estimates. We

note that a homogeneous function which vanishes at ξ = ξ

may be written in the

form

|ξ|

− ξ

r(x, ξ) =

√

on A

∩ Γ

and this gives the ﬁrst bound for β

= 1. We also have ∂

∂

ϕ(x, ξ

) = 0 by Euler’s

identity, therefore the former remark yields ∂

∂

ϕ(x, ξ) =

√

h) which is the ﬁrst

bound for β

= 1 (as well as the second bound for β

= 0). It remains to prove

the second bound for β

= 0: by the mean value theorem and the bounds we have

already obtained

|∂

ϕ(x, ξ)

− ∂

ϕ(x, ξ

)

√

|ξ|

− ξ

The estimates on b

are consequences of (1.13), (1.14) and (1.15) and of Leibniz’s

rule.

1.2. TOOLS IN PROVING L

BOUNDEDNESS

1.2.3. Phase reduction. In our deﬁnition of class L

∞

we have only re-

quired control of those frequency derivatives of the phase function which are greater
or equal to k. This restriction is motivated by the simple model case phase func-
tion ϕ(x, ξ) = t(x)

|ξ| + x, ξ, t(x) ∈ L

∞

, for which the ﬁrst order ξ-derivatives of

the phase are not bounded but all the derivatives of order equal or higher than 2
are indeed bounded and so ϕ(x, ξ)

∈ L

∞

. However in order to deal with low

frequency portions of Fourier integral operators one also needs to control the ﬁrst
order ξ derivatives of the phase. The following phase reduction lemma will reduce
the phase of the Fourier integral operators to a linear term plus a phase for which
the ﬁrst order frequency derivatives are bounded.

Lemma

1.10. Any Fourier integral operator T of the type (0.1) with amplitude

σ(x, ξ)

∈ L

∞

and phase function ϕ(x, ξ)

∈ L

∞

, can be written as a ﬁnite sum

of operators of the form

(1.16)

(2π)

a(x, ξ) e

iθ(x,ξ)+i

∇

ϕ(x,ζ),ξ

u(ξ) dξ

where ζ is a point on the unit sphere S

−1

, θ(x, ξ)

∈ L

∞

, and a(x, ξ)

∈ L

∞

is localized in the ξ variable around the point ζ.

Proof.

We start by localizing the amplitude in the ξ variable by introducing

an open convex covering

}

M
l=1

, with maximum of diameters d, of the unit sphere

−1

. Let Ξ

be a smooth partition of unity subordinate to the covering U

and set

(x, ξ) = σ(x, ξ) Ξ

(

|ξ|

). We set

(1.17)

u(x) :=

(2π)

(x, ξ) e

iϕ(x,ξ)

u(ξ) dξ,

and ﬁx a point ζ

∈ U

. Then for any ξ

∈ U

, Taylor’s formula and Euler’s homo-

geneity formula yield

ϕ(x, ξ) = ϕ(x, ζ) +

∇

ϕ(x, ζ), ξ

− ζ + θ(x, ξ)

= θ(x, ξ) +

∇

ϕ(x, ζ), ξ

Furthermore, for ξ

∈ U

, ∂

θ(x, ξ) = ∂

ϕ(x,

|ξ|

)

− ∂

ϕ(x, ζ), so the mean value

theorem and the deﬁnition of class L

∞

yield

|∂

θ(x, ξ)

| ≤ Cd and for |α| ≥ 2,

|∂

θ(x, ξ)

| ≤ C|ξ|

−|α|

. Here we remark in passing that in dealing with function

θ(x, ξ), we only needed to control the second and higher order ξ

−derivatives of the

phase function ϕ(x, ξ) and this gives a further motivation for the deﬁnition of the
class L

∞

. We shall now extend the function θ(x, ξ) to the whole of R

× R

\ 0,

preserving its properties and we denote this extension by θ(x, ξ) again. Hence the
Fourier integral operators T

deﬁned by

(1.18)

u(x) :=

(2π)

(x, ξ) e

iθ(x,ξ)+i

∇

ϕ(x,ζ),ξ

u(ξ) dξ,

are the localized pieces of the original Fourier integral operator T and therefore
T =

M
l=1

as claimed.

1.2.4. Necessary and suﬃcient conditions for the non-degeneracy of

smooth phase functions. The smoothness of phases of Fourier integral operators
makes the study of boundedness considerably easier in the sense that the conditions
of a phase being strongly non-degenerate and belonging to the class Φ

are enough

1. PROLEGOMENA

to secure L

boundedness for a wide range of rough amplitudes. The following

proposition which is useful in proving global L

boundedness of Fourier integral

operators, establishes a relationship between the strongly non-degenerate phases
and the lower bound estimates for the gradient of the phases in question.

Proposition

1.11. Let ϕ(x, ξ)

∈ C

∞

×R

\0) be a real valued phase then

the following statements hold true:

(i) Assume that

det

∂

ϕ(x, ξ)

∂x

∂ξ

≥ C

for all (x, ξ)

∈ R

× R

\ 0, and that

∂

ϕ(x, ξ)

∂x∂ξ

≤ C

for all (x, ξ)

∈ R

×R

\0 and some constant C

> 0, where

· denotes

matrix norm. Then

(1.19)

|∇

ϕ(x, ξ)

− ∇

ϕ(y, ξ)

| ≥ C|x − y|,

for x, y

∈ R

and ξ

∈ R

\ 0 and some C > 0.

(ii) Assume that

|∇

ϕ(x, ξ)

−∇

(y, ξ)

| ≥ C|x−y| for x, y ∈ R

and ξ

∈ R

and some C > 0. Then there exists a constant C

such that

det

∂

ϕ(x, ξ)

∂x

∂ξ

≥ C

for all (x, ξ)

∈ R

× R

\ 0.

Proof.

(i) We consider the map R

x → ∇

ϕ(x, ξ)

∈ R

and using our

assumptions on ϕ, Schwartz’s global inverse function theorem [Sch] yields that this
map is a global C

-diﬀeomorphism whose inverse λ

satisﬁes

(1.20)

|λ

(z)

− λ

(w)

| ≤ sup

[z,w]

× |z − w|.

Furthermore, λ

(z) = [(λ

−1

)

]

−1

◦ λ

(z) = [∂

x,ξ

ϕ(λ

(z), ξ)]

−1

. Therefore using the

wellknown matrix inequality

−1

≤ c

| det A|

−1

which is valid for all

∈ GL (n, R), we obtain using the assumption

∂

ϕ(x,ξ)

∂x∂ξ

≤ C

that

(z)

≤ c

| det[∂

x,ξ

ϕ(λ

(z), ξ)]

−1

∂

x,ξ

ϕ(λ

(z), ξ)

−1

≤

This yields that

|λ

(z)

− λ

(w)

| ≤ C|z − w| and setting z = ∇

ϕ(x, ξ) and w =

∇

ϕ(y, ξ), we obtain (1.19).

(ii) Given the lower bound on the diﬀerence of the gradients as in the statement

of the second part of the proposition, setting y = x + hv with v

∈ R

yields,

|∇

ϕ(x + hv, ξ)

− ∇

ϕ(x, ξ)

≥ C|v|

and letting h tend to zero we have for any v

∈ R

(1.21)

|∂

x,ξ

ϕ(x, ξ)

· v| ≥ C|v|.

1.2. TOOLS IN PROVING L

BOUNDEDNESS

This means that ∂

x,ξ

ϕ(x, ξ) is invertible and

|[∂

x,ξ

ϕ(x, ξ)]

−1

· w| ≤

|w|

. Therefore,

taking the supremum we obtain

[∂

x,ξ

ϕ(x,ξ)]

−1

≥

. Now using the wellknown

matrix inequality

(1.22)

−1

≤ | det A| ≤ γ

which is a consequence of the Hadamard inequality, yields for A =

∂

ϕ(x,ξ)

∂x

∂ξ

(1.23)

det

∂

ϕ(x, ξ)

∂x

∂ξ

≥

This completes the proof.

Remark

1.12. Proposition 1.11 gives a motivation for our rough non-degeneracy

condition in Deﬁnition 1.5, when there is no diﬀerentiability in the spatial variables.

1.2.5. Necessity of strong non-degeneracy for global regularity. We

shall now discuss a simple example which illustrates the necessity of the strong non-
degeneracy condition for the validity o global L

boundedness of Fourier integral

operators. To this end, we take a smooth diﬀeomorphism κ : R

→ R

with

everywhere nonzero Jacobian determinant, i.e. det κ

(x)

= 0 for all x ∈ R

. Now,

if we let ϕ(x, ξ) =

κ(x), ξ and take a(x, ξ) = 1 ∈ S

1,0

, then the Fourier integral

operator T

a,ϕ

u(x) is nothing but the composition operator u

◦ κ(x). Therefore

(1.24)

a,ϕ

u ◦ κ

{

|u(y)|

| det κ

−1

(κ

−1

(y))

| dy}

1
p

from which we see that T

a,ϕ

is L

bounded for any p, if and only if there exists a

constant C > 0 such that

| det κ

−1

(x)

| ≤ C for all x ∈ R

. The latter is equivalent

| det κ

(x)

| ≥

> 0. Now since

| det

∂

ϕ(x,ξ)

∂x∂ξ

| = | det κ

(x)

| it follows at once that

a necessary condition for the L

boundedness of the operator T

a,ϕ

is the strong

non-degeneracy of the phase function ϕ. We observe that if we instead had chosen
a(x, ξ) to be equal to a smooth compactly supported function in x, then the L

boundedness of T

a,ϕ

would have followed from the mere non-degeneracy condition

| det

∂

ϕ(x,ξ)

∂x∂ξ

| = | det κ

(x)

| = 0.

1.2.6. Non smooth changes of variables. In dealing with rough Fourier

integral operators we would need at some point to make changes of variables when
the substitution is not diﬀerentiable. This issue is problematic in general but in our
setting, thanks to the rough non-degeneracy assumption on the phase, we can show
that the substitution is indeed valid and furthermore has the desired boundedness
properties. The discussion below is an abstract approach to the problem of non
smooth substitution and we refer the reader interested in related substitution results
to De Guzman [Guz].

Lemma

1.13. Let U be a measurable set and let t : U

→ R

be a bounded

measurable map satisfying

|t(x) − t(y)| ≥ c|x − y|

(1.25)

1. PROLEGOMENA

for almost every x, y

∈ U. Then there exists a function J

∈ L

∞

) supported in

t(U ) such that the substitution formula

◦ t(x) dx =

u(z)J

(z) dz

(1.26)

holds for all u

∈ L

) and the Jacobian J

satisﬁes the estimate

∞

≤

√

Remark

1.14. If one works with a representative t in the equivalence class

of functions equal almost everywhere, then possibly after replacing U with U

\ N

(where N is a null-set where (1.25) does not hold), one may assume that t is an
injective map with (1.25) holding everywhere on U .

For the convenience of the reader, we provide a proof of this simple lemma.

Proof.

As observed in Remark 1.14, we may assume that t is an injective map

from U to R

for which (1.25) holds on U . The formula

(f ) =

◦ t(x) dx, f ∈ C

)

deﬁnes a non-negative Radon measure, which by the Riesz representation theorem
is associated to a Borel measure. In this case, the latter measure is explicitly given
by

(A) =

−1

(A)

∩ U|

on all Lebesgue measurable sets A

⊂ R

, where we use the notation

| · | for the

Lebesgue measure of a set. By the Lebesgue decomposition theorem, this measure
can be split into an absolutely continuous and a singular part, i.e.

= μ

ac
t

+ μ

sing
t

Now assumption (1.25) yields

−1

∞

(w, r)

⊂ B

∞

(x, 2

√

nr/c),

if t(x)

∈ B

∞

(w, r)

where B

∞

(w, r) is a ball of center w and radius r for the supremum norm. This

implies that whenever

∩ t(U) ⊂

∞

k=0

∞

, r

)

it follows that

−1

(A)

∩ U ⊂

∞

k=0

∞

, 2

√

/c)

where the centers x

have been chosen in t

−1

∞

, r

)) when this set is nonempty.

Furthermore, it is wellknown that the Lebesgue measure of a set can be computed
using

|Ω| = inf

∞

k=0

|, Ω ⊂

∞

k=0

where the inﬁmum is taken over all possible sequences (Q

)

∈N

of cubes with faces

parallel to the axes. Therefore

(A)

≤

√

|A ∩ t(U)| ≤

√

|A|

(1.27)

1.2. TOOLS IN PROVING L

BOUNDEDNESS

for all Lebesgue measurable sets A in R

. In particular, Lebesgue null-sets are also

null-sets with respect to μ

, which in turn implies that the measure μ

is absolutely

continuous with respect to the Lebesgue measure. By the Radon-Nikodym theorem,
there exists a positive Lebesgue measurable function J

∈ L

1
loc

such that μ

has

density J

(A) =

(x) dx.

By Lebesgue’s diﬀerentiation theorem, we may compute the Jacobian function J

from the measure μ

by a limiting process on balls B, namely

(x) =

lim

→{x}

|B|

(y) dy =

lim

→{x}

(B)

|B|

(1.28)

Equality (1.28) together with the estimate (1.27) yields that J

is bounded and

∞

≤

√

Moreover, from the deﬁnition of μ

it is clear that it is supported in t(U ). Finally,

(1.26) follows from

◦ t(x) dx = μ

(u) =

u dμ

u(z)J

(z) dz

for all u

∈ C

(U ), and this extends to functions u

∈ L

Remark

1.15. Note that if there is a representative t in the equivalence class

such that (1.25) holds everywhere on U and such that t(U ) is an open subset of
R

, then t

−1

: t(U )

→ U is a Lipschitz bijection. Furthermore, any open subset

⊂ t(U) is open in R

and by Brouwer’s theorem on the invariance of the domain

−1

(W ) is open. This means that the map t is actually continuous.

Corollary

1.16. Let t : R

→ R

be a map satisfying the assumptions in

Lemma 1.13 with U = R

, then u

→ u ◦ t is a bounded map on L

for p

∈ [1, ∞].

Proof.

This easily follows from Lemma 1.13:

|u ◦ t(x)|

dx =

|u(z)|

(z) dz

≤ J

∞

when p

∈ [1, ∞). The L

∞

estimate is similar.

1.2.7. L

boundedness of the low frequency portion of rough Fourier

integral operators. Here we will prove the L

boundedness for p

∈ [1, ∞] of

Fourier integral operators whose amplitude contains a smooth compactly supported
function factor, the support of which lies in a neighbourhood of the origin. There
are a couple of diﬃculties to overcome here, the ﬁrst being the singularity of the
phase function in the frequency variable at the origin. The second problem is the
one caused by the lack of smoothness in the spatial variables. In order to handle
these problems we need the following lemma

Lemma

1.17. Let b(x, ξ) be a bounded function which is C

n+1

n
ξ

\ 0) and

compactly supported in the frequency variable ξ and L

∞

) in the space variable

x satisfying

sup

∈R

|ξ|

−1+|α|

∂

· , ξ)

∞

< +

∞, |α| ≤ n + 1.

1. PROLEGOMENA

Then for all 0

≤ μ < 1 we have

sup

x,y

∈R

n+μ

−iy,ξ

b(x, ξ) dξ

< +∞.

(1.29)

Proof.

Since b(x, ξ) is assumed to be bounded, the integral in (1.29) which

we denote by B(x, y), is uniformly bounded and therefore it suﬃces to consider the
case

|y| ≥ 1. Integrations by parts yield

B(x, y) =

|y|

−2n

−iy,ξ

y, D

b(x, ξ) dξ

and therefore we have the estimate

|B(x, y)| ≤ C|y|

−n

|ξ|<M

dξ

|ξ|

−1

We would like to gain an extra factor of

|y|

−μ

; to this end consider the function

β(x, y, ξ) =

|y|

−n

y, D

b(x, ξ) which is smooth in ξ on R

\ 0 and satisﬁes

sup

∈R

|ξ|

|α|−1

∂

β(

·, ·, ξ)

∞

< +

∞, |α| ≤ 1.

Let χ be a C

∞

) function which is one on the unit ball and zero outside the ball

of radius 2. Taking 0 < ε

≤ 1, we have

|y|

B(x, y) =

−iy,ξ

χ(ξ/ε)β(x, y, ξ) dξ

−iy,ξ

− χ(ξ/ε)

β(x, y, ξ) dξ

the ﬁrst term is bounded by a constant times ε, while the second term is equal to

|y|

−2

−iy,ξ

−1

y, ∂

χ(ξ/ε)β −

− χ(ξ/ε)

y, ∂

dξ

which may be bounded by

|y|

−1

− C

log ε).

We minimize the bound C

ε +

|y|

−1

− C

log ε) by taking ε =

|y|

−1

, and obtain

|B(x, y)| ≤ C|y|

−n−1

1 + log

|y|

≤ C

|y|

−n−μ

for all 0

≤ μ < 1. This is the desired estimate.

Having this in our disposal we can show that the low frequency portion of the

Fourier integral operators are L

bounded, more precisely we have

Theorem

1.18. Let a(x, ξ)

∈ L

∞

with m

∈ R and ∈ [0, 1] and let

the phase function ϕ(x, ξ)

∈ L

∞

satisfy the rough non-degeneracy condition

(according to Deﬁnition 1.6). Then for all χ

(ξ)

∈ C

∞

supported around the origin,

the Fourier integral operator

u(x) =

(2π)

iϕ(x,ξ)

a(x, ξ)χ

(ξ)

u(ξ) dξ

is bounded on L

for p

∈ [1, ∞].

1.3. LINKS BETWEEN NONSMOOTHNESS AND GLOBAL BOUNDEDNESS

Proof.

In proving the L

boundedness, according to the reduction of the

phase procedure in Lemma 1.16, there is no loss of generality to assume that our
Fourier integral operator is of the form

u(x) =

(2π)

a(x, ξ) χ

(ξ)e

iθ(x,ξ)+i

∇

ϕ(x,ζ),ξ

u(ξ) dξ,

for some ζ

∈ S

−1

, a

∈ L

∞

and θ

∈ L

∞

. In the proof of the L

boundedness

of T

we only need to analyze the kernel of the operator

a(x, ξ)χ

(ξ)e

iθ(x,ξ)+i

∇

ϕ(x,ζ),ξ

u(ξ) dξ,

which is given by

(x, y) :=

∇

ϕ(x,ζ)

−y,ξ

iθ(x,ξ)

a(x, ξ)χ

(ξ) dξ.

