5441337180

Now, let us change the variables according to the mapping G: (M₊)ⁿ —> (M₊)ⁿ, G(xx_n) =    .....x²n).

We obtain

v_n (F^_1(A)) = f e^-^"⁼¹®^łda;.

Jg(A)

Since G(A) is an ideał iff so is A, we infer that for any unconditional ideał

ifclⁿ

X_n(K) = v_n(K), where K := G~^lF~^l (K n (M+)ⁿ) •

Moreover, for a strip P = {x € Mⁿ | |xi| < p}, the set P C Cⁿ is a cylinder. Notę also that tK = y/tK. These observations combined with Theorem 1 yield the assertion.    □

Following the method of [LOl, Corollary 3] we obtain the result concerning the comparison of moments.

Corollary 2. Let || • || be a norm on Rⁿ which is unconditional, i.e.

B(ei*l,...,Cn*n)II = ||Ol,---,Z„)||,

for any Xj € M and ej G {—1,1}. Then for p > q > 0

{L ^M”^dXr

(x)

l/p

f IWI’dA„(:r)

J R"

1/9

(9)

where the constant

₍r(p+ !))»/»’

^M (AWdA_l(*))^1/ł    (^r(« ^{+ 1}»^1/?

is the best possible.

Proof. The proof hinges on the fact that a bali K = {x E Kⁿ ||x|| < t} with respect to the norm || • || is a closed convex unconditional set, so that Theorem 2 can be applied.    □

Acknowledgements

We would like to thank R. Adamczak for his remark regarding Lemma 1, which led to the present generał formulation. We also thank B. Maurey for pointing out the change of variables used in the proof of Theorem 2.

The work was done while the second named author was participating in The Kupcinet-Getz International Summer Science School at the Weizmann Institute of Science in Rehovot, Israel.

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