Sparse Rademacher chaos in symmetric spaces

Abstract

In this paper we study properties of series with respect to orthogonal systems {r i (t)r j (t)} i =j and {r i (s)r j (t)} ∞ i,j=1 in symmetric spaces on interval and square, respectively. Necessary and sufficient conditions for the equivalence of these systems with the canonical base in l 2 and also for the complementability of the corresponding generated subspaces, usually called Rademacher chaos, are derived. The results obtained allow, in particular, to establish the unimprovability of the exponential integrability of functions from Rademacher chaos. Besides, it is shown that for spaces that are "close" to L ∞ , the systems considered, in contrast to the ordinary Rademacher system, do not possess the property of unconditionality. The degree of this non-unconditionality is explained in the space L ∞ .

Rademacher Chaos in Symmetric Spaces

“…If now U n is the complement of the set from the last estimate, then µ(U n ) > 1 − 2(e/2) −dn and for all u ∈ U n we have (9). Thus, the claim is proved.…”

“…, and hence the right-hand side inequality in (iii) follows. Since X ⊂ L 1 , the left-hand side of this inequality is fulfilled in every symmetric space X (see also [9,Lemma 6]). Thus, the implication (iv)⇒ (iii) is proven.…”

“…The next step in the study of the behaviour of the Rademacher chaos in symmetric spaces was made by the authors of this paper by using the important concept of combinatorial dimension developed earlier by Blei (see [10,11,12,13,14]) . Namely, in [9], it was shown that the above results in [7] and [8] can be extended to a non-complete chaos…”

“…, are the Rademacher functions, numbered as usual by positive integers (see § 1). Observe that the system {r  } ∈△ d , considered in the lexicographic order of  ∈ △ d , is basic in any symmetric space X (see [9,Theorem 2]). However, in this paper, the numbering order of the system {r  } ∈△ d will do not matter.…”

“…..,j d )∈A can be characterized also, as well as in [6,7,9], in terms of continuous embeddings of certain exponential Orlicz spaces into L M (Theorem 3). Note that somewhat close results to Theorem 3 were obtained earlier by Blei and Ge [15,16].…”