Now the estimates on the ξ derivatives of θ(x, ξ) above, yield

sup

|ξ|=0

|ξ|

−1+|α|

|∂

θ(x, ξ)

| < ∞,

for

|α| ≥ 1 uniformly in x, and therefore setting b(x, ξ) := a(x, ξ)χ

(ξ)e

iθ(x,ξ)

have that b(x, ξ) is bounded and sup

|ξ|=0

|ξ|

−1+|α|

|∂

b(x, ξ)

| < ∞, for |α| ≥ 1

uniformly in x and using Lemma 1.17, we have for all μ

∈ [0, 1)

(x, y)

| ≤ C∇

ϕ(x, ζ)

− y

−n−μ

From this it follows that

sup

(x, y)

| dy < ∞,

and using our rough non-degeneracy assumption and Corollary 1.16 in the case
p = 1, we also have

(x, y)

| dx

∇

ϕ(x, ζ)

− y

−n−μ

dz <

∞,

uniformly in y. This estimate and Young’s inequality yield the L

boundedness of

the operator T

1.3. Links between nonsmoothness and global boundedness

In this paragraph, we illustrate some of the relations between boundedness

for rough Fourier integral operators and the global boundedness of operators with
smooth amplitudes and phases. Our observation is that local estimates for non-
smooth Fourier integral operators imply global estimates for certain classes of
Fourier integral operators. This can be done either by compactiﬁcation or by using
a dyadic decomposition. To see the relation between compactiﬁcation and global
boundedness, consider the operator

T u(x) = (2π)

−n

iϕ(x,ξ)

a(x, ξ)

u(ξ) dξ.

(1.30)

Let χ

∈ C

∞

(B(0, 2)) be equal to one on the unit ball B(0, 1), and ω = 1

− χ be

supported away from zero. Then

T = T

+ T

= χT,

= ωT.

1. PROLEGOMENA

For the global continuity of T , we are only interested in T

since the amplitude of T

is compactly supported in the space variable and the boundedness of that operator
follows from the local theory. Concerning T

, we make the change of variables

z =

|x|

1
θ

x =

|z|

1+θ

∈ (0, 1]

(1.31)

so that

u(x)

dx = θ

|z|

1+θ

|z|

−n(1+θ)

dz.

(1.32)

Therefore it suﬃces to study the L

boundedness of the Fourier integral operator

u(z) = (2π)

−n

iϕ(z/

|z|

1+θ

,ζ)

|z|

−

(1+θ)

ωa

|z|

1+θ

, ζ

=˜

a(z,ζ)

u(ζ) dζ.

(1.33)

The amplitude ˜

a(z, ζ) is compactly supported (in the unit ball), and for a suitable

choice of θ belongs to L

∞

provided

∗

a(x, ξ)

∈ L

∞

s >

(1.34)

Now suppose that ϕ satisﬁes the following (global) non-degeneracy assumption:

|∇

ϕ(x, ξ)

− ∇

ϕ(y, ξ)

| ≥ c|x − y|

(1.35)

for all x, y

∈ R

. Then since

†

|z|

1+θ

−

|w|

1+θ

|z|

−2θ

|w|

−2θ

−

w, z

|z|

1+θ

|w|

1+θ

(1.36)

≥

max(

|w|, |z|)

1+θ

|z − w|

the phase ˜

ϕ(z, ζ) = ϕ(z/

|z|

1+θ

, ζ) satisﬁes a similar non-degeneracy condition,

namely

|∇

ϕ(z, ζ)

− ∇

ϕ(w, ζ)

| =

∇

|z|

1+θ

, ζ

− ∇

|w|

1+θ

, ζ

(1.37)

≥

max(

|w|, |z|)

1+θ

|z − w| ≥ c|z − w|,

when

|w|, |z| ≤ 1. In order to improve the decay assumption on the amplitude

(1.34), one can consider more general changes of variables which do not aﬀect the
angular coordinate in the polar decomposition, i.e. coordinate changes of the form

z = f (

|x|)

|x|

where f : (0,

∞) → (0, 1) is a diﬀeomorphism.

∗

This decay assumption is due to the singularity at 0 of

|z|

−n(1+θ)/p

of the Jacobian. Note

that any improvement on the regularity of ˜

a, ˜

ϕ should translate into decay properties at inﬁnity

of the original amplitude and phase a, ϕ.

†

In the case of the Kelvin transform θ = 1, it is easy to get a better lower bound (in fact an

equality):

|z|

−

|w|

|z|

−2

|w|

−2

−

w, z

|z|

|w|

|z − w|

|z|

|w|

1.3. LINKS BETWEEN NONSMOOTHNESS AND GLOBAL BOUNDEDNESS

Then x = g(

|z|)z/|z| where g is the inverse function of f, and the Jacobian of such

a change of variables is given by

(

|z|)|g

−1

(

|z|)|z|

−n

We would like to choose g in such a way that the singularities of its Jacobian become
weaker than those in the case of g(s) = s

−θ

. One possible choice is to take

g(s) = log(1

− s)

for which we have

(s)

−1

(s)s

−n

log

−1

− s)

−1

− s)

−1−θ

if s

∈ (0, 1). For this choice, we need the following decay

a(x, ξ)

∈ L

∞

s >

(1.38)

Furthermore, if one assumes that g/s is decreasing (or increasing) then the phase

ϕ satisﬁes our non-degeneracy assumption, because

|z|

|z|) −

|w|

|w|)

(1.39)

= g(

|z|)

+ g(

|w|)

−

w, z

|z||w|

|z|)g(|w|)

≥

min(

|w|, |z|)

|z − w|

≥ g

(0)

|z − w|

Alternatively, in order to investigate global boundedness using a dyadic decompo-
sition, one takes a Littlewood-Paley partition of unity 1 = χ(x) +

∞
j=1

ψ(2

−j

x),

which yields

T = χT +

∞

j=1

:= ψ(2

−j

·)T.

(1.40)

Once again we are only interested in T

and following a change of variables, we

want to prove

u(2

−j

≤ C

|u(z)|

dz.

(1.41)

This leads us to the study of the operator

u(z) = T

u(2

−j

(1.42)

= (2π)

−n

−j

ϕ(2

z,ζ)

ψ(z)a

z, 2

−j

=˜

(z,ζ)

u(ζ) dζ.

The estimate

|∂

(z, ζ)

| ≤ 2

−j|α|

(1 + 2

|z|)

j(m

−|α|)

(1 + 2

−j

|ζ|)

−|α|

(1.43)

≤ C

(1 +

|ζ|)

−|α|

yields that the amplitude ˜

(z, ζ) belongs (uniformly with respect to j) to the class

∞

provided

−m

a(x, ξ)

∈ L

∞

(1.44)

1. PROLEGOMENA

The phase ˜

(z, ζ) = 2

−j

ϕ(2

z, ζ) satisﬁes the non-degeneracy assumption

|∇

(z, ζ)

− ∇

(w, ζ)

| ≥ c|z − w|.

(1.45)

Therefore, once again the problem of establishing the global L

boundedness is

reduced to a local problem concerning operators with rough amplitudes.

CHAPTER 2

Global Boundedness of Fourier Integral Operators

In this chapter, partly motivated by the investigation in [KS] of the L

bound-

edness of the so called pseudo-pseudodiﬀerential operators where the symbols of
the aforementioned operators are only bounded and measurable in the spatial vari-
ables x, we consider the global and local boundedness in Lebesgue spaces of Fourier
integral operators of the form

T u(x) = (2π)

−n

iϕ(x,ξ)

a(x, ξ)

u(ξ) dξ,

(2.1)

in case when the phase function ϕ(x, ξ) is smooth and homogeneous of degree 1 in
the frequency variable ξ, and the amplitude a(x, ξ) is either in some H¨

ormander

class S

,δ

, or is a L

∞

function in the spatial variable x and belongs to some L

∞

class. We shall also investigate the L

boundedness problem for Fourier integral

operators with rough phases that are L

∞

functions in the spatial variable. In the

case of the rough phase, the standard notion of non-degeneracy of the phase func-
tion has no meaning due to lack of diﬀerentiability in the x variables. However,
there is a non-smooth analogue of the non-degeneracy condition which has already
been introduced in Deﬁnition 1.6 which will be exploited further here.
We start by investigating the question of L

boundedness of Fourier integral oper-

ators with rough amplitudes but smooth phase functions satisfying the strong non-
degeneracy condition. Thereafter we turn to the problem of L

boundedness of the

Fourier integral operators with smooth phases, but rough or smooth amplitudes. In
the case of smooth amplitudes, we show the analogue of the Calder´

on-Vaillancourt’s

boundedness of pseudodiﬀerential operators in the realm of Fourier integral op-

erators. Next, we consider Fourier integral operators with rough amplitudes and
rough phase functions and show a global and a local L

result in that context. We

also give a fairly general discussion of the symplectic aspects of the L

boundedness

of Fourier integral operators.
After concluding our investigation of the L

boundedness, we proceed by proving

an L

∞

boundedness theorem for Fourier integral operators with rough amplitudes

and rough phases in class L

∞

, without any non-degeneracy assumption on the

phase. Finally, we close this chapter by proving L

− L

and L

− L

estimates for

operators with smooth phase function, and smooth or rough amplitudes.

2.1. Global L

boundedness of rough Fourier integral operators

As will be shown below, the global L

boundedness of Fourier integral opera-

tors is a consequence of Theorem 1.18, the Seeger-Sogge-Stein decomposition, and
elementary kernel estimates.

Theorem

2.1. Let T be a Fourier integral operator given by (0.1) with ampli-

tude a

∈ L

∞

and phase function ϕ

∈ L

∞

satisfying the rough non-degeneracy

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

condition. Then there exists a constant C > 0 such that

T u

≤ Cu

∈ S (R

)

provided m <

−

−1

+ n(

− 1) and 0 ≤ ≤ 1. Furthermore, in the case = 0 i.e.

for operators with amplitude in a

∈ L

∞

one can obtain the L

boundedness with

an improved decay m <

−n.

Proof.

Using semiclassical reduction of Subsection 1.2.1, we decompose T

into low and high frequency portions T

and T

. Then we use the Seeger-Sogge-

Stein decomposition of Subsection 1.2.2 to decompose T

into the sum

J
ν=1

The boundedness of T

follows at once from Theorem 1.18, so it remains to estab-

lish suitable semiclassical estimates for T

. To this end we consider the following

diﬀerential operator

L = 1

− ∂

− h∂

for which we have according to Lemma 1.9

(2.2)

sup

(

·, ξ, h)

∞

−m−2N(1−)

Integrations by parts yield

(x, y)

| ≤ (2πh)

−n

1 + g(y

− ∇

ϕ(x, ξ

)

−N

(x, ξ, h)

| dξ

for all integers N , with

(2.3)

g(z) = h

−2

+ h

−1

This further gives

(x, y)

| ≤ C

−m−

n+1

−2N(1−)

1 + g(y

− ∇

ϕ(x, ξ

)

−N

since the volume of the portion of cone

|A ∩ Γ

| is of the order of h

−1)/2

. By

interpolation, it is easy to obtain the former bound when the integer N is replaced
by M/2 where M is any given positive number ; indeed write M/2 = N + θ where
N = [

] and θ

∈ [0, 1) and

(x, y)

| = |T

(x, y)

−θ

≤ C

−θ

N +1

−m−

n+1

−(1−)M

1 + g(y

− ∇

ϕ(x, ξ

)

−

(2.4)

This implies that for any real number M > n

sup

(x, y)

| dy ≤ C

−m−M(1−)

Furthermore the rough non-degeneracy assumption on the phase function ϕ(x, ξ)
and Corollary 1.16 with p = 1 yield

sup

(x, y)

| dx ≤ C

−θ

N +1

1 + g(

∇

ϕ(x, ξ

)

−

≤ C

−m−M(1−)

thus using Young’s inequality and summing in ν

≤

ν=1

≤ C

−m−

−1

−M(1−)

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

since J is bounded (from above and below) by a constant times h

−

−1

. By Lemma

1.8 one has

T u

provided m <

−

−1

− M(1 − ) and M > n, i.e. if m < −

−1

+ n(

− 1).

Now to see how m <

−n yields the L

boundedness for a(x, ξ)

∈ L

∞

, we

just use Lemma 1.10, and representation 1.16 and observe that if we set b(x, ξ) :=
a(x, ξ)e

iθ(x,ξ)

, we can essentially reduce the study of T u(x) to that of the pseudo-

pseudodiﬀerential operator (i.e. Ψ-pseudodiﬀerential operator in the sense of Kenig-
Staubach), (b(x, D)u)(

∇

ϕ(x, ζ)), with symbol b(x, ξ)

∈ L

∞

. The L

bounded-

ness of T for m <

−n follows from Corollary 1.16 and Proposition 2.3 in [KS]. This

completes the proof of Theorem 2.1

2.2. Local and global L

boundedness of Fourier integral operators

In this section we study the local and global L

boundedness properties of

Fourier integral operators. Here we complete the global L

theory of Fourier in-

tegral operators with smooth strongly non-degenerate phase functions in class Φ

and smooth amplitudes in the H¨

ormander class S

,δ

for all ranges of ρ’s and δ’s.

As a ﬁrst step we establish global L

boundedness of Fourier integral operators

with smooth phases and rough amplitudes in L

∞

, then we proceed by investi-

gating the L

boundedness of Fourier integral operators with smooth phases and

amplitudes and ﬁnally we consider the L

regularity of the operators with rough

amplitudes in L

∞

and rough non-degenerate phase functions in L

∞

2.2.1. L

boundedness of Fourier integral operators with phases in

. The global L

boundedness of Fourier integral operators which we aim to prove

below, yields on one hand a global version of Eskin’s and H¨

ormander’s local L

boundedness theorem for amplitudes in S

1,0

, and on the other hand generalises the

global L

result of Fujiwara’s for amplitudes in S

0,0

to the case of rough amplitudes.

Furthermore, as we shall see later, our result is sharp.

Theorem

2.2. Let a(x, ξ)

∈ L

∞

and the phase ϕ(x, ξ)

∈ Φ

be strongly non

degenerate. Then the Fourier integral operator

a,ϕ

u(x) =

(2π)

a(x, ξ) e

iϕ(x,ξ)

u(ξ) dξ

is a bounded operator from L

to itself provided m <

(

− 1). The bound on m is

sharp.

Proof.

In light of Theorem 1.18, we can conﬁne ourselves to deal with the high

frequency component T

of T

a,ϕ

, hence we can assume that ξ

= 0 on the support

of the amplitude a(x, ξ). Here we shall use a T

∗

argument, and therefore, the

kernel of the operator S

= T

∗

reads

(x, y) =

(2πh)

(ϕ(x,ξ)

−ϕ(y,ξ))

(ξ)a(x, ξ/h)a(y, ξ/h) dξ.

Now the strong non degeneracy assumption on the phase and Proposition 1.11 yield
that there is a constant C > 0 such that

|∇

ϕ(x, ξ)

− ∇

ϕ(y, ξ)

| ≥ C|x − y|, for

x, y

∈ R

and ξ

∈ R

\ 0. This enables us to use the non-stationary phase estimate

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

in [H1] Theorem 7.7.1, and the smoothness of the phase function ϕ(x, ξ) in the
spatial variable, yield that for all integers N

(x, y)

| ≤ C

−2m−n−(1−)N

−1

− y)

−N

for some constant C

> 0. Let M be a positive real number, we have M = N + θ

where N is the integer part of M and θ

∈ [0, 1) and therefore

(x, y)

| = |S

(x, y)

−θ

≤ C

−θ

N +1

−2m−n−(1−)M

−1

− y)

−M

(2.5)

This implies

sup

(x, y)

| dy ≤ C

−2m−(1−)M

(2.6)

for all M > n. By Cauchy-Schwarz and Young inequalities, we obtain

∗

2
L

≤ S

≤ Ch

−2m−(1−)M

2
L

(2.7)

Therefore, by Lemma 1.8 we have the L

bound

T u

provided m <

−(1 − )M/2 and M > n. This completes the proof of Theorem 2.2.

For the sharpness of this result we consider the phase function ϕ(x, ξ) =

x, ξ ∈ Φ

which is strongly non-degenerate. It was shown in [Rod] that for m =

(

− 1)

there are symbols a(x, ξ)

∈ S

such that the pseudodiﬀerential operator

a(x, D)u(x) =

(2π)

a(x, ξ) e

x,ξ

u(ξ) dξ

is not L

bounded. Since S

⊂ L

∞

, it turns out that there are amplitudes in

∞

which yield an L

unbounded operator for a non-degenerate phase function

in class Φ

. Hence the order m in the theorem is sharp.

As a consequence, we obtain an alternative proof for the L

boundedness of

pseudo-pseudodiﬀerential operators introduced in [KS]. More precisely we have

Corollary

2.3. Let a(x, D) be a pseudo-pseudodiﬀerential operator, i.e. an

operator deﬁned on the Schwartz class, given by

(2.8)

a(x, D)u =

(2π)

x,ξ

a(x, ξ)

u(ξ) dξ,

with symbol a

∈ L

∞

, 0

≤ ≤ 1. If m < n( − 1)/2, then a(x, D) extends as an

bounded operator.

Theorem 2.2 can be used to show a simple local L

boundedness result for

Fourier integral operators with smooth symbols in the H¨

ormander class S

,δ

those cases when the symbolic calculus of the Fourier integral operators, as deﬁned
in [H3] breaks down (e.g. in case δ

≥ ), more precisely we have

Corollary

2.4. Let a(x, ξ)

∈ S

,δ

with compact support in x variable and let

ϕ(x, ξ)

∈ Φ

be strongly non-degenerate. Then for m <

(

−δ −1) and 0 ≤ ≤ 1,

≤ δ ≤ 1. Then the corresponding Fourier integral operator is bounded on L

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

Proof.

By Sobolev embedding theorem, for a function f (x, y) one has

|f(x, x)|

≤ C

|α|≤N

|∂

f (x, y)

dx dy

with N > n/2. Now let f (x, y) :=

a(y, ξ) e

iϕ(x,ξ)

u(ξ) dξ. Since a(x, ξ)

∈ S

,δ

, we

have that ∂

a(y, ξ)

∈ L

∞

m+δ

|α|

. Therefore, Theorem 2.2 yields

|∂

f (x, y)

2
L

provided that m + δ

|α| <

(

− 1). Since |α| ≤ N and N > n/2, one sees that it

suﬃces to take m <

(

− δ − 1). We also note that in the argument above, the

integration in the y variable will not cause any problem due to the compact support
assumption of the amplitude.

