S. V. Astashkin- A Generalized Khintchine Inequality in Rearrangement Invariant Spaces

8/3/2019 S. V. Astashkin- A Generalized Khintchine Inequality in Rearrangement Invariant Spaces

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144 00162663/08/42020144 c2008 Springer Science+Business Media, Inc.

Functional Analysis and Its Applications, Vol. 42, No. 2, pp. 144147, 2008

Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 42, No. 2, pp. 7881, 2008

Original Russian Text Copyright c by S. V. Astashkin

A Generalized Khintchine Inequalityin Rearrangement Invariant Spaces

S. V. Astashkin

Received October 25, 2006

Abstract. Let X be a separable or maximal rearrangement invariant space on [0, 1]. Necessaryand sufficient conditions are found under which the generalized Khintchine inequality

k=1

fk

X

C

k=1

f2k

1/2X

holds for an arbitrary sequence {fk}

k=1 X of mean zero independent variables. Moreover, thesubspace spanned in a rearrangement invariant space by the Rademacher system with independentvector coefficients is studied.

Key words: Khintchine inequality, rearrangement invariant space, Rademacher system, indepen-dent functions, Kruglov property, Boyd indices.

1. Introduction. Let rk(t) = sign sin 2kt (k = 1, 2, . . . ) be the Rademacher functions on the

interval [0, 1]. By the classical Khintchine inequality [1], for every p > 0 there exists a constantCp > 0 such that

k=1 akrkLp Cp(

k=1 a2k)

1/2 for arbitrary real ak , k = 1, 2, . . . .This relation has led to a number of studies and generalizations and found numerous applications

in various fields of analysis [2]. In particular, Rodin and Semenov [3] proved in 1975 that theinequality

k=1

akrk

X

C

k=1

a2k

1/2(1)

holds for a rearrangement invariant space X on [0, 1] if and only if X contains the separable partG of the Orlicz space LN2 generated by the function N2(u) = e

u2 1. In this note, we solve asimilar problem in which scalar multiples of Rademacher functions are replaced by any mean zeroindependent functions. More precisely, we characterize rearrangement invariant spaces X for whichthere exists a constant C > 0 such that

k=1

fk

X

C

k=1

f2k

1/2X

(2)

for all sequences {fk}

k=1 X of independent functions satisfying the conditions1

0 fk(t) dt = 0

(k = 1, 2, . . . ) and (

k=1 f2k )

1/2 X. Note that such a characterization is known if, instead of theclass of sequences of mean zero independent functions, the larger class of all martingale differences

is considered: inequality (2) holds in X if and only if the lower Boyd index of X is positive, X > 0[4].

In our case, the central role is played by the so-called Kruglov property, which was introducedand intensively studied by Braverman [5]. The main result of this note (Theorem 1) provides adescription, in terms of this property, of separable or maximal rearrangement invariant spacesfor which the generalized Khintchine inequality (2) holds for arbitrary sequences of mean zeroindependent functions. The second result (Theorem 2) shows the same property to be useful whenstudying the subspace spanned in a rearrangement invariant space by the Rademacher system withindependent vector coefficients.


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2. Definitions and notation. A Banach space X of real-valued Lebesgue measurable functionson the interval [0, ) (0 < ) is called a rearrangement invariant space if the relations y Xand x(t) y(t) (t [0, )) imply that x X and xX yX . Here x

(t) is the nonincreasingright-continuous rearrangement of the function |x(s)|; i.e., x(t) = inf{ 0 : ({s [0, ) :|x(s)| > }) t}, t > 0 (where stands for the Lebesgue measure).

Given a rearrangement invariant space X on [0, ), the Kothe dual (or associate) space X

consists of all measurable functions y such that yX := sup{

0 x(t)y(t) dt : x X, xX

1} < . A rearrangement invariant space X is said to be maximal (or have the Fatou property) ifthe natural embedding ofX in the Kothe bidual X is a surjective isometry. This is equivalent to thefollowing property ofX: if xn X (n = 1, 2, . . . ), xn x a.e. on [0, ), and supn=1,2,... xnX < ,then x X and x lim infn xn. Note that all rearrangement invariant spaces importantin applications (Orlicz, Lorentz, and Marcinkiewicz spaces etc.) are separable or maximal. TheBoyd indices X and X of a rearrangement invariant space X are an important characteristic ofX; they are related to the dilation operator and always satisfy 0 X X 1 [6, Theorem2.4.4]. For any rearrangement invariant space X on [0, 1], one has the continuous embeddingsL[0, 1] X L1[0, 1] [6, Theorem 2.4.1].

If X is a rearrangement invariant space on [0, 1], then the space Z2X introduced in [7] consistsof all functions f (L1 + L)(0,) such that

fZ

2X :=

f[0,1]X +

f[1,

)L2[1,) 0 such that

k=1

fk

X

C

k=1

fk

Z2X

(3)

for any sequence {fk}

k=1 X of independent functions satisfying the conditions1

0 fk(t) dt = 0,

k = 1, 2, . . . , and

k=1 fk Z2X .

(c) There exists a constant C > 0 such that inequality (2) holds for any sequence {fk}

k=1 X

of independent functions satisfying the conditions1

0 fk(t) dt = 0, k = 1, 2, . . . , and (

k=1 f2k )

1/2 X.


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[9] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.[10] V. M. Kruglov, Teor. Veroyatnost. Primenen., 15:2 (1970), 331336.[11] S. V. Astashkin and M. Sh. Braverman, in: Operator Equations in Function Spaces, Voronezh

Gos. Univ., Voronezh, 1986, 310.[12] S. V. Astashkin and F. A. Sukochev, Mat. Zametki, 76:4 (2004), 483489; English transl.:

Math. Notes, 76:4 (2004), 449454.[13] S. V. Astashkin and F. A. Sukochev, Israel J. Math., 145 (2005), 125156.

Samara State University

e-mail: [email protected]

Translated by S. V. Astashkin

S. V. Astashkin- A Generalized Khintchine Inequality in Rearrangement Invariant Spaces

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