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Disjointification of martingale differences and conditionally independent random variables with some applications

Sergey AstashkinFedor SukochevChin Pin Wong — 2011

Studia Mathematica

Disjointification inequalities are proven for arbitrary martingale difference sequences and conditionally independent random variables of the form f k ( s ) x k ( t ) k = 1 , where f k ’s are independent and xk’s are arbitrary random variables from a symmetric space X on [0,1]. The main results show that the form of these inequalities depends on which side of L₂ the space X lies on. The disjointification inequalities obtained allow us to compare norms of sums of martingale differences and non-negative random variables with...

Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. AstashkinLech Maligranda — 2015

Studia Mathematica

The structure of the closed linear span of the Rademacher functions in the Cesàro space C e s is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in C e s , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of C e s if 1 < p < ∞.

Interpolation of Cesàro sequence and function spaces

Sergey V. AstashkinLech Maligranda — 2013

Studia Mathematica

The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that C e s p ( I ) is an interpolation space between C e s p ( I ) and C e s p ( I ) for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, C e s p [ 0 , 1 ] is not an interpolation space between Ces₁[0,1] and C e s [ 0 , 1 ] .

Structure of Cesàro function spaces: a survey

Sergey V. AstashkinLech Maligranda — 2014

Banach Center Publications

Geometric structure of Cesàro function spaces C e s p ( I ) , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that C e s p [ 0 , 1 ] contains isomorphic and complemented copies of l q -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces C e s p [ 0 , 1 ] .

Rademacher functions in BMO

Sergey V. AstashkinMikhail LeibovLech Maligranda — 2011

Studia Mathematica

The Rademacher sums are investigated in the BMO space on [0,1]. They span an uncomplemented subspace, in contrast to the dyadic B M O d space on [0,1], where they span a complemented subspace isomorphic to l₂. Moreover, structural properties of infinite-dimensional closed subspaces of the span of the Rademacher functions in BMO are studied and an analog of the Kadec-Pełczyński type alternative with l₂ and c₀ spaces is proved.

New examples of K-monotone weighted Banach couples

Some new examples of K-monotone couples of the type (X,X(w)), where X is a symmetric space on [0,1] and w is a weight on [0,1], are presented. Based on the property of w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X,X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is t 1 / p for some p ∈ [1,∞], then X = L p . At...

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