However, as was shown by D. Fujiwara in [Fuji], Fourier integral operators

with phases in Φ

and amplitudes in S

0,0

are bounded in L

. This result suggests

the possibility of the existence of an analog of the Calder´

on-Vaillancourt theorem

[CV], concerning L

boundedness of pseudodiﬀerential operators with symbols in

with

∈ [0, 1), in the realm of smooth Fourier integral operators. That this

is indeed the case will be the content of Theorem 2.7 below. However, before
proceeding with the statement of that theorem, we will need two lemmas, the ﬁrst
of which is a continuous version of the Cotlar-Stein lemma, due to A. Calder´

on and

R. Vaillancourt, see i.e. [CV] for a proof.

Lemma

2.5. Let

H be a Hilbert space, and A(ξ) a family of bounded linear

endomorphisms of

H depending on ξ ∈ R

. Assume the following three conditions

hold:

(i) the operator norm of A(ξ) is less than a number C independent of ξ.

(ii) for every u

∈ H the function ξ → A(ξ)u from R

→ H is continuous

for the norm topology of

H .

(iii) for all ξ

and ξ

in R

(2.9)

∗

(ξ

)A(ξ

)

≤ h(ξ

, ξ

)

, and

A(ξ

∗

(ξ

)

≤ h(ξ

, ξ

)

with h(ξ

, ξ

)

≥ 0 is the kernel of a bounded linear operator on L

with

norm K.

Then for every E

⊂ R

, with

|E| < ∞, the operator A

A(ξ) dξ deﬁned by

u, v

A(ξ)u, v

dξ, is a bounded linear operator on

H with norm less

than or equal to K.

We shall also use the following useful lemma.

Lemma

2.6. Let

(2.10)

Lu(x) := D

−2

− is(x)∇

∇

)u(x),

with D := (1 + s(x)

|∇

)

1/2

. Then

(i) L(e

iF (x)

) = e

iF (x)

(ii) if

L denotes the formal transpose of L, then for any positive integer N,

(

u(x) is a ﬁnite linear combination of terms of the form

(2.11)

−k

{

μ=1

∂

s(x)

}{

ν=1

∂

F (x)

}∂

u(x),

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

with

(2.12)

≤ k ≤ 4N; k − 2N ≤ p ≤ k − N; |α

| ≥ 0;

μ=1

|α

| ≤ N

− 2N ≤ q ≤ k − N; |β

| ≥ 1;

μ=1

|β

| ≤ q + N; |γ| ≤ N.

Proof.

First one notes that

∂

−N

−

−N−2

k=1

{2s(x) ∂

F ∂

F + ∂

s (∂

F )

This and Leibniz’s rule yield

Lu(x) = D

−2

u(x) + i

j=1

∂

−2

s(x) u(x) ∂

F )

= D

−2

u(x)

− iD

−4

k, j=1

u(x) s(x) ∂

2s(x) ∂

F ∂

+ ∂

s (∂

F )

+ iD

−2

j=1

u(x) ∂

s(x) ∂

+ iD

−2

j=1

s(x) u(x) ∂

F + iD

−2

j=1

s(x) ∂

u(x) ∂

From this it follows that

L is a linear combination of operators of the form

(2.13)

−2

(2.14)

−4

(x) ∂

F ∂

(2.15)

−4

s ∂

F (∂

F )

(2.16)

−2

∂

s(x) ∂

(2.17)

−2

s(x) ∂

(2.18)

−2

s(x) ∂

F ∂

If we conventionally say that the term (2.11) is of the type

k, p,

μ=1

|α

|, q,

ν=1

|β

|, |γ|

then

L is sum of terms of the types (2, 0, 0, 0, 0, 0), (4, 2, 0, 3, 4, 0), (4, 2, 1, 3, 3, 0),

(2, 1, 1, 1, 1, 0), (2, 1, 0, 1, 2, 0) and (2, 1, 0, 1, 1, 1). Now operating the operators in
(2.13), (2.14), (2.15), (2.16), (2.17) on a term (2.11) of type

k, p,

μ=1

|α

|, q,

ν=1

|β

|, |γ|

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

increases the types by (2, 0, 0, 0, 0, 0), (4, 2, 0, 3, 4, 0), (4, 2, 1, 3, 3, 0), (2, 1, 1, 1, 1, 0),
(2, 1, 0, 1, 2, 0) respectively. To see how operating a term of form the 2.18 on (2.11)
changes the type we use Leibniz rule to obtain

−2

s(x) ∂

F ∂

−k

μ=1

∂

s(x)

ν=1

∂

F (x)

∂

u(x)

−

−k−4

l=1

∂

2s(x) ∂

F ∂

F + ∂

s (∂

F )

μ=1

∂

s(x)

ν=1

∂

F (x)

∂

u(x)

+ D

−k−2

∂

=μ

∂

s(x)

∂

s(x)

ν=1

∂

F (x) ∂

u(x)

+ D

−k−2

∂

μ=1

∂

s(x)

=ν

∂

F (x)

∂

F (x)

∂

u(x)

+ D

−k−2

μ=1

∂

s(x)

ν=1

∂

F (x)

∂

u(x).

Therefore, upon application of

L to (2.11), the types of the resulting terms increase

by (2, 0, 0, 0, 0, 0), (4, 2, 0, 3, 4, 0), (4, 2, 1, 3, 3, 0), (2, 1, 1, 1, 1, 0), (2, 1, 0, 1, 2, 0) and
(2, 1, 0, 1, 1, 1). Iteration of this process yields

(

u(x) =

C D

−k

μ=1

∂

s(x)

ν=1

∂

F (x)

∂

u(x),

where the summation is taken over all non-negative integers N

, N

with

6
j=1

= N and

(2.19)

k, p,

μ=1

|α

|, q,

ν=1

|β

|, |γ|

= N

(2, 0, 0, 0, 0, 0) + N

(4, 2, 0, 3, 4, 0)+

(4, 2, 1, 3, 3, 0) + N

(2, 1, 1, 1, 1, 0) + N

(2, 1, 0, 1, 2, 0) + N

(2, 1, 0, 1, 1, 1).

Hence,

(2.20)

k = 2N

+ 4N

+ 2N

(2.21)

p = 2N

+ 2N

+ N

(2.22)

μ=1

|α

| = N

+ N

(2.23)

q = 3N

+ 3N

+ N

(2.24)

ν=1

|β

| = 4N

+ 3N

+ N

+ 2N

+ N

(2.25)

|γ| = N

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

From this it also follows that (k, p,

p
μ=1

|α

|, q,

q
ν=1

|β

|, |γ|) satisfy (2.12).

Theorem

2.7. If m = min(0,

(

− δ)), 0 ≤ ≤ 1, 0 ≤ δ < 1, a ∈ S

,δ

and

∈ Φ

, satisﬁes the strong non-degeneracy condition, then the operator T

u(x) =

a(x, ξ) e

iϕ(x,ξ)

u(ξ) dξ is bounded on L

Proof.

First we observe that since for δ

≤ , S

,δ

⊂ S

, it is enough to show

the theorem for 0

≤ ≤ δ < 1 and m =

(

− δ). Also, as we have done previously,

we can assume without loss of generality that a(x, ξ) = 0 when ξ is in a a small
neighbourhood of the origin. Using the T T

∗

argument, it is enough to show that

the operator

(2.26)

u(x) =

b(x, y, ξ) e

iϕ(x,ξ)

−iϕ(y,ξ)

u(y) dy dξ,

where b satisﬁes the estimate

(2.27)

|∂

∂

b(x, y, ξ)

| ≤ C

α β γ

−|α|+δ(|β|+|γ|)

with m

= n(

− δ) and 0 ≤ ≤ δ < 1, is bounded on L

We introduce a diﬀerential operator

L := D

−2

− iξ

∇

ϕ(x, ξ)

− ∇

ϕ(y, ξ),

∇

with D = (1 +

|∇

ϕ(x, ξ)

− ∇

ϕ(y, ξ)

)

1
2

. It follows from Lemma 2.6 that

L(e

iϕ(x,ξ)

−iϕ(y,ξ)

) = e

iϕ(x,ξ)

−iϕ(y,ξ)

and that (

(b(x, y, ξ)) is a ﬁnite sum of terms of the form

(2.28)

−k

μ=1

∂

ν=1

∂

ϕ(x, ξ)

− ∂

ϕ(y, ξ)

∂

b(x, y, ξ).

Furthermore since ϕ

∈ Φ

is assumed to be strongly non-degenerate, we can use

Proposition 1.11 to deduce that

(2.29)

|∇

ϕ(x, ξ)

− ∇

ϕ(y, ξ)

| ≥ c

|x − y|

(2.30)

|∇

ϕ(z, ξ

)

− ∇

ϕ(z, ξ

)

| ≥ c

|ξ

− ξ

Using (2.29), (2.12) and (2.27), we have

(2.31)

|∂

(

(b(x, y, ξ))

| ≤ CΛ(ξ

− y))ξ

+δ

|σ|

where Λ is an integrable function with

Λ(x) dx

1. Integration by parts using L,

N times, in (2.26) one has

(2.32)

u(x) =

c(x, y, ξ) e

iϕ(x,ξ)

−iϕ(y,ξ)

u(y) dy dξ,

with c(x, y, ξ) = (

(b(x, y, ξ)) and

(2.33)

|∂

c(x, y, ξ)

| ≤ CΛ(ξ

− y))ξ

+δ

|σ|

and the same estimate is valid for ∂

c(x, y, ξ). From this we get the representation

(2.34)

A(ξ) dξ,

where A(ξ)u(x) :=

c(x, y, ξ) e

iϕ(x,ξ)

−iϕ(y,ξ)

u(y) dy. Noting that A(ξ) = 0 for ξ

outside some compact set, we observe that condition (1) of Lemma 2.5 follows

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

from Young’s inequality and (2.33) with σ = 0, and condition (2) of Lemma 2.5
follows from the assumption of the compact support of the amplitude. To verify
condition (3) we conﬁne ourselves to the estimate of

∗

(ξ

)A(ξ

)

, since the one

for

A(ξ

∗

(ξ

)

is similar. To this end, a calculation shows that the kernel of

∗

(ξ

)A(ξ

) is given by

(2.35)

K(x, y, ξ

, ξ

) :=

c(z, x, ξ

) c(z, y, ξ

) e

i[ϕ(z,ξ

)

−ϕ(z,ξ

)+ϕ(x,ξ

)

−ϕ(y,ξ

)]

dz.

The estimate (2.33) yields

(2.36)

|K(x, y, ξ

, ξ

)

Λ(

− z)) Λ(ξ

− z)) dz.

Therefore by choosing N large enough, Young’s inequality and using the fact that

Λ(x) dx

1 yield

(2.37)

∗

(ξ

)A(ξ

)

−n

At this point we introduce another ﬁrst order diﬀerential operator M := G

−2

{1 −

∇

ϕ(z, ξ

)

−∇

ϕ(z, ξ

∇

)}, with G = (1+|∇

ϕ(z, ξ

)

−∇

ϕ(z, ξ

)

1
2

. Using

the fact that M e

i(ϕ(z,ξ

)

−ϕ(z,ξ

))

= e

i(ϕ(z,ξ

)

−ϕ(z,ξ

))

, integration by parts in (2.35)

yields

(2.38)

(

M )

{c(z, x, ξ

) c(z, y, ξ

)

} e

i[ϕ(z,ξ

)

−ϕ(z,ξ

)+ϕ(x,ξ

)

−ϕ(y,ξ

)]

dz.

Using the second part of Lemma 2.6, we ﬁnd that (

M )

{c(z, x, ξ

) c(z, y, ξ

)

} is

a linear combination of terms of the form

(2.39)

−k

ν=1

(∂

ϕ(z, ξ

)

− ∂

ϕ(z, ξ

))

∂

c(z, x, ξ

) ∂

c(z, y, ξ

where k, q, β

satisfy the inequalities in 2.12 and

|γ

| + |γ

| ≤ N

. Now, (2.30),

(2.33) and (2.39), yield the following estimate for K(x, y, ξ

, ξ

)

|K(x, y, ξ

, ξ

)

| ξ

(1 +

|ξ

| + |ξ

δN

|ξ

− ξ

−N

(2.40)

Λ(

− z)) Λ(ξ

− z)) dz.

Once again, choosing N large enough, Young’s inequality yields

(2.41)

∗

(ξ

)A(ξ

)

−n

(1 +

|ξ

| + |ξ

δN

|ξ

− ξ

Using the fact that for x > 0, inf(1, x)

∼ (1 +

1
x

)

−1

, one optimizes the estimates

(2.37) and (2.41) by

∗

(ξ

)A(ξ

)

−n

1 +

|ξ

− ξ

(1 +

|ξ

| + |ξ

δN

−1

(2.42)

:= h

(ξ

, ξ

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

Therefore recalling that m

= n(

− δ), in applying Lemma 2.5, we need to show

that

(2.43)

K(ξ

, ξ

) = (1 +

|ξ

−nδ

(1 +

|ξ

−

nδ

1 +

|ξ

− ξ

(1 +

|ξ

| + |ξ

δN

−

1
2

is the kernel of a bounded operator in L

. At this point we use Schur’s lemma,

which yields the desired conclusion provided that

sup

K(ξ

, ξ

) dξ

sup

K(ξ

, ξ

) dξ

are both ﬁnite. Due to the symmetry of the kernel, we only need to show the
ﬁniteness of one of these quantities.
To this end, we ﬁx ξ

and consider the domains

A = {(ξ

, ξ

);

|ξ

| ≥ 2|ξ

|}, B =

{(ξ

, ξ

);

|ξ

≤ |ξ

| ≤ 2|ξ

|}, and C = {(ξ

, ξ

);

|ξ

| ≤

|ξ

}. Now we observe that

on the set

A, K(ξ

, ξ

) is dominated by

(2.44)

(1 +

|ξ

−

nδ

(1 +

|ξ

−

nδ

(δ

−1)

B, K(ξ

, ξ

) is dominated by

(2.45)

(1 +

|ξ

−nδ

1 +

|ξ

− ξ

(1 +

|ξ

δN

−

1
2

and on

C, K(ξ

, ξ

) is dominated by

(2.46)

(1 +

|ξ

−

nδ

(1 +

|ξ

−

nδ

(δ

−1)

Therefore, if I

K(ξ

, ξ

) dξ

, then choosing

(δ

− 1) < −n, which is only

possible if δ < 1, we have that I

∞ uniformly in ξ

. Also,

(2.47)

≤ (1 + |ξ

−

nδ

(δ

−1)

≤ C,

which is again possible by the fact that δ < 1 and a suitable choice of N

. In I

let

us make a change of variables to set ξ

− ξ

= (1 +

|ξ

y, then

(2.48)

≤

(1 +

|y|

)

−

1
2

dy <

∞,

by taking N

large enough. These estimates yield the desired result and the proof

of there theorem is therefore complete.

2.2.2. L

boundedness of Fourier integral operators with phases in

∞

. Next we shall turn to the problem of L

boundedness of Fourier integral op-

erators with non-smooth amplitudes and phases. As was mentioned in the introduc-
tion, a motivation for considering fully rough Fourier integral operators stems from
a ”linearisation” procedure which reduces certain maximal operators to Fourier
integral operators with a non-smooth phase and sometimes also a non-smooth am-
plitude. For instance, estimates for the maximal spherical average operator

Au(x) = sup

∈[0,1]

−1

u(x + tω) dσ(ω)

are related to those for the maximal wave operator

W u(x) = sup

∈[0,1]

√

−Δ

u(x)

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

and can for instance be deduced from those of the linearized operator

it(x)

√

−Δ

u = (2π)

−n

it(x)

|ξ|+ix,ξ

u(ξ) dξ,

(2.49)

where t(x) is a measurable function in x, with values in [0, 1] and the phase here
belongs to the class L

∞

. As will be demonstrated later, the validity of the re-

sults in the rough case depend on the geometric conditions (imposed on the phase
functions) which are the rough analogues of the non-degeneracy and corank condi-
tions for smooth phases. In trying to understand the subtle interrelations between
boundedness, smoothness and geometric conditions, we remark that even if one as-
sumes the phase of the linearized operator (2.49) to be smooth, there are cases for
which the canonical relation of this operator ceases to be the graph of a symplec-
tomorphism. Indeed, contrary to the wave operator e

√

−Δ

at ﬁxed time t

∈ [0, 1],

the phase ϕ(x, ξ) =

x, ξ+t(x)|ξ| of the linearized operator cannot be a generating

function of a canonical transformation, (see [D]), in certain cases since

∂

∂x∂ξ

(x, ξ) = Id +

∇t(x) ⊗

|ξ|

ker

∂

∂x∂ξ

(x, ξ) = span

∇t(x)

when

ξ, ∇t(x) + |ξ| = 0,

and this happens when

|∇t(x)| ≥ 1 and ξ = (−

∇t(x)

|∇t(x)|

+ η) with

∈ R

∗

and η

is a vector orthogonal to

∇t(x) of norm (1 − |∇t(x)|

−2

)

1/2

. Therefore, one can not

expect L

boundedness of (2.49) even when the function t(x) is smooth. Never-

theless, in this case the rank of the Hessian ∂

ϕ/∂x∂ξ drops by one with respect

to its maximal possible value, and one could still establish L

estimates with loss

of derivatives (see section 2.2.3 for more details). The operators that we intend to
study will fall into this category. Before we investigate the local L

boundedness

of operators based on geometric conditions on their phase, we state and prove a
purely analytic global L

boundedness result which will be used later.

Theorem

2.8. Let T be a Fourier integral operator given by (0.1) with ampli-

tude a

∈ L

∞

, 0

≤ ≤ 1 and a phase function ϕ(x, ξ) ∈ L

∞

satisfying the

rough non-degeneracy condition. Then there exists a constant C > 0 such that

T u

≤ Cu

provided m < n(

− 1)/2 − (n − 1)/4.

Proof.

Using semiclassical reduction of Subsection 1.2.1, we decompose T into

low and high frequency portions T

and T

. The boundedness of T

follows at once

from Theorem 1.18, so it remains to establish suitable semiclassical estimates for
T

. Once again we use the T T

∗

argument. The kernel of the operator S

= T

∗

reads

(x, y) = (2πh)

−n

(ϕ(x,ξ)

−ϕ(y,ξ))

(ξ)a(x, ξ/h)a(y, ξ/h) dξ.

We now use the Seeger-Sogge-Stein decomposition (section 1.2.2) and split this
operator as the sum

N
j=1

where the kernel of S

takes the form

(x, y) = (2πh)

−n

∇

ϕ(x,ξ

)

−∇

ϕ(y,ξ

),ξ

(x, ξ, h)b

(y, ξ, h) dξ.

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

We consider the following diﬀerential operator

L = 1

− ∂

− h∂

for which we have according to Lemma 1.9

(2.50)

sup

(

·, ξ, h)

∞

−m−2N(1−)

Integration by parts yields

(x, y)

| ≤ (2πh)

−n

1 + g

∇

ϕ(y, ξ

)

− ∇

ϕ(x, ξ

)

−N

(x, ξ, h)b

(y, ξ, h)

dξ

for all integers N , with

(2.51)

g(z) = h

−2

+ h

−1

The standard interpolation trick gives the same inequality for for all positive num-
bers M > 0 and thus we have

(x, y)

| ≤ Ch

−2m−

n+1

−2M(1−)

1 + g

∇

ϕ(y, ξ

)

− ∇

ϕ(x, ξ

)

−M

since the volume of the portion of cone

|A ∩ Γ

| is of the order of h

−1)/2

. By the

non-degeneracy assumption and Lemma 1.13, we get

(x, y)

| dy ≤ Ch

−2m−

n+1

−2M(1−)

1 + g(z)

−M

=ch

n+1

By Young’s inequality (remembering that the kernel S

(x, y) is symmetric), we

obtain

≤ Ch

−2m−2M(1−)

and summing the inequalities

∗

2
L

≤

ν=1

≤ Ch

−2m+

−1

−2(1−)M

since there are roughly h

−(n−1)/2

terms in the sum. By Lemma 1.8, we have the

bound

T u

provided m <

−(n − 1)/4 + ( − 1)M and M > n/2, which yields the desired

result.

Remark

2.9. The reason why we were led to perform the Seeger-Sogge-Stein

decomposition is that under the rough non-degeneracy assumption (Deﬁnition 1.6),
the non-stationary phase (Theorem 7.7.1 [H1]) provides the bound

(x, y)

| ≤ C

−2m−n+N

|x − y|

(2.52)

≤ C

−2m−n

1 + h

−1

|x − y|

)

−N

leading, when say = 1, to a loss of n/4 derivatives instead of (n

− 1)/4 derivatives

in our case. This however can be improved to no loss of derivatives when one also
assumes that there is a Lipschitz bound on the higher order derivatives

|∂

ϕ(x, ξ)

− ∂

ϕ(y, ξ)

| ≤ C

|x − y|, |α| ≥ 2.

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

This is indeed the case in dimension n = 1, or if the phase can be decomposed as
ϕ(x, ξ) = ϕ

(x, ξ) + ϕ

(x, ξ) where ϕ

is linear in ξ and ϕ

∈ Φ

Let π

denote the projection onto the spatial variables, i.e.

: T

∗

→ R

(x, ξ)

→ x.

A geometric condition suﬃcient for the local L

boundedness of rough Fourier

integral operators with phase functions ϕ(x, ξ) and amplitudes a(x, ξ) is as follows:

Rough corank condition.

For each x

∈ π

(supp a) and all ξ

∈ S

−1

there

exists a linear subspace V

x,ξ

belonging to the Grassmannian Gr(n, n

− k) varying

continuously with (x, ξ), and constants c

, c

> 0 such that if π

x,ξ

denotes the

projection onto V

x,ξ

, then

|∂

ϕ(x, ξ)

− ∂

ϕ(y, ξ)

| + c

|x − y|

≥ c

|π

x,ξ

− y)|

for all x, y

∈ π

(supp a).

Theorem

2.10. Let T be a Fourier integral operator given by (0.1) with ampli-

tude a

∈ L

∞

and phase function ϕ

∈ L

∞

. Suppose that the phase satisﬁes the

rough corank condition 2.2.2, then T can be extended as a bounded operator from
L

comp

to L

2
loc

provided m <

−

n+k

−1

−k)(−1)

Proof.

Since we aim to prove a local L

boundedness result, we may assume

that the amplitude a is compactly supported in the spatial variable x. Then since
S

= T

∗

has a bounded compactly supported kernel, it extends to a bounded

operator on L

. It remains to deal with the high frequency part of the operator.

Given (x

, ξ

)

∈ R

× R

, μ = 1, . . . , J, we consider a partition of unity

μ=1

(x, ξ) = 1,

= 0

given by functions ψ

homogeneous of degree 0 in the frequency variable ξ supported

in cones

(x, ξ)

∈ T

∗

;

|x − x

|ξ|

− ξ

≤ ε

where ε is yet to be chosen. We decompose the operator as

μ=1

(2.53)

where the kernel of T

is given by

(x, y) = (2πh)

−n

ϕ(x,ξ)

−

y,ξ

(x, ξ)χ(ξ)a(x, ξ/h) dξ.

We have the direct sum

= V

,ξ

⊕ V

⊥

,ξ

dim V

,ξ

= n

− k

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

and we decompose vectors x = x

+ x

(i.e. x = (x

, x

)) according to this sum.

Assumption 2.2.2 implies

|∂

ϕ(x

, x

, ξ)

− ∂

ϕ(y

, x

, ξ)

≥ c

|π

x,ξ

− y

)

| − c

− y

≥ c

− y

− π

x,ξ

− π

xμ ,ξμ

−

− y

Now since (x, ξ)

→ π

x,ξ

is continuous, we can choose ε in the deﬁnition of the cone

small enough so that

x,ξ

− π

xμ ,ξμ

≤

− y

| ≤ |x

− x

| + |y

− x

| ≤

and therefore we have

|∂

ϕ(x

, x

, ξ)

− ∂

ϕ(y

, x

, ξ)

| ≥

− y

(2.54)

when (x, ξ) and (y, ξ) belong to Γ

. We ﬁx the x

variable and use a T T

∗

argument

on the operator acting in the x

variables. We consider

, x

, y

) = (2πh)

−n

(ϕ(x,ξ)

−ϕ(y

,ξ))

μ
h

(x, ξ)a

μ
h

, x

, ξ) dξ.

Because of (2.54), performing a Seeger-Sogge-Stein decomposition and reasoning
as in the proof of Theorem 2.8 we get

xμ ,ξμ

, x

, y

)u(y

) dy

1
2

≤ Ch

−2m−

−k−1

−k−2M(1−)

|u(y

)

1
2

with a constant C that is independent of x

, provided M >

−k

and therefore

xμ ,ξμ

, x

, y)u(x) dx

≤ Ch

−2m−

−1

−

k
2

−2M(1−)

2
L

Hence by Minkowski’s integral inequality

∗

≤

⊥

xμ ,ξμ

, x

, y)u(x) dx

1
2

≤ Ch

−m−

−1

−

k
4

−M(1−)

provided M >

−k

and the amplitude is compactly supported in x

. This yields

the L

bound for m <

−(n−1+k)/4−(1−)M provided M >

−k

, and completes

the proof of Theorem 2.10

Remark

2.11. The phase of the linearized maximal wave operator which is

ϕ(x, ξ) = t(x)

|ξ| + x, ξ, satisﬁes the assumptions of Theorem 2.10 since it belongs

to L

∞

and it also satisﬁes the rough corank condition 2.2.2. Indeed if ξ

∈ S

−1

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

we can take V

x,ξ

= ξ

⊥

and if π

, π

⊥

denote the projections onto span ξ and V

x,ξ

respectively then it is clear that

|∂

ϕ(x, ξ)

− ∂

ϕ(y, ξ)

|t(x) − t(y)|

|x − y|

+ 2(t(x)

− t(y)) ξ, x − y

±|π

−y)|

≥

⊥

− y)

t(x)

− t(y)| − |π

− y)|

≥ |π

⊥

− y)|

Therefore, as mentioned earlier, the Fourier integral operators under consideration
include the linearized maximal wave operator.

A consequence of this is a local L

boundedness result for Fourier integral

operators with smooth phase functions and rough symbols.

Corollary

2.12. Suppose that ϕ(x, ξ) is a smooth phase function satisfying

the non-degeneracy condition

(2.55)

rank

∂

∂x

∂ξ

≥ n − k,

on supp a

and the entries of the Hessian matrix have bounded derivatives with respect to
both x and ξ separately. Assume also that the symbol a belongs to L

∞

, 0

≤

≤ 1. Then the associated Fourier integral operator is bounded from L

comp

to L

2
loc

provided m <

−

k
2

−k)(−1)

This is sharp, for example in the case k = 0 (i.e. pseudodiﬀerential operators),

since there exists m

with m

> n(

−δ)/2 such that the pseudodiﬀerential operator

with symbol belonging to S

,δ

is not bounded from L

comp

to L

2
loc

, see [H4]. Now

since the phase of a pseudodiﬀerential operator satisﬁes the condition of the above
corollary with k = 0 and since obviously m

≥ n( − 1)/2 and S

,δ

⊂ L

∞

, it

follows that the above L

boundedness is sharp.

2.2.3. Symplectic aspects of the L

boundedness. Here we shall discuss

the symplectic aspects of the L

boundedness of Fourier integral operators which

aims to highlight the essentially geometric nature of the problem of L

regularity

of Fourier integral operators. We begin by recalling some of the well known L

continuity results in the case of smooth phases and amplitudes. The kernel of the
Fourier integral operator

T u(x) = (2π)

−n

iϕ(x,ξ)

a(x, ξ)

u(ξ) dξ

(2.56)

is an oscillatory integral whose wave front set is contained in the closed subset of

∗

= T

∗

\ 0

WF(T )

⊂

(x, ∂

ϕ(x, ξ), ∂

ϕ(x, ξ),

−ξ) : (x, ξ) ∈ supp a, ξ = 0

(2.57)

The cotangent space T

∗

is endowed with the symplectic form

σ =

j=1

dξ

∧ dx

A canonical relation is a Lagrangian submanifold of the product T

∗

× T

∗

endowed with the symplectic form σ

⊕ (−σ), this means that the aforementioned

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

symplectic form vanishes on the canonical relation. In particular, by rearranging
the terms in the closed cone (2.57), one obtains a canonical relation

(x, ∂

ϕ(x, ξ), ∂

ϕ(x, ξ), ξ) : (x, ξ)

∈ supp a

in T

∗

×T

∗

. If

C is a canonical relation, we consider the two maps π

: (x, ξ)

→

(x, ∂

ϕ) and π

: (x, ξ)

→ (∂

ϕ, ξ),

C ⊂ T

∗

× T

∗

{{www

www

##G

∗

The canonical relation

C is (locally) the graph of a smooth function χ if and only

if π

is a (local) diﬀeomorphism, and in this case χ = π

◦ π

−1

. This function χ

is a diﬀeomorphism if and only if π

is a diﬀeomorphism. Note that if this is the

case, χ is a symplectomorphism because the submanifold

C is Lagrangian for the

symplectic form, i.e. σ

⊕ (−σ)

dξ

∧ dx − dη ∧ dy = 0

when (y, η) = χ(x, ξ).

The canonical relation

is locally the graph of a symplectomorphism in the neigh-

bourhood of (x

, ∂

ϕ(x

, ξ

), ∂

ϕ(x

, ξ

), ξ

) if and only if

det

∂

∂x∂ξ

, ξ

)

= 0.

(2.58)

It is well-known that the Fourier integral operators of order 0 whose canonical
relation

is locally the graph of a symplectic transformation χ, are locally L

bounded. More precisely

Theorem

2.13. Let a

∈ S

1,0

and ϕ be a real valued function in C

∞

\ 0) which is homogeneous of degree 1 in ξ. Assume that the homogeneous

canonical relation

is locally the graph

∗

of a symplectomorphism between two

open neighbourhoods in ˙

∗

= T

∗

\ 0. Then the Fourier integral operator

(2.56) deﬁnes a bounded operator from L

comp

to L

2
loc

Proof.

This is Theorem 25.3.1 in [H2].

But in fact, there are boundedness results even when

C is not the graph of a

symplectomorphism, i.e. when either the projection π

or π

is not a diﬀeomor-

phism. There is an important instance for which this is the case and one could still
prove local L

boundedness with loss of derivatives. A suggestive example for this

situation is the restriction operator to a linear subspace H =

x = (x

, x

)

∈ R

× R

: x

= 0

u =

u(x

, 0) = (2π)

−n

,ξ

u(ξ) dξ

where m

≤ 0. We know that this operator is bounded from L

comp

to L

2
loc

; indeed

for all a

∈ C

∞

) there exists a constant C

m,n

such that

≤ C

m,n

∗

Or equivalently that (2.58) holds on supp a.

2.2. LOCAL AND AND GLOBAL L

BOUNDEDNESS OF FIO’S

provided m

≤ − codim H/2. The canonical relation of the Fourier integral operator

is given by

(x, ξ

, 0; x

, 0, ξ), (x, ξ)

∈ T

∗

||yyy

yyy

""E

= 0

⊂ T

∗

= 0

⊂ T

∗

By σ

we denote the pullback of the symplectic form σ, by π

, to

(of course

we could equally well consider the pullback π

∗

σ without changing anything)

= π

∗

σ = dξ

∧ dx

Then we have

corank σ

= 2n

= 2 codim H

and the condition of L

boundedness is therefore m

≤ − corank σ

/4. In fact, this

example models the general situation, and this is Theorem 25.3.8 in [H2].

Theorem

2.14. Let a

∈ S

1,0

and ϕ be a real valued function in C

∞

\ 0) which is homogeneous of degree 1 in ξ such that

†

dϕ

= 0 on supp a. Then

the Fourier integral operator (2.56) deﬁnes a bounded operator from L

comp

to L

2
loc

provided m

≤ − corank σ

/4. Here σ

is the two form on

obtained by lifting

to C

the symplectic form σ on ˙

∗

by one of the projections π

or π

The fact that the canonical relation is parametrised by

F : (x, ξ)

→ (x, ∂

ϕ(x, ξ), ∂

ϕ(x, ξ), ξ)

allows us to compute

∗

(π

∗

σ) = d(π

◦ F )

∗

(ξ dx) = d

∂

ϕ(x, ξ) dx

j,k=1

∂

ϕ(x, ξ) dx

∧ dx

j,k=1

∂

ϕ(x, ξ) dξ

∧ dx

Therefore we have

∗

j,k=1

∂

ϕ(x, ξ) dξ

∧ dx

which yields

corank σ

= 2 corank

∂

∂x∂ξ

The geometric assumption in Theorem 2.14 (which is valid for general Fourier
integral operators, not necessarily of the form (2.56)) is therefore equivalent to

≤ −

corank

∂

∂x∂ξ

(2.59)

†

This ensures that

is a homogeneous canonical relation to which the radial vectors of

∗

× 0 and 0 × ˙T

∗

are never tangential.

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

Remark

2.15. If the function t(x) in the linearized maximal wave operator

(2.49) were smooth, then that operator would fall into the category of Fourier
integral operators satisfying the assumptions of Theorem 2.14. Indeed as already
noted in the introduction, the corank of ∂

ϕ/∂x∂ξ when ϕ(x, ξ) = t(x)

|ξ| + x, ξ

is at most 1. Therefore e

it(x)

√

−Δ

deﬁnes a bounded operator from H

1/2

comp

to L

2
loc

when t(x) is a smooth function on R

Theorem 2.8 for = 1 is the non-smooth analogue of Theorem 2.13 where the

non-degeneracy condition (2.58) which requires smoothness in x has been replaced
by Deﬁnition 1.6. Note nevertheless that Theorem 2.8 is a global L

result. Similarly

Theorem 2.10 for = 1 is the non-smooth analogue of Theorem 2.14 with (2.59)
replaced by assumption 2.2.2.

2.3. Global L

∞

boundedness of rough Fourier integral operators

In this section, we establish the L

∞

boundedness of Fourier integral operators.

To prove the L

∞

boundedness of the high frequency portion of the operator, we

need to use the semiclassical estimates of Subsection 1.2.2. However, using only
the Seeger-Sogge-Stein decomposition yields a loss of derivatives no better than
m <

−

−1

+n(

−1), and to obtain the L

∞

boundedness result claimed in Theorem

2.16, further analysis is needed.

Theorem

2.16. Let T be a Fourier integral operator given by (0.1) with ampli-

tude a

∈ L

∞

and phase function ϕ

∈ L

∞

. Then there exists a constant C > 0

such that

T u

∞

∈ S (R

)

provided m <

−

−1

(

− 1) and 0 ≤ ≤ 1. Furthermore, in the case = 0

i.e. for operators with amplitude in a

∈ L

∞

one can obtain the L

∞

boundedness

with an improved decay m <

−n

Proof.

As a ﬁrst step, we use the semiclassical reduction of Subsection 1.2.1

to decompose T into T

and T

. Thereafter we split the semiclassical piece T

fur-

ther into

J
ν=1

using the Seeger-Sogge-Stein decomposition of Subsection 1.2.2

applied to the amplitude a(x, ξ) and the phase ϕ(x, ξ). Once again, the boundedness
of T

follows from Theorem 1.18, but here we don’t need the rough non-degeneracy

of the phase function due to the fact that we are dealing with the L

∞

boundedness

of T

which only requires that the integral with respect to the y variable of the

Schwartz kernel T

(x, y) being ﬁnite. See Theorem 1.18 for further details.

From equation 1.12 one deduces that the kernel of the semiclassical high fre-

quency operator T

is given by

(x, y) = (2πh)

−n

∇

ϕ(x,ξ

)

−y,ξ

(x, ξ, h) dξ,

with

(x, ξ, h) = e

∇

ϕ(x,ξ)

−∇

ϕ(x,ξ

),ξ

(ξ)χ(ξ)a(x, ξ/h).

Now since

∞

≤ u

∞

(x, y)

| dy,

2.3. GLOBAL L

∞

BOUNDEDNESS OF ROUGH FIO’S

it remains to show a suitable estimate for

(x, y)

| dy. As in the proof of L

boundedness, we use the diﬀerential operator

L = 1

− ∂

− h∂

for which we have according to Lemma 1.9

(2.60)

sup

(

·, ξ, h)

∞

−m−2N(1−)

Setting

(2.61)

g(z) = h

−2

+ h

−1

we have

∇

ϕ(x,ξ

)

−y,ξ

1 + g(y

− ∇

ϕ(x, ξ

)

∇

ϕ(x,ξ

)

−y,ξ

for all integers N . Now we observe that

(2πh)

(x, y)

| dy = (2πh)

(x, y +

∇

ϕ(x, ξ

))

| dy

(x, y, h)

| dy

√

g(y)

≤h

√

g(y)>h

(x, y, h)

| dy := I

+ I

where

(x, y, h) = (2πh)

−

y,ξ

(x, ξ, h) dξ

is the semiclassical Fourier transform of b

. To estimate I

we use the Cauchy-

Schwarz inequality, the semiclassical Plancherel theorem, the deﬁnition of g in
(2.61) and (2.60). Hence remembering the fact that the measure of the ξ-support

of b

(x, ξ, h) is O(h

−1)

) we have

≤

√

g(y)

≤h

1
2

(x, y, h)

1
2

n+1

|y|≤h

1
2

(x, ξ, h)

dξ

1
2

n+1

−m+

−1

−m+

Before we proceed with the estimate of I

, we observe that if l is a non-negative

integer then the semiclassical Plancherel theorem and (2.60) yield

(x, y, h)

(1 + g(y))

1
2

≤

(x, ξ, h)

dξ

1
2

(2.62)

≤ h

−m−2l(1−)+

−1

Moreover, any positive real number l which is not an integer can be written as
[l] +

{l} where [l] denotes the integer part of l and {l} its fractional part, which

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

is 0 <

{l} < 1. Therefore, H¨older’s inequality with conjugate exponents

{l}

and

−{l}

yields

(x, y, h)

(1 + g(y))

{l}

2(1

−{l})

(1 + g(y))

{l}([l]+1)

(1 + g(y))

2[l](1

−{l})

≤

(1 + g(y))

2([l]+1)

{l}

(1 + g(y))

2[l]

−{l}

Therefore, using (2.62) we obtain

(x, y, h)

(1 + g(y))

1
2

≤

[l]+1

(x, ξ, h)

dξ

{l}

[l]

(x, ξ, h)

dξ

−{l}

≤ h

{l}(−m−2([l]+1)(1−)+

−1

)

−{l})(−m−2[l](1−)+

−1

)

≤ h

−m−2l(1−)+

−1

and hence (2.62) is actually valid for all non-negative real numbers l. Turning now
to the estimates for I

, we use the same tools as in the case of I

and (2.62) for

∈ [0, ∞). This yields for any l >

≤

√

g(y)>h

(1 + g(y))

−2l

1
2

(x, y, h)

(1 + g(y))

1
2

n+1

|y|>h

|y|

−4l

1
2

−m−2l(1−)+

−1

n+1

(

−2l)

−m−2l(1−)+

−1

−m+

−2l

Therefore

sup

(x, y)

| dy ≤ C

−m+

−2l

(2.63)

and summing in ν yields

∞

≤

ν=1

∞

≤ C

−m+

−2l−

−1

∞

since J is bounded (from above and below) by a constant times h

−

−1

. By Lemma

1.8 one has

T u

∞

provided m <

−

−1

− 2l and l >

, i.e. if m <

−

−1

(

− 1).

Now to see how m <

−n

yields the L

∞

boundedness for a(x, ξ)

∈ L

∞

we use Lemma 1.10, and representation 1.16 and observe that if we set b(x, ξ) :=
a(x, ξ)e

iθ(x,ξ)

, we can reduce the study of T u(x) to that of the Ψ-pseudodiﬀerential

operator (b(x, D)u)(

∇

ϕ(x, ζ)), with symbol b(x, ξ)

∈ L

∞

. The L

∞

boundedness

of T for m <

−n

follows now at once from Proposition 2.5 in [KS]. This completes

the proof of Theorem 2.16.

2.4. GLOBAL L

-L

AND L

-L

BOUNDEDNESS OF FIO’S

2.4. Global L

-L

and L

-L

boundedness of Fourier integral operators

In this section we shall state and prove our main boundedness results for Fourier

integral operators. Here, we prove results both for smooth and rough operators with
phases satisfying various non-degeneracy conditions. As a ﬁrst step, interpolation
yields the following global L

results:

Theorem

2.17. Let T be a Fourier integral operator given by (0.1) with am-

plitude a

∈ S

,δ

, 0

≤ ≤ 1, 0 ≤ δ ≤ 1, and a phase function ϕ(x, ξ) ∈ Φ

satisfying

the strong non-degeneracy condition. Setting λ := min(0, n(

− δ)), suppose that

either of the following conditions hold:

(a) 1

≤ p ≤ 2 and

m < n(

− 1)

− 1

−

+ λ

−

;

(b) 2

≤ p ≤ ∞ and

m < n(

− 1)

−

+ (n

− 1)

−

;

≤ ≤ 1, 0 ≤ δ < 1, and

m =

Then there exists a constant C > 0 such that

T u

≤ Cu

Proof.

The proof is a direct consequence of interpolation of the the L

bound-

edness result of Theorem 2.1 with the L

boundedness of Theorem 2.7 on one hand,

and the interpolation of the latter with the L

∞

boundedness result of Theorem 2.16.

The details are left to the reader.

Theorem

2.18. Let T be a Fourier integral operator given by (0.1) with am-

plitude a

∈ L

∞

, 0

≤ ≤ 1 and a strongly non-degenerate phase function

ϕ(x, ξ)

∈ Φ

. Suppose that either of the following conditions hold:

(a) 1

≤ p ≤ 2 and

m <

(

− 1) +

− 1

−

;

(b) 2

≤ p ≤ ∞ and

m <

(

− 1) + (n − 1)

−

Then there exists a constant C > 0 such that

T u

≤ Cu

Proof.

The proof follows once again from interpolation of the L

boundedness

result of Theorem 2.1 with the L

boundedness of Theorem 2.2 on one hand, and

the interpolation of the latter with the L

∞

boundedness result of Theorem 2.16.

As an immediate consequence of the theorem above one has

2. GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

Corollary

2.19. For a Fourier integral operator T with amplitude a

∈ L

∞

and a strongly non-degenerate phase function ϕ(x, ξ)

∈ Φ

, one has L

boundedness

for p

∈ [1, ∞] provided m < −(n − 1)|

1
p

−

1
2

Using Sobolev embedding theorem one can also show the following L

− L

estimates for rough Fourier integral operators.

Theorem

2.20. Suppose that

(1) T is a Fourier integral operator with an amplitude a

∈ S

,δ

, 0

≤ ≤ 1,

≤ δ ≤ 1 and a strongly non-degenerate phase function ϕ(x, ξ) ∈ Φ

with either of the following conditions:

(a) 1

≤ p ≤ q ≤ 2 and

m < n(

− 1)

− 1

− (n − 1)

−

+ λ

−

;

(b) 2

≤ p ≤ q ≤ ∞ and

m < n(

− 1)

−

+ (n

− 1)

−

(2) T is a Fourier integral operator with an amplitude a

∈ L

∞

, 0

≤ ≤ 1

and a strongly non-degenerate phase function ϕ(x, ξ)

∈ Φ

, with either of

the following conditions:

(a) 1

≤ p ≤ q ≤ 2 and

m <

(

− 1) − (n − 1)

−

;

(b) 2

≤ p ≤ q ≤ ∞ and

m <

(

− 1) + (n − 1)

−

Then there exists a constant C > 0 such that

T u

≤ Cu

Proof.

We give the details of the proof only for (1) (a). The rest of the proof

is similar to that of (1)(a), through the use of Theorem 2.17 part (b) or Theorem
2.18. Condition m < n(

− 1)(

2
q

− 1) − (n − 1)(

1
p

−

1
2

) + λ(1

−

1
q

) +

1
q

−

1
p

yields the

existence of of a real number s with

(2.64) n

−

≤ s < n( − 1)

− 1

−

+ λ

−

− m.

Therefore writing T = T (1

−Δ)

s
2

−Δ)

−

s
2

, Leibniz rule reveals that the amplitude

of T (1

− Δ)

s
2

belongs to L

∞

m+s

and since

m + s < n(

− 1)

− 1

−

+ λ

−

Theorem 2.17 part (a) yields

T u

T (1 − Δ)

s
2

− Δ)

−

s
2

(1 − Δ)

−

s
2

where the very last estimate is a direct consequence of (2.64) and the Sobolev
embedding theorem. Hence

T u

for the above ranges of p, q and m,

and the proof is complete.

CHAPTER 3

Global and Local Weighted L

Boundedness of

Fourier Integral Operators

The purpose of this chapter is to establish boundedness results for a fairly wide

class of Fourier integral operators on weighted L

spaces with weights belonging to

Muckenhoupt’s A

class. We also prove these results for Fourier integral operators

whose phase functions and amplitudes are only bounded and measurable in the spa-
tial variables and exhibit suitable symbol type behavior in the frequency variable.
We will start by recalling some facts from the theory of A

weights which will be

needed in this section. Thereafter we prove a couple of uniform stationary phase
estimates for oscillatory integrals and then proceed with the weighted boundedness
for the low frequency portions of Fourier integral operators. Before proceeding with
our claims about the weighted boundedness of the high frequency part of Fourier
integral operators, a discussion of a counterexample leads us to a rank condition on
the phase function ϕ(x, ξ) which is crucial for the validity of the weighted bound-
edness (with A

weights) of Fourier integral operators. Using interpolation and

extrapolation, we can prove an endpoint weighted L

boundedness theorem for op-

erators within a speciﬁc class of amplitudes and all A

weights, which is shown

to be sharp in a case of particular interest and can also be invariantly formulated
in the local case. Finally we show the L

boundedness of a much wider class of

operators for some subclasses of the A

weights.

3.1. Tools in proving weighted boundedness

The following results are well-known and can be found in their order of appear-

ance in [GR], [J] and [S].

Theorem

3.1. Suppose p > 1 and w

∈ A

. There exists an exponent q < p,

which depends only on p and [w]

, such that w

∈ A

. There exists ε > 0, which

depends only on p and [w]

, such that w

1+ε

∈ A

Theorem

3.2. For 1 < q <

∞, the Hardy-Littlewood maximal operator is

bounded on L

if and only if w

∈ A

. Consequently, for 1

≤ p < ∞, M

is bounded

on L

if and only if w

∈ A

q/p

Theorem

3.3. Suppose that ϕ : R

→ R is integrable non-increasing and ra-

dial. Then, for u

∈ L

, we have

ϕ(y)u(x

− y) dy ≤ ϕ

M u(x)

for all x

∈ R

The following result of J.Rubio de Francia is also basic in the context of

weighted norm inequalities.

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

Theorem

3.4 (Extrapolation Theorem). If

T u

≤ Cu

for some ﬁxed

∈ (1, ∞) and all w ∈ A

, then one has in fact

T u

≤ Cu

for all

∈ (1, ∞) and w ∈ A

3.1.1. A pointwise uniform bound on oscillatory integrals. Before we

proceed with the main estimates we would need a stationary-phase estimate which
will enable us to control certain integrals depending on various parameters uni-
formly with respect to those parameters. Here and in the sequel we denote the
Hessian in ξ of the phase function ϕ(x, ξ) by ∂

ξξ

ϕ(x, ξ).

Lemma

3.5. For λ

≥ 1, let a

(x, ξ)

∈ L

∞

with seminorms that are uniform

in λ and let supp

(x, ξ)

⊂ B(0, λ

) for some μ

≥ 0. Assume that ϕ(x, ξ) ∈ L

∞

and

| det ∂

ξξ

ϕ(x, ξ)

| ≥ c > 0 for all (x, ξ) ∈ supp a

. Then one has

(3.1)

sup

∈R

iλϕ(x,ξ)

(x, ξ) dξ

nμ

−

Proof.

We start with the case μ = 0.

The matrix inequality

−1

≤

| det A|

−1

with A = ∂

ξξ

ϕ(x, ξ) and the assumptions on ϕ, yield the uni-

form bound (in x and ξ)

(3.2)

[∂

ξξ

ϕ(x, ξ)]

−1

≤

| det ∂

ξξ

ϕ(x, ξ)

−1

∂

ξξ

ϕ(x, ξ)

−1

Looking at the map κ

: ξ

→ ∇

ϕ(x, ξ), we observe that Dκ

(ξ) = ∂

ξξ

ϕ(x, ξ),

where Dκ

(ξ) denotes the Jacobian matrix of the map κ

, and that κ

is a diﬀeo-

morphism due to the condition on ϕ in the lemma. Therefore

Dκ

−1

( ˜

ξ) =

∂

ξξ

ϕ(x, κ

−1

( ˜

ξ))

−1

which using (3.2) yields uniform bounds for

Dκ

−1

( ˜

ξ)

, hence

|κ

−1

( ˜

ξ)

− κ

−1

(˜

η)

| ≤

Dκ

−1

× |˜

− ˜η| |˜ξ− ˜η|.

This applied to ˜

ξ = κ

(ξ), ˜

η = κ

(η) implies that

|ξ − η| |κ

(ξ)

− κ

(η)

| = |∇

ϕ(x, ξ)

− ∇

ϕ(x, η)

(3.3)

We set

I(λ, x) :=

iλϕ(x,ξ)

(x, ξ) dξ

and compute

|I(λ, x)|

iλ(ϕ(x,ξ)

−ϕ(x,ξ+η))

(x, ξ) a

(x, ξ + η) dξ dη.

We decompose the integral in η into two integrals, one on

|η| ≤ δ and the other on

|η| > δ, and this yields the estimate

|I(λ, x)|

∞

−1

iλr

ϕ(x,ξ)

−ϕ(x,ξ+rθ)

(x, ξ) a

(x, ξ + rθ) dξ

dθ r

−1

dr.

Using the uniform lower bound on the gradient of the phase in (3.3), we get the
uniform lower bound

∇

ϕ(x, ξ)

− ∇

ϕ(x, ξ + rθ)

3.1. TOOLS IN PROVING WEIGHTED BOUNDEDNESS

Therefore, applying the non-stationary phase estimate of [H1, Theorem 7.7.1] to
the right-hand side integral yields

|I(λ, x)|

+ λ

−n−1

∞

−n−1

−1

+ δ

−1

−n−1

We now optimize this estimate by choosing δ = λ

−1

and obtain the bound

|I(λ, x)| ≤ Cλ

−n

with a constant uniform in x.

In the case μ > 0, we cover the ball B(0, λ

) with balls of radius 1 and in

doing that, one would need O(λ

nμ

) balls of unit radius. This will obviously pro-

vide a covering of the ξ support of a

with balls of radius 1 and we take a ﬁnite

smooth partition of unity θ

(ξ), j = 1, . . . , O(λ

nμ

), subordinate to this covering

with

|∂

| ≤ C

. Now by the ﬁrst part of this proof we have

(3.4)

iλϕ(x,ξ)

(x, ξ) θ

(ξ) dξ

≤ Cλ

−

with a constant that is uniform in x and j. Finally summing in j and remembering
that there are roughly O(λ

nμ

) terms involved yields the desired estimate.

3.1.2. Weighted local and global low frequency estimates. For the low

frequency portion of the Fourier integral operators studied in this section we are
once again able to handle the L

boundedness for all p

∈ [1, ∞], using Lemma 1.17

and imposing suitable conditions on the phases.

Proposition

3.6. Let

∈ [0, 1] and suppose either:

(a) a(x, ξ)

∈ L

∞

is compactly supported in the x variable, m

∈ R and

ϕ(x, ξ)

∈ L

∞

; or

(b) a(x, ξ)

∈ L

∞

, m

∈ R and ϕ(x, ξ) − x, ξ ∈ L

∞

Then for all χ

(ξ)

∈ C

∞

supported near the origin, the Fourier integral operator

u(x) =

(2π)

a(x, ξ) χ

(ξ) e

iϕ(x,ξ)

u(ξ) dξ

is bounded on L

for 1 < p <

∞ and all w ∈ A

Proof.

(a) The operator T

can be written as T

u(x) =

(x, y) u(x

−y) dy

with

(x, y) =

(2π)

iψ(x,ξ)

−iy,ξ

(ξ) a(x, ξ) dξ,

where ψ(x, ξ) := ϕ(x, ξ)

− x, ξ satisﬁes the estimate

sup

|ξ|=0

|ξ|

−1+|α|

|∂

ψ(x, ξ)

| ≤ C

for

|α|≥1, on support of the amplitude a. Therefore setting b(x,ξ):=a(x, ξ)χ

(ξ)e

iψ(x,ξ)

we have that b is bounded and

sup

|ξ|=0

|ξ|

−1+|α|

|∂

b(x, ξ)

| < ∞,

for

|α| ≥ 1 uniformly in x and using Lemma 1.17, we have for all μ ∈ [0, 1)

(3.5)

(x, y)

| ≤ C

−n−μ

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

for all x. From this and Theorem 3.3, it follows that

u(x)

| Mu(x) and Theorem

3.2 yields the L

boundedness of T

(b) The only diﬀerence from the local case, is that instead of the assumption of

compact support in x, the assumption ϕ(x, ξ)

− x, ξ ∈ L

∞

yields that b(x, ξ) in

the previous proof satisﬁes the very same estimate, whereupon the same argument
will conclude the proof.

3.2. Counterexamples in the context of weighted boundedness

The following counterexample going back to [KW], shows that for smooth

Fourier integral operators (smooth phases as well as amplitudes), the non-degeneracy
of the phase function i.e. the non-vanishing of the determinant of the mixed hes-
sian of the phase, is not enough to yield weighted L

boundedness, unless one is

prepared to loose a rather unreasonable amount of derivatives.

Counterexample

1. Let ϕ(x, ξ) =

x, ξ + ξ

which is non-degenerate but

rank ∂

ξξ

ϕ = 0,

and let a(x, ξ) =

with

−n < m < 0. Then it has been shown in [Yab] that for

1 < p <

∞ there exists w ∈ A

and f

∈ L

such that the Fourier integral operator

T u(x) = (2π)

−n

x,ξ+iξ

u(ξ) dξ, does not belong to L

However, as the following proposition shows, even with a phase of the type

above, one can prove weighted L

boundedness provided certain (comparatively

large) loss of derivatives.

Proposition

3.7. Let a(x, ξ)

∈ L

∞

, m

≤ −n and ϕ(x, ξ) − x, ξ ∈ L

∞

Then T

a,ϕ

u(x) :=

iϕ(x,ξ)

σ(x, ξ)

u(ξ) dξ is bounded on L

for w

∈ A

and 1 <

p <

∞. This result is sharp.

Proof.

For the low frequency part of the Fourier integral operator we could

for example use Proposition 3.6. For the high frequency part we may assume that
a(x, ξ) = 0 when ξ is in a neighborhood of the origin. The proof in the case m <

−n

can be done by a simple integration by parts argument in the integral deﬁning the
Schwartz kernel of the operator. Hence the main point of the proof is to deal with
the case m =

−n. Now the Fourier integral operator T

a,ϕ

can be written as

(3.6)

a,ϕ

u(x) =

iϕ(x,ξ)

a(x, ξ)

u(ξ) dξ =

σ(x, ξ)e

x,ξ

u(ξ) dξ

with σ(x, ξ) = a(x, ξ)e

i(ϕ(x,ξ)

−x,ξ)

and we can assume that σ(x, ξ) = 0 in a neigh-

borhood of the origin. Therefore, since we have assumed that ϕ(x, ξ)

− x, ξ ∈

∞

, the operator T

a,ϕ

= σ(x, D) is a pseudo-pseudodiﬀerential operator in the

sense of [KS], belonging to the class OPL

∞

−n

. We use the Littlewood-Paley par-

tition of unity and decompose the symbol as

σ(x, ξ) =

∞

k=1

(x, ξ)

with σ

(x, ξ) = σ(x, ξ)ϕ

(ξ), k

≥ 1. We have

(x, D)(u)(x) =

(2π)

(x, ξ)

u(ξ)e

x,ξ

dξ

3.2. COUNTEREXAMPLES IN THE CONTEXT OF WEIGHTED BOUNDEDNESS

for k

≥ 1. We note that σ

(x, D)u(x) can be written as

(x, D)u(x) =

(x, y)u(x

− y) dy

with

(x, y) =

(2π)

(x, ξ)e

y,ξ

dξ = ˇ

(x, y),

where ˇ

here denotes the inverse Fourier transform of σ

(x, ξ) with respect to ξ.

One observes that

|σ

(x, D)u(x)

(x, y)u(x

− y) dy

(x, y)ω(y)

ω(y)

u(x

− y) dy

with weight functions ω(y) which will be chosen momentarily. Therefore, H¨

older’s

inequality yields

(3.7)

|σ

(x, D)u(x)

≤

(x, y)

|ω(y)|

|u(x − y)|

|ω(y)|

where

1
p

= 1. Now for an l >

, we deﬁne ω by

ω(y) =

|y| ≤ 1;

|y|

|y| > 1.

By Hausdorﬀ-Young’s theorem and the symbol estimate, ﬁrst for α = 0 and then
for

|α| = l, we have

(x, y)

≤

|σ

(x, ξ)

dξ

|ξ|∼2

−npk

dξ

(3.8)

(

−n)

and

(x, y)

|y|

|∇

l
ξ

(x, ξ)

dξ

(3.9)

|ξ|∼2

−kpn

dξ

(

−n)

Hence, splitting the integral into

|y| ≤ 1 and , |y| > 1 yields

(x, y)

|ω(y)|

(

−n)

= 2

kp(

−n)

Furthermore, using Theorem 3.1.3 we have

|u(x − y)|

|ω(y)|

u(x)

with a constant that only depends on the dimension n. Thus (3.7) yields

(3.10)

|σ

(x, D)u(x)

−n)

u(x)

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

Summing in k using (3.10) we obtain

a,ϕ

u(x)

| = |σ(x, D)u(x)|

∞

k=1

|σ

(x, D)u(x)

1
p

u(x)

∞

k=1

−n)

1
p

We observe that the series above converges for any p > 1 and therefore an applica-
tion of Theorem 3.2 ends the proof. The sharpness of the result follows from the
Counterexample 1.

The above discussion suggests that without further conditions on the phase,

which as it will turn out amounts to a rank condition, the weighted norm inequalities
of the type considered in this paper will be false. The following counterexample
shows that, even if the phase function satisﬁes an appropriate rank condition, there
is a critical threshold on the loss of derivatives, below which the weighted norm
inequalities will fail.

Counterexample

2. We consider the following operator

= e

|D|

and the following functions

(x) =

|x|

−b

(x) =

−i|ξ|+ix·ξ

−μ

dξ.

As was mentioned in Example 1 in Chapter 1, w

∈ A

if 0

≤ b < n, from

which it also follows that w := w

|x|<2

∈ A

for 0

≤ b < n, were χ

denotes

the characteristic function of the set A. Now if μ < m + n then we claim that
|T

(x)

| ≥ C

mμ

|x|

−m−n

|x| ≤ 2. Indeed, we have

(x) =

·ξ

−μ

dξ

which is a radial function equal to

−1

| |x|

−m−n

∞

dω(r)

|x|

+ r

−μ
2

−1

dr.

If we denote by g

μm

the function given by the integral, and take a dyadic partition

of unity 1 = ψ

∞
j=1

ψ(2

−j

·) then

μm

(s) =

dω(r)(s

+ r

)

−μ
2

−1

(r) dr

∞

j=1

∞

dω(2

r)(s

+ 2

)

−μ
2

−1

ψ(r) dr.

The ﬁrst term is continuous if m

− μ + n > 0 and writing down the integral

representing

dω(2

r) and integrating by parts yields that the series in the second

term of g

μm

is a smooth function of s. Moreover

μm

(0) =

ξ,e

|ξ|

−μ

dξ = C

−n−m+μ

= 0,

3.2. COUNTEREXAMPLES IN THE CONTEXT OF WEIGHTED BOUNDEDNESS

since the inverse Fourier transform of a radial homogeneous distribution of degree α
is a radial homogeneous distribution of order

−n − α. This proves the claim. From

this claim it follows that

w dx

≥ C

μm

|x|≤2

|x|

(μ

−m−n)p−b

dx,

and therefore T

∈ L

if (μ

− m − n)p − b ≤ −n.

Now, we also have

(x)

| ≤ A

− |x|

−

n+1

+ B

|x| ≤ 2. This is because

the function f

is radial

(x) =

−1

∞

dω(r

|x|)e

−ir

(1 + r

)

−1

and using the information on the Fourier transform of the measure of the sphere,

(x) =

−1

∞

−ir(1±|x|)

|x|)(1 + r

)

−1

(3.11)

where

|∂

(r)

| ≤ C

−

−1

−α

We now use a dyadic partition of unity 1 = ψ

∞
k=1

ψ(2

−k

·) on the integral and

obtain a sum of terms of the form

∞

−2

ir(1

±|x|)

|x|)ψ(r)(1 + 2

)

−1

±
k

(r,

|x|)

with

|∂

(r,

|x|)| ≤ C

−(

−1

−μ+α)k

Integration by parts yields

| ≤ C

+ C

|1−|x||≤1

−(

n+1

−μ)k

+ C

|1−|x||>1

−(

n+1

−μ+N)k

− |x|

−N

≤ C

+ C

− |x|

−

n+1

Hence one has

w dx

≤ A

|x|≤2

− |x|

μp

−

n+1

|x|

−b

dx + B

|x|≤2

|x|

−b

dx,

which in turn yields f

∈ L

if μ >

n+1

−

1
p

and 0

≤ b < n. From the estimates

above it follows that if 1 < p <

∞ and T

is bounded on L

then

≤ −

− 1

−

(3.12)

Indeed if T

is bounded on L

then we have

−m >

− n

+ n

− μ

for all 0

≤ b < n and all μ >

n+1

−

1
p

. Letting μ tend to

n+1

−

1
p

we obtain

≤ −

− n

−

− 1

−

for all 0

≤ b < n, and letting b tend to n yields (3.12).

Now by Theorem 3.4 if T

is bounded on L

for a ﬁxed q > 1 and for all w

∈ A

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

then by extrapolation it is bounded on all L

for all w

∈ A

and all 1 < p <

∞,

therefore since w

∈ A

⊂ A

, we conclude that T

is bounded on L

for all

1 < p <

∞, which implies that m has to satisfy the inequality

≤ −

− 1

−

for all 1 < p <

∞. Letting p tend to 1, we obtain

≤ −

n + 1

which is the desired critical threshold for the validity of the weighted L

bounded-

ness of the class of Fourier integral operators under consideration in this paper.

We end up with an example showing that in the most general situation one

cannot expect global L

weighted estimates unless the phase satisﬁes some slightly

stronger property than a rank condition.

Counterexample

3. Let B be the unit ball in R

, we consider the operator

T u(x) = (1

∗ u)(2x)

and suppose that this operator is bounded on L

with bound C

= C

[w]

only

depending on [w]

T u

≤ C

∈ S (R

Note that the A

-constant [w]

is scale invariant. If we apply the estimate to the

function u(ε

·) and the weight w(ε·) and scale it, we obtain

≤ C

∈ S (R

)

with T

u = ε

−n

εB

∗ u)(2x). Since T

u tends to u(2x) in L

, by letting ε tend to

0 we deduce from the former

u(2 · )

≤ C

∈ S (R

After a change of variable, this would imply

−n

|u(x)|

w(x/2) dx

≤ C

|u(x)|

w(x) dx

for all u

∈ S (R

), hence

w(x/2)

≤ C

w(x).

This means that one can expect weighted L

estimates for T only if w satisﬁes

a doubling property. Note that T can be written as a sum of Fourier integral

operators with amplitudes in S

−

n+1

1,0

with phases of the form

= 2

x, ξ ± |ξ|

which satisfy a rank condition. Nevertheless, one has

− x, ξ /∈ L

∞

In particular, one cannot generally expect global weighted estimate for Fourier
integral operators with phases such that ϕ

− x, ξ /∈ L

∞

unless the weight

satisﬁes some doubling property.

3.3. INVARIANT FORMULATION IN THE LOCAL BOUNDEDNESS

3.3. Invariant formulation in the local boundedness

The aim of this section is to give an invariant formulation of the rank condition

on the phase function, which will be used to prove our weighted norm inequalities
for Fourier integral operators. In Counterexample 1 we saw that a rank condition
is necessary for the validity of weighted L

estimates. The following discussion will

enable us to give an invariant formulation of the local weighted norm inequalities

for operators with amplitudes in S

−(n+1)

+n(

−1)

−

∈ [

1
2

, 1]. We refer the reader

to [H3] for the properties of Fourier integral operators with amplitudes in S

−

∈ (

1
2

, 1] and the paper by A. Greenleaf and G. Uhlmann for the case when =

1
2

The central object in the theory of Fourier integral operators is the canonical

relation

(x, ∂

ϕ(x, ξ), ∂

ϕ(x, ξ), ξ) : (x, ξ)

∈ supp a

in T

∗

× T

∗

. We consider the following projection on the space variables

⊂ T

∗

× T

∗

x,y

)

(

C) ⊂ R

with

(x, ξ, y, η) = (x, y).

The diﬀerential of this projection is given by

dπ

, t

) = (t

, t

= ∂

ϕ t

+ ∂

xξ

ϕ t

= ∂

ξx

ϕ t

+ ∂

ξξ

ϕ t

so that its kernel is given by

(0, ∂

xξ

ϕ t

, 0, t

) : t

∈ ker ∂

ξξ

This implies

rank dπ

= codim ker dπ

= codim ker ∂

ξξ

ϕ = n + rank ∂

ξξ

ϕ.

Our assumption on the phase rank ∂

ξξ

ϕ = n

− 1 can be invariantly formulated as

rank dπ

= 2n

− 1.

Using these facts, we will later on be able to show that if T is a Fourier integral

operator with amplitude in S

−

n+1

+n(

−1)

−

with

∈ [

1
2

, 1] whose canonical relation

C is locally the graph of a symplectomorphism, and if

rank dπ

= 2n

− 1

everywhere on

C, with π

C → R

deﬁned by π

(x, ξ, y, η) = (x, η), then there

exists a constant C > 0 such that

T u

p
w,loc

≤ u

w,comp

for all w

∈ A

and all 1 < p <

∞. However, we will actually prove local weighted

boundedness of operators with amplitudes in the class L

∞

−

n+1

+n(

−1)

with

∈ [0, 1] for which the invariant formulation above lacks meaning, and therefore to

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

keep the presentation of the statements as simple as possible, we will not always
(with the exception of Theorem 3.12) state the local boundedness theorems in an
invariant form.

3.4. Weighted local and global L

boundedness of Fourier integral

operators

We start this by showing the local weighted L

boundedness of Fourier integral

operators. In Counterexample 1 which was related to Fourier integral operators
with linear phases, the Hessian in the frequency variable ξ of the phase function
vanishes identically. This suggests that some kind of condition on the Hessian
of the phase might be required. It turns out that the condition we need can be
formulated in terms of the rank of the Hessian of the phase function in the frequency
variable. Furthermore Counterexample 2 which was related to the wave operator,
suggests a condition on the order of the amplitudes involved. It turns out that these
conditions, appropriately formulated, will indeed yield weighted boundedness of a
wide range of Fourier integral operators even having rough phases and amplitudes.

Theorem

3.8. Let a(x, ξ)

∈ L

∞

with m <

−

n+1

+ n(

− 1) and ∈ [0, 1]

be compactly supported in the x variable. Let the phase function ϕ(x, ξ)

∈ C

∞

\ 0) homogeneous of degree 1 in ξ satisfy rank ∂

ξξ

ϕ(x, ξ) = n

− 1. Then the

corresponding Fourier integral operator is L

bounded for 1 < p <

∞ and all

∈ A

Proof.

The low frequency part of the Fourier integral operator is handled

by Proposition 3.6 part (a). For the high frequency portion, once again we use a
Littlewood-Paley decomposition in the frequency variables as in Subsection 1.2.1
to reduce the operator to its semiclassical piece

u(x) = (2π)

−n

i(ϕ(x,ξ)

−y,ξ)

χ(hξ) a(x, ξ) u(y) dy dξ

with χ(ξ) smooth and supported in the annulus

1
2

≤ |ξ| ≤ 2. Now if we let θ(x, ξ) :=

ϕ(x, ξ)

− x, ξ, then we have

(3.13)

|∇

| 1

on the support of a. Furthermore, if

(3.14)

(x, y) = (2π)

−n

i(θ(x,ξ)

−y,ξ)

χ(hξ)a

x, ξ

dξ,

then in order to get useful pointwise estimates for the operator T

we would need to

estimate the kernel T

(x, y). Localising in frequency and rotating the coordinates

we may assume that a(x, ξ) is supported in a small conic neighbourhood of a ξ

. At this point we split the modulus of T

into

u(x)

| ≤

|y|>1+2∇

L∞

|y|≤1+2∇

L∞

(x, y)

| |u(x − y)| dy

:= I + II.

where there are obviously no critical points on the domain of integration for I.
Estimate of I. Making the change of variables ξ

→ h

−1

ξ we obtain

u(x) = (2π)

−n

ih(θ(x,ξ)

−y,ξ)

χ(ξ) a

x, ξ/h

u(y) dy dξ.

3.4. WEIGHTED LOCAL AND GLOBAL L

BOUNDEDNESS OF FIO’S

Here, since 2

∇

∞

< 1 + 2

∇

∞

|y|, we have

(3.15)

|∇

θ(x, ξ)

− y| ≥ |y| − ∇

∞

|y|

Also,

|∂

(θ(x, ξ)

−y, ξ)| ≤ C

for all

|α| ≥ 2 uniformly in x and y. Therefore using

the non-stationary phase estimate in [H1] Theorem 7.7.1 to (3.14), the derivative
estimates on a(x, ξ/h) and (3.15) yield for N > 0

(x, y)

| h

−n

≤N

sup

∂

(χa(x, ξ/h)

|∇

θ(x, ξ)

− y|

|α|−2N

−m−n+N

|y|

−2N

−m−n+N

−2N

where we have used the fact that

|y| > 1 in I. Hence taking N >

, Theorem 3.3

yields

(3.16)

−m−n+N

−N

|u(x − y)| dy h

−m−n+N

M u(x).

Estimate of II. Making a change of variables ξ

→ h

−

ξ we obtain

(x, y) = (2π)

−n

−

(θ(x,ξ)

−y,ξ)

χ(h

−+1

ξ) a

x, h

−

dξ

:= (2π)

−n

−

(ϕ(x,ξ)

−y,ξ)

x, ξ

dξ

where a

(x, ξ) is compactly supported in x, supported in ξ in the annulus

1
2

−1

≤

|ξ| ≤ 2h

−1

and is uniformly bounded together with all its derivatives in ξ, by h

−m

Here the assumption, rank ∂

ξξ

ϕ(x, ξ) = n

− 1 for all ξ, yields that ker ∂

ξξ

ϕ(x, ξ

) =

span

{ξ

} = span {e

}. Therefore by the deﬁnition of θ(x, ξ)

(3.17)

det ∂

θ(x, e

)

= 0.

The assumption that a has its ξ-support in a small conic neighborhood of e

implies

that if that support is suﬃciently small, then

(3.18)

| det ∂

θ(x, ξ)

| ≥ 0, (x, ξ) ∈ supp a

Finally, due to the restriction 1 + 2

∇

∞

≥ |y| and (3.13), one has

(3.19)

|∂

(θ(x, ξ)

− y, ξ)| ≤ C

for all

|α| ≥ 1 uniformly in x and y.

Hence θ(x, ξ)

− y

, ξ

and h

satisfy all the assumptions of the stationary

phase estimate in Lemma 3.5 with λ = h

−

and λ

= h

−1

, we obtain

−

(ϕ(x,ξ)

−y,ξ)

x, ξ

dξ

−m

−1

−1)(−1)

and using the fact that the integral in ξ

lies on a segment of size h

−1

, we get

(3.20)

(x, y)

| h

−n

−m

−1

−(1−)n

−m−

n+1

−(1−)n

This yields that

−m−

n+1

−(1−)n

|y|≤1+2∇

L∞

|u(x − y)| dy

−m−

n+1

−(1−)n

M u(x)

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

Now adding I and II, taking N > n large enough, using Lemma 1.8, the assumption
m <

−

n+1

+ n(

− 1) and Theorem 3.2, we will obtain the desired result.

Here we remark that the condition on the rank of the Hessian on the metric is

quite natural and is satisﬁed by phases like

x, ξ + |ξ| and x, ξ +

|ξ

− |ξ

where ξ = (ξ

, ξ

) (with an amplitude supported in

|ξ

| > |ξ

|), but if we put a

slightly stronger condition than the rank condition on the phase, then it turns out
that we would not only be able to extend the local result to a global one but also
lower the regularity assumption on the phase function to the sole assumption of
measurability and boundedness in the spatial variable x. Therefore the estimates
we provide below will aim to achieve this level of generality. Having the uniform
stationary phase above in our disposal we will proceed with the main high frequency
estimates, but before that let us ﬁx a notation.

Notation. Given an n

× n matrix M, we denote by det

−1

M the determinant

of the matrix P M P where P is the projection to the orthogonal complement of
kerM .

Theorem

3.9. Let a(x, ξ)

∈ L

∞

with m <

−

n+1

+ n(

− 1) and ∈ [0, 1].

Let the phase function ϕ(x, ξ) satisfy

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0. Furthermore,

suppose that ϕ(x, ξ)

− x, ξ ∈ L

∞

, then the associated Fourier integral operator

is bounded on L

, for 1 < p <

∞ and w ∈ A

Proof.

As before, the low frequency part of the Fourier integral operator is

treated using Proposition 3.6 part (b). For the high frequency part we follow the
same line of argument as in the proof of Theorem 3.8. More speciﬁcally at the
level of showing the estimate (3.13), the lack of compact support in x variable
lead us to use our assumption ϕ(x, ξ)

− x, ξ ∈ L

∞

instead, which yields that

∇

θ(x, ξ)

∞

1. Splitting the kernel of the Fourier integral operator into the

same pieces I and II as in the proof of Theorem 3.8. We estimate the I piece
exactly in the same way as before but for piece II we proceed as follows. First of
all, the assumption

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0 for all (x, ξ), yields in particular that

|det

−1

(∂

ξξ

(θ(x, ξ)

− y, ξ)| ≥ c > 0. Due to the restriction 1 + 2∇

∞

≥ |y|

and (3.13), one also has

|∂

(θ(x, ξ)

− y, ξ)| ≤ C

for all

|α| ≥ 1 uniformly in x and y, which yields that θ(x, ξ) − y, ξ ∈ L

∞

. This

means that all the assumptions in Lemma 3.5 are satisﬁed and therefore we get

−m−

n+1

−(1−)n

|y|≤1+2∇

L∞

|u(x − y)| dy h

−m−

n+1

−(1−)n

M u(x).

Now adding I and II, using Lemma 1.8 and the assumptions N > n, m <

−

n+1

− 1) and Theorem 3.2 all together yield the desired result.

3.4.1. Endpoint weighted boundedness of Fourier integral operators.

The Following interpolation lemma is the main tool in proving the endpoint weighted
boundedness of Fourier integral operators.

Lemma

3.10. Let 0

≤ ≤ 1, 1 < p < ∞ and m

< m

. Suppose that

3.4. WEIGHTED LOCAL AND GLOBAL L

BOUNDEDNESS OF FIO’S

(a) the Fourier integral operator T with amplitude a(x, ξ)

∈ L

∞

and the

phase ϕ(x, ξ) are bounded on L

for a ﬁxed w

∈ A

, and

(b) the Fourier integral operator T with amplitude a(x, ξ)

∈ L

∞

and the

same type of phases as in (a) are bounded on L

where the bounds depend only on a ﬁnite number of seminorms in Deﬁnition 1.2.
Then, for each m

∈ (m

, m

), operators with amplitudes in L

∞

are bounded on

p
w

, and

(3.21)

ν =

− m

Proof.

For a

∈ L

∞

we introduce a family of amplitudes a

(x, ξ) :=

a(x, ξ), where z

∈ Ω := {z ∈ C; m

− m ≤ Re z ≤ m

− m}. It is easy to see

that, for

|α| ≤ C

with C

large enough and z

∈ Ω,

|∂

(x, ξ)

| (1 + |Im z|)

Re z+m

−|α|

for some C

. We introduce the operator

u := w

m2−m−z

p(m2−m1)

,ϕ

−

m2−m−z

p(m2−m1)

First we consider the case of p

∈ [1, 2]. In this case, A

⊂ A

which in turn

implies that both w

1
p

and w

−

1
p

belong to L

p
loc

and therefore for z

∈ Ω, T

an analytic family of operators in the sense of Stein-Weiss [SW]. Now we claim
that for z

∈ C with Re z

= m

− m, the operator (1 + |Im z

−C

,ϕ

bounded on L

with bounds uniform in z

. Indeed the amplitude of this operator

is (1 +

|Im z

−C

(x, ξ) which belongs to L

∞

with constants uniform in z

Thus, the claim follows from assumption (a). Consequently, we have

p
L

= (1 +

|Im z

(1 + |Im z

−C

m2−m−z1
p(m2−m1)

,ϕ

−

m2−m−z1
p(m2−m1)

(1 + |Im z

−

m2−m−z1
p(m2−m1)

= (1 +

|Im z

p
L

where we have used the fact that

m2−m−z1

(m2−m1)

= w.

Similarly if z

∈ C with Re z

= m

− m, then

m2−m−z2

(m2−m1)

= 1 and the

amplitude of the operator (1 +

|Im z

−C

belongs to L

∞

with constants

uniform in z

. Assumption (b) therefore implies that

p
L

(1 + |Im z

p
L

Therefore the complex interpolation of operators in [BS] implies that for z = 0 we
have

p
L

m2−m

p(m2−m1)

a,ϕ

−

m2−m

p(m2−m1)

≤ Cu

p
L

Hence, setting v = w

−

m2−m

p(m2−m1)

u this reads

(3.22)

a,ϕ

p
L

p
wν

≤ Cu

p
L

p
wν

where ν = (m

− m)/(m

− m

). This ends the proof in the case 1

≤ p ≤ 2. At

this point we recall the fact that if a linear operator T is bounded on L

, then

its adjoint T

∗

is bounded on L

−p

. Therefore, in the case p > 2, we apply the

above proof to T

∗

a,ϕ

, with p

∈ [1, 2) and v = w

−p

, which yields that T

∗

a,ϕ

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

bounded on L

and since w

∈ A

, we have v

∈ A

and so T

is bounded on

p
v

−p)ν

= L

p
w

−p)(1−p)ν

= L

p
w

, which concludes the proof of the theorem.

Now we are ready to prove our main result concerning weighted boundedness

of Fourier integral operators. This is done by combining our previous results with
a method based on the properties of the A

weights and complex interpolation.

Theorem

3.11. Let a(x, ξ)

∈ L

∞

−

n+1

+n(

−1)

and

∈ [0, 1]. Suppose that

either

(a) a(x, ξ) is compactly supported in the x variable and the phase function

ϕ(x, ξ)

∈ C

∞

× R

\ 0), is positively homogeneous of degree 1 in ξ

and satisﬁes, det ∂

xξ

ϕ(x, ξ)

= 0 as well as rank ∂

ξξ

ϕ(x, ξ) = n

− 1; or

(b) ϕ(x, ξ)

− x, ξ ∈ L

∞

, ϕ satisﬁes the rough non-degeneracy condition

as well as

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0.

Then the operator T

a,ϕ

is bounded on L

for p

∈ (1, ∞) and all w ∈ A

. Further-

more, for = 1 this result is sharp.

Proof.

The sharpness of this result for = 1 is already contained in Coun-

terexample 2 discussed above. The key issue in the proof is that both assump-
tions in the statement of the theorem guarantee the weighted boundedness for
m <

−

n+1

+ n(

− 1). The rest of the argument is rather abstract and goes as

follows. By the extrapolation Theorem 3.4, it is enough to show the boundedness
of T

a,ϕ

in weighted L

spaces with weights in the class A

. Let us ﬁx m

such

that

−

n+1

+ n(

− 1) < m

(

− 1). By Theorem 3.1, given w ∈ A

choose

ε such that w

1+ε

∈ A

. For this ε take m

−

n+1

+ n(

− 1) in such a way

that the straight line L that joins points with coordinates (m

, 1 + ε) and (m

, 0),

intersects the line x =

−

n+1

+ n(

− 1) in the (x, y) plane in a point with coordi-

nates (

−

n+1

+ n(

− 1), 1). Clearly this procedure is possible due to the fact that

we can choose the point m

on the negative x axis as close as we like to the point

−

n+1

+ n(

− 1). By Theorem 3.8, given ϕ(x, ξ) ∈ C

∞

× R

\ 0), positively

homogeneous of degree 1 in ξ, satisfying rank ∂

ξξ

ϕ(x, ξ) = n

− 1, and a ∈ L

∞

the Fourier integral operators T

a,ϕ

are bounded operators on L

2
w

1+ε

for w

∈ A

and, by Theorem 2.2, or rather its proof, the Fourier integral operators with am-
plitudes in L

∞

compactly supported in the spatial variable, and phases that

are positively homogeneous of degree 1 in the frequency variable and satisfying the
non-degeneracy condition det ∂

xξ

ϕ(x, ξ)

= 0, are bounded on L

. Therefore, by

Lemma 3.10, the Fourier integral operators T

a,ϕ

with phases and amplitudes as

in the statement of the theorem are bounded operators on L

. The proof of part

(b) is similar and uses instead the interpolation between the weighted boundedness
of Theorem 3.9 and the unweighted L

boundedness result of Theorem 2.8. The

details are left to the interested reader.

If we don’t insist on proving weighted boundedness results valid for all A

weights then, it is possible to improve on the number of derivatives in the estimates
and push the numerology almost all the way to those that guaranty unweighted L

boundedness. Therefore, there is the trade-oﬀ between the generality of weights
and loss of derivatives as will be discussed below.

Theorem

3.12. Let

C ⊂ (R

×R

\0)×(R

×R

\0), be a conic manifold which

is locally a canonical graph, see e.g. [H3] for the deﬁnitions. Let π :

C → R

× R

3.4. WEIGHTED LOCAL AND GLOBAL L

BOUNDEDNESS OF FIO’S

denote the natural projection. Suppose that for every λ

= (x

, ξ

, y

, η

)

∈ C there

is a conic neighborhood U

⊂ C of λ

and a smooth map π

C ∩ U

→ C,

homogeneous of degree 0, with rank dπ

= 2n

− 1, such that

π = π

◦ π

Let T

∈ I

,comp

× R

;

C) (see [H3]) with

1
2

≤ ≤ 1 and m < ( − n)|

1
p

−

1
2

Then for all w

∈ A

there exists α

∈ (0, 1) depending on m, , δ, p and [w]

such

that, for all ε

∈ [0, α], the Fourier integral operator T

a,ϕ

is bounded on L

p
w

where

1 < p <

∞.

Proof.

By the equivalence of phase function theorem, which for

1
2

≤ 1

is due to H¨

ormander [H3] and for =

1
2

is due to Greenleaf-Uhlmann [GU], we

reduce the study of operator T to that of a ﬁnite linear combination of operators
which in appropriate local coordinate systems have the form

(3.23)

u(x) = (2π)

−n

iϕ(x,ξ)

−iy,ξ

a(x, ξ) u(y)dy dξ,

where a

∈ S

−

with compact support in x variable, and ϕ homogeneous of degree

1 in ξ, with det ∂

xξ

ϕ(x, ξ)

= 0 and rank ∂

ξξ

ϕ(x, ξ) = n

−1. If m ≤ −

n+1

+n(

−1),

then Theorem 3.11 case (a) yields the result, so we assume that m >

−

n+1

+n(

−

1). Also by assumption of the theorem we can ﬁnd a m

, which we shall ﬁx from

now on, such that m < m

< (

− n)|

1
p

−

1
2

| and m

−

n+1

+ n(

− 1). Now

a result of Seeger-Sogge-Stein, namely Theorem 5.1 in [SSS] yields that operators
T

with amplitudes compactly supported in the x variable in the class S

−

and phase functions ϕ(x, ξ) satisfying rank ∂

ξξ

ϕ(x, ξ) = n

− 1 are bounded on L

Furthermore by Theorem 3.11 case (a), the operators T

with a

∈ S

,δ

are bounded

on L

, p

∈ (1, ∞). Therefore, Lemma 3.10 yields the desired result.

A similar result also holds for operators with amplitudes in S

,δ

with without

any rank condition on the phase function.

Theorem

3.13. Let a(x, ξ)

∈ S

,δ

, ϕ(x, ξ) be a strongly non-degenerate phase

function with ϕ(x, ξ)

− x, ξ ∈ Φ

, and λ := min(0, n(

− δ)), with either of the

following ranges of parameters:

(1) 0

≤ ≤ 1, 0 ≤ δ ≤ 1, 1 ≤ p ≤ 2 and

m < n(

− 1)

− 1

−

+ λ

−

;

(2) 0

≤ ≤ 1, 0 ≤ δ ≤ 1, 2 ≤ p ≤ ∞ and

m < n(

− 1)

−

+ (n

− 1)

−

Then for all w

∈ A

there exists α

∈ (0, 1) depending on m, , δ, p and [w]

such

that, for all ε

∈ [0, α], the Fourier integral operator T

a,ϕ

is bounded on L

p
w

where

1 < p <

∞.

Proof.

The proof is similar to that of Theorem 3.12 and we only consider the

proof in case (1), since the other case is similar. We observe that since Φ

⊂ Φ

and

x, ξ ∈ Φ

, the assumption ϕ(x, ξ)

− x, ξ ∈ Φ

, implies that ϕ(x, ξ)

∈ Φ

To proceed with the proof we can assume that m >

−n because otherwise by

3. GLOBAL AND LOCAL WEIGHTED BOUNDEDNESS OF FIO’S

Proposition 3.7 there is nothing to prove. The assumption of the theorem, enables
us to ﬁnd m

such that

m < m

< n(

− 1)

− 1

−

+ λ

−

and m

−n. Now Theorem 2.17 yields that operators T

with amplitudes in the

class S

,δ

, and strongly non-degenerate phase functions ϕ(x, ξ)

∈ Φ

are bounded

on L

. Furthermore Proposition 3.7 yields that operators T

with b

∈ S

,δ

are

bounded on L

. Therefore, Lemma 3.10 yields once again the desired result for the

range 1 < p

≤ 2.

Finally, for operators with non-smooth amplitudes we can prove the following:

Theorem

3.14. Let a(x, ξ)

∈ L

∞

, 0

≤ ≤ 1, and let ϕ(x, ξ) − x, ξ ∈ Φ

with a strongly non-degenerate ϕ and either of the following ranges of parameters:

(a) 1

≤ p ≤ 2 and

m <

(

− 1) +

− 1

−

;

(b) 2

≤ p ≤ ∞ and

m <

(

− 1) + (n − 1)

−

Then for all w

∈ A

there exists α

∈ (0, 1) depending on m, , p and [w]

such

that, for all ε

∈ [0, α], the Fourier integral operator T

a,ϕ

is bounded on L

p
w

Proof.

The proof is a modiﬁcation of that of Theorem 3.13, where one also

uses Theorem 2.18. The straightforward modiﬁcations are left to the interested
reader.

Here we remark that if in the proofs of Theorems 3.13 and 3.14 we would have

used Theorem 3.11 case (b) instead of Proposition 3.7 in the proof above, then
we would obtain a similar result, under the condition

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0

on the phase, but with an improved α as compared to those in the statements of
Theorems 3.13 and 3.14.

CHAPTER 4

Applications in Harmonic Analysis and Partial

Diﬀerential Equations

In this chapter, we use our weighted estimates proved in the previous chap-

ter to show the boundedness of constant coeﬃcient Fourier integral operators in
weighted Triebel-Lizorkin spaces. This is done using vector-valued inequalities for
the aforementioned operators. We proceed by establishing weighted and unweighted
L

boundedness of commutators of a wide class of Fourier integral operators with

functions of bounded mean oscillation (BMO), where in some cases we also show
the weighted boundedness of iterated commutators. The boundedness of commu-
tators are proven using the weighted estimates of the previous chapter and a rather
abstract complex analytic method. Finally in the last section, we prove global un-
weighted and local weighted estimates for the solutions of the Cauchy problem for
m-th and second order hyperbolic partial diﬀerential equations on R

4.1. Estimates in weighted Triebel-Lizorkin spaces

In this section, we investigate the problem of the boundedness of certain classes

on Fourier integral operators in weighted Triebel-Lizorkin spaces. The result ob-
tained here can be viewed as an example of the application of weighted norm
inequalities for FIO’s. The main reference for this section is [GR] and we will
refer the reader to that monograph for the proofs of the statements concerning
vector-valued inequalities.

Definition

4.1. An operator T deﬁned in L

(μ) ( this denotes L

spaces with

measure dμ) is called linearizable if there exits a linear operator U deﬁned on L

(μ)

whose values are Banach space-valued functions such that

(4.1)

|T u(x)| = Uu(x)

, u

∈ L

(μ)

We shall use the following theorem due to Garcia-Cuerva and Rubio de Francia,

whose proof can be found in [GR].

Theorem

4.2. Let T

be a sequence of linearizable operators and suppose that

for some ﬁxed r > 1 and all weights w

∈ A

(4.2)

u(x)

w(x) dx

≤ C

(w)

|u(x)|

w(x) dx,

with C

(w) depending on the weight w. Then for 1 < p, q <

∞ and w ∈ A

one

has the following weighted vector-valued inequality

(4.3)

1
q

≤ C

p,q

(w)

1
q

4. APPLICATIONS IN HARMONIC ANALYSIS AND PDE’S

Next we recall the deﬁnition of the weighted Triebel-Lizorkin spaces, see e.g.

[T].

Definition

4.3. Start with a partition of unity

∞
j=0

(ξ) = 1, where ψ

(ξ)

is supported in

|ξ| ≤ 2, ψ

(ξ) for j

≥ 1 is supported in 2

−1

≤ |ξ| ≤ 2

j+1

and

|∂

(ξ)

| ≤ C

−j|α|

, for j

≥ 1. Given s ∈ R, 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, and w ∈ A

a tempered distribution u belongs to the weighted Triebel-Lizorkin space F

s,p

q, w

(4.4)

s,p

q, w

∞

j=0

(D)u

1
q

∞.

From this it follows that for a linear operator T the estimate

(4.5)

T u

s,p

q, w

s,p

q, w

is implied by

(4.6)

∞

j=0

(D)T u

1
q

∞

j=0

(D)u

1
q

Now if one is in the situation where [T, ψ

] = 0, then (4.6) is equivalent to

(4.7)

∞

j=0

j(s

−s)

T (2

(D)u)

1
q

∞

j=0

(D)u

1
q

Therefore, setting T

:= 2

j(s

−s)

T and u

:= 2

u and assuming that s

≥ s

, (4.7)

has the same form as the the vector-valued inequality (4.3) and follows from (4.2).
Using these facts yields the following result,

Theorem

4.4. Let a(ξ)

∈ S

−

n+1

1,0

and ϕ

∈ Φ

with

|det

−1

∂

ξξ

ϕ(ξ)

| ≥ c > 0.

Then for s

≥ s

, 1 < p <

∞, 1 < q < ∞, and w ∈ A

, the Fourier integral operator

T u(x) =

(2π)

iϕ(ξ)+i

x,ξ

a(ξ)ˆ

u(ξ) dξ

satisﬁes the estimate

(4.8)

T u

s ,p

q, w

s,p

q, w

Proof.

We only need to check that T

= 2

j(s

−s)

T satisﬁes (4.2). But this

follows from the assumption s

≥ s

and Theorem 3.11 part (b) concerning the global

weighted boundedness of Fourier integral operators.

4.2. Commutators with BMO functions

In this section we show how our weighted norm inequalities can be used to derive

the L

boundedness of commutators of functions of bounded mean oscillation with

a wide range of pseudodiﬀerential operators. We start with the precise deﬁnition
of a function of bounded mean oscillation.

Definition

4.5. A locally integrable function b is of bounded mean oscillation

(4.9)

BMO

:= sup

|B|

|b(x) − b

| dx < ∞,

4.2. COMMUTATORS WITH BMO FUNCTIONS

where the supremum is taken over all balls in R

. We denote the set of such

functions by BMO.

For b

∈ BMO it is well-known that for any γ <

, there exits a constant C

n,γ

so that for u

∈ BMO and all balls B,

(4.10)

|B|

|b(x)−b

|/u

BMO

≤ C

n,γ

For this see [G, p. 528]. The following abstract lemma will enable us to prove the
L

boundedness of the BMO commutators of Fourier integral operators.

Lemma

4.6. For 1 < p <

∞, let T be a linear operator which is bounded on

p
w

for all w

∈ A

for some ﬁxed ε

∈ (0, 1]. Then given a function b ∈ BMO, if

Ψ(z) :=

zb(x)

T (e

−zb(x)

u)(x)v(x) dx is holomorphic in a disc

|z| < λ, then the

commutator [b, T ] is bounded on L

Proof.

Without loss of generality we can assume that

BMO

= 1. We take

u and v in C

∞

with

≤ 1 and v

≤ 1, and an application of H¨older’s

inequality to the holomorphic function Ψ(z) together with the assumption on v
yield

|Ψ(z)|

≤

p Re zb(x)

|T (e

− z b(x)

dx.

Our ﬁrst goal is to show that the function Ψ(z) deﬁned above is bounded on a
disc with centre at the origin and suﬃciently small radius. At this point we recall a
lemma due to Chanillo [Chan] which states that if

BMO

= 1, then for 2 < s <

∞,

there is an r

depending on s such that for all r

∈ [−r

, r

], e

rb(x)

∈ A

s
2

Taking s = 2p in Chanillo’s lemma, we see that there is some r

depending on p

such that for

|r| < r

, e

rb(x)

∈ A

. Then we claim that if R := min (λ,

εr

) and

|z| < R then |Ψ(z)| 1. Indeed since R <

εr

we have

|Re z| <

εr

and therefore

p Re z

| < r

. Therefore Chanillo’s lemma yields that for

|z| < R, w := e

p Re z

b(x)

∈

and since e

p Re zb(x)

= w

, the assumption of weighted boundedness of T and

the L

bound on u, imply that for

|z| < R

|Ψ(z)|

≤

p Re zb(x)

|T (e

− z b(x)

|T (e

− z b(x)

dx =

−ε

|u|

and therefore

|Ψ(z)| 1 for |z| < R. Finally, using the holomorphicity of Ψ(z) in

the disc

|z| < R, Cauchy’s integral formula applied to the circle |z| = R

< R, and

the estimate

|Ψ(z)| 1, we conclude that

|Ψ

(0)

| ≤

2π

|z|=R

|Ψ(ζ)|

|ζ

|dζ| 1.

By construction of Ψ(z), we actually have that Ψ

(0) =

v(x)[b, T ]u(x) dx and the

deﬁnition of the L

norm of the operator [b, T ] together with the assumptions on u

and v yield at once that [f, T ] is a bounded operator from L

to itself for p.

4. APPLICATIONS IN HARMONIC ANALYSIS AND PDE’S

The following lemma guarantees the holomorphicity of

Ψ(z) :=

zb(x)

a,ϕ

−zb(x)

u)(x)v(x) dx,

when T

a,ϕ

is a L

bounded Fourier integral operator.

Lemma

4.7. Assume that ϕ is a strongly non-degenerate phase function in the

class Φ

and suppose that either:

(a) T

a,ϕ

is a Fourier integral operator with a

∈ S

,δ

, 0

≤ ≤ 1, 0 ≤ δ < 1,

m = min(0,

(

− δ)) or

(b) T

a,ϕ

is a Fourier integral operator with a

∈ L

∞

, 0

≤ ≤ 1, m <

(

− 1).

Then given b

∈ BMO with b

BMO

= 1 and u and v in C

∞

, there exists λ > 0

such that the function Ψ(z) :=

zb(x)

a,ϕ

−zb(x)

u)(x)v(x) dx is holomorphic in

the disc

|z| < λ.

Proof.

(a) From the explicit representation of Ψ(z)

(4.11)

Ψ(z) =

a(x, ξ) e

iϕ(x,ξ)

−iy,ξ

zb(x)

−zb(y)

v(x) u(y) dy dξ dx

we can without loss of generality assume that a(x, ξ) has compact x

−support. For

∈ S and ε ∈ (0, 1) we take χ(ξ) ∈ C

∞

) such that χ(0) = 1 and set

(4.12)

,ϕ

f (x) =

a(x, ξ) χ(εξ) e

iϕ(x,ξ)

f (ξ) dξ.

Using this and the assumption of the compact x

−support of the amplitude, one

can see that for f

∈ S , lim

→0

,ϕ

f = T

a,ϕ

f in the Schwartz class

S and also

lim

→0

,ϕ

− T

a,ϕ

= 0. Since a(x, ξ) χ(εξ)

∈ S

,δ

with seminorms that are

independent of ε, it follows from our assumptions on the amplitude and the phase
and Theorem 2.7 that

,ϕ

with a L

bound that is independent of

ε. Therefore, the density of

S in L

yields

(4.13)

lim

→0

,ϕ

− T

a,ϕ

= 0,

for all f

∈ L

. Now if we deﬁne

(z) :=

zb(x)

,ϕ

−zb(x)

u)(x)v(x) dx

(4.14)

a(x, ξ) χ(εξ) e

iϕ(x,ξ)

−iy,ξ

zb(x)

−zb(y)

v(x) u(y) dy dξ dx,

then the integrand in the last expression is a holomorphic function of z. Further-
more, from (4.10) and the assumption

BMO

= 1 one can deduce that for all

∈ [1, ∞) and |z| <

γ
p

, and all compact sets K one has

(4.15)

±p Re z b(x)

≤ C

(K).

This fact shows that ue

−z b

and ve

z b

both belong to L

for all p

∈ [1, ∞) provided

|z| <

γ
p

. These together with the compact support in ξ of the integrand deﬁning

(z) and uniform bounds on the amplitude in x, yield the absolute convergence

4.2. COMMUTATORS WITH BMO FUNCTIONS

of the integral in (4.14) and therefore Ψ

(z) is a holomorphic function in

|z| < 1.

Now we claim that for γ as in (4.10),

lim

→0

sup

|z|<

|Ψ

(z)

− Ψ(z)| = 0.

Indeed, since

1
2

, one has for

|z| <

|Ψ

(z)

− Ψ(z)| =

v(x)e

zb(x)

,ϕ

− T

a,ϕ

−zb

u)(x) dx

≤ v e

,ϕ

− T

a,ϕ

](u e

−zb

)

≤ v

∞

supp v

2Re zb(x)

1
2

,ϕ

− T

a,ϕ

](u e

−zb

)

Therefore, using (4.15) with p = 2 and (4.13) yield that

lim

→0

sup

|z|<

|Ψ

(z)

− Ψ(z)| = 0

and hence Ψ(z) is a holomorphic function in

|z| < λ with λ ∈ (0,

(b) Using the semiclassical reduction in the proof of Theorem 2.2, we decompose

the operator T

a,ϕ

into low and high frequency parts, T

and T

. From this it follows

that Ψ

(z) :=

zb(x)

−zb(x)

u)(x)v(x) dx can be written as

(4.16)

(z) =

iϕ(x,ξ)

−iy,ξ

(ξ) a(x, ξ) u(y) e

−zb(y)

dy dξ

v(x) e

zb(x)

dx,

and Ψ

(z) :=

zb(x)

−zb(x)

u)(x)v(x) dx is given by

(4.17)

(z) =

−n

ϕ(x,ξ)

−

y,ξ

χ(ξ) a(x, ξ/h) u(y) e

−zb(y)

dy dξ

v(x) e

zb(x)

dx.

Now we claim that for Ψ

(z) and Ψ

(z) are holomorphic in

|z| < 1. To see this, we

reason in a way similar to the proof of part (a). Namely, using the compact support
in ξ of the integrands of (4.16) and (4.17) and uniform bounds on the amplitude in
x, yield the absolute convergence of the integrals in (4.16) and (4.17) and therefore
Ψ

(z) and Ψ

(z) are holomorphic functions in

|z| < 1. Next we proceed with a

uniform estimate (in z) for Ψ

(z). For this we use once again that ue

−z b

and ve

z b

both belong to L

provided

|z| <

. Therefore the Cauchy-Schwartz inequality and

(2.7) yield

|Ψ

(z)

| =

v(x)e

zb(x)

−zb

u)(x) dx

(4.18)

≤ u e

−zb

∗

(v e

)

≤ u e

−zb

∗

(v e

)

1/2
L

v e

1/2
L

≤ h

−m−(1−)M/2

u e

−zb

v e

−m−(1−)M/2

Hence,

|Ψ

(z)

| h

−m−(1−)M/2

and setting h = 2

−j

, using m <

(

− 1) and

summing in j we would have a uniformly convergent series of holomorphic functions
which therefore converges to a holomorphic function and by taking a λ in the
interval (0,

) we conclude the holomorphicity of Ψ(z) in

|z| < λ.

4. APPLICATIONS IN HARMONIC ANALYSIS AND PDE’S

Lemmas 4.6 and 4.7 yield our main result concerning commutators with BMO

functions, namely

Theorem

4.8. Suppose either

(a) T

∈ I

,comp

×R

;

C) with

1
2

≤ ≤ 1 and m < (−n)|

1
p

−

1
2

|, satisﬁes

all the conditions of Theorem 3.12 or;

(b) T

a,ϕ

with a

∈ S

,δ

, 0

≤ ≤ 1, 0 ≤ δ ≤ 1, λ = min(0, n(−δ)) and ϕ(x, ξ)

is a strongly non-degenerate phase function with ϕ(x, ξ)

− x, ξ ∈ Φ

where in the range 1 < p

≤ 2,

m < n(

− 1)

− 1

−

+ λ

−

;

and in the range 2

≤ p < ∞

m < n(

− 1)

−

+ (n

− 1)

−

;

a,ϕ

with a

∈ L

∞

, 0

≤ ≤ 1 and ϕ is a strongly non-degenerate phase

function with ϕ(x, ξ)

− x, ξ ∈ Φ

, where in the range 1 < p

≤ 2,

m <

(

− 1) +

− 1

−

and for the range 2

≤ p < ∞

m <

(

− 1) + (n − 1)

−

Then for b

∈ BMO, the commutators [b, T ] and [b, T

a,ϕ

] are bounded on L

with

1 < p <

∞.

Proof.

(a) One reduces T to a ﬁnite sum of operators of the form T

as in

the proof of Theorem 3.12. That theorem also yields the existence of an ε

∈ (0, 1)

such that T

with a

∈ S

−

and m < (

− n)|

1
p

−

1
2

| is L

p
w

−bounded. Moreover,

since m < (

− n)|

1
p

−

1
2

| ≤ 0, and 1 − ≤ , Theorem 2.7 yields that T

is L

bounded. Hence, if

Ψ(z) =

zb(x)

−zb(x)

u)(x)v(x) dx,

with u and v in C

∞

, then Lemma 4.7 yields that Ψ(z) is holomorphic in a neigh-

bourhood of the origin. Therefore, Lemma 4.6 implies that the commutator [b, T

]

is bounded on L

and the linearity of the commutator in T allows us to conclude

the same result for a ﬁnite linear combinations of operators of the same type as T

This ends the proof of part (a).

(b) The proof of this part is similar to that of part (a). Here we observe that

for any ranges of p in the statement of the theorem, the order of the amplitude
m < min(0,

(

−δ)) and so T

a,ϕ

is L

bounded. Now, application of 3.13, Theorem

2.7 and Lemma 4.7 part (a), concludes the proof.

the order of the amplitude m <

(

− 1) and so T

a,ϕ

is L

bounded. Therefore,

Theorem 3.14, Theorem 2.2 and Lemma 4.7 part (b), yield the desired result.

4.3. APPLICATIONS TO HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Finally, the weighted norm inequalities with weights in all A

classes have the

advantage of implying weighted boundedness of repeated commutators. Namely,
one has

Theorem

4.9. Let a(x, ξ)

∈ L

∞

−

n+1

+n(

−1)

and

∈ [0, 1]. Suppose that

either

(a) a(x, ξ) is compactly supported in the x variable and the phase function

ϕ(x, ξ)

∈ C

∞

× R

\ 0), is positively homogeneous of degree 1 in ξ

and satisﬁes, det ∂

xξ

ϕ(x, ξ)

= 0 as well as rank ∂

ξξ

ϕ(x, ξ) = n

− 1; or

(b) ϕ(x, ξ)

− x, ξ ∈ L

∞

, ϕ satisﬁes either the strong or the rough non-

degeneracy condition (depending on whether the phase is spatially smooth
or not), as well as

|det

−1

∂

ξξ

ϕ(x, ξ)

| ≥ c > 0.

Then, for b

∈ BMO and k a positive integer, the k-th commutator deﬁned by

a,b,k

u(x) := T

(b(x)

− b(·))

(x)

is bounded on L

for each w

∈ A

and p

∈ (1, ∞).

Proof.

The claims in (a) and (b) are direct consequences of Theorem 3.11

and Theorem 2.13 in [ABKP].

4.3. Applications to hyperbolic partial diﬀerential equations

It is wellknown, see e.g. [D], that the Cauchy problem for a strictly hyperbolic

partial diﬀerential equation

(4.19)

m
j=1

(x, t, D

−j

, t

= 0

∂

t=0

= f

(x), 0

≤ j ≤ m − 1

can be solved locally in time and modulo smoothing operators by

(4.20)

u(x, t) =

−1

j=0

k=1

iϕ

(x,ξ,t)

(x, ξ, t)

(ξ) dξ,

where a

(x, ξ, t) are suitably chosen amplitudes depending smoothly on t and

belonging to S

−j

1,0

, and the phases ϕ

(x, ξ, t) also depend smoothly on t, are strongly

non-degenerate and belong to the class Φ

. This yields the following:

Theorem

4.10. Let u(x, t) be the solution of the hyperbolic Cauchy problem

(4.19) with initial data f

. Let m

= (n

− 1)|

1
p

−

1
2

|, for a given p ∈ [1, ∞]. If

∈ H

s+m

−j,p

) and T

∈ (0, ∞) is ﬁxed, then for any ε > 0 the solution

·, t) ∈ H

−ε,p

), satisﬁes the global estimate

(4.21)

u(·, t)

−ε,p

≤ C

−1

j=0

s+mp−j,p

, t

∈ [−T, T ], p ∈ [1, ∞],

and in fact for p = 2, we have the improved estimate

(4.22)

u(·, t)

≤ C

−1

j=0

−j

, t

∈ [−T, T ].

Proof.

The results follow at once from the Fourier integral operator represen-

tation (4.20), Corollary 2.19 and 2.7.

4. APPLICATIONS IN HARMONIC ANALYSIS AND PDE’S

The representation formula (4.20) also yields the following local weighted es-

timate for the solution of the Cauchy problem for the second order hyperbolic
equation above and in particular for variable coeﬃcient wave equation. In this
connection we recall that H

{u ∈ S

; (1

− Δ)

s
2

∈ L

, w

∈ A

Theorem

4.11. Let u(x, t) be the solution of the hyperbolic Cauchy problem

(4.19) with m = 2 and initial data f

. For p

∈ (1, ∞), if f

∈ H

n+1

−j,p

) with

∈ A

, and if T

∈ (0, ∞) is small enough, then the solution u(·, t) is in H

s,p

)

and satisﬁes the weighted estimate

(4.23)

χu(·, t)

s,p

≤ C

j=0

n+1

−j,p

, t

∈ [−T, T ] \ {0}, ∀w ∈ A

and all χ

∈ C

∞

Proof.

In the case when m = 2 then one has the important property that

rank ∂

ξξ

ϕ(x, ξ, t) = n

− 1,

for t

∈ [−T, T ] \ {0} and small T. This fact and the localization of the solution

u(x, t) is the spatial variable x, enable us to use Theorem 3.11 in the case = 1,
from which the theorem follows.

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1054 Kening Lu, Qiudong Wang, and Lai-Sang Young, Strange Attractors for

Periodically Forced Parabolic Equations, 2013

1053 Alexander M. Blokh, Robbert J. Fokkink, John C. Mayer, Lex G.

Oversteegen, and E. D. Tymchatyn, Fixed Point Theorems for Plane Continua with
Applications, 2013

1052 J.-B. Bru and W. de Siqueira Pedra, Non-cooperative Equilibria of Fermi Systems

with Long Range Interactions, 2013

1051 Ariel Barton, Elliptic Partial Diﬀerential Equations with Almost-Real Coeﬃcients, 2013

1050 Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono, The Poset

of k-Shapes and Branching Rules for k-Schur Functions, 2013

1049 David I. Stewart, The Reductive Subgroups of F

, 2013

1048 Andrzej Nag´

orko, Characterization and Topological Rigidity of N¨

obeling Manifolds,

2013

1047 Joachim Krieger and Jacob Sterbenz, Global Regularity for the Yang-Mills

Equations on High Dimensional Minkowski Space, 2013

1046 Keith A. Kearnes and Emil W. Kiss, The Shape of Congruence Lattices, 2013

1045 David Cox, Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, A Study of

Singularities on Rational Curves Via Syzygies, 2013

1044 Steven N. Evans, David Steinsaltz, and Kenneth W. Wachter, A

Mutation-Selection Model with Recombination for General Genotypes, 2013

For a complete list of titles in this series, visit the

AMS Bookstore at www.ams.org/bookstore/memoseries/.

ISBN 978-0-8218-9119-3

9 780821 891193

MEMO/229/1074

Document Outline

Introduction
Chapter 1. Prolegomena
Chapter 2. Global Boundedness of Fourier Integral Operators
Chapter 3. Global and Local Weighted 𝐿^{𝑝} Boundedness of Fourier Integral Operators
Chapter 4. Applications in Harmonic Analysis and Partial Differential Equations
Bibliography

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