Rademacher Chaos in Symmetric Spaces

Abstract

We study density estimates of an index set A, under which unconditionality (or even a weaker property of the random unconditional divergence) of the corresponding Rademacher fractional chaos..,j d )∈A in a symmetric space X implies its equivalence in X to the canonical basis in ℓ 2 . In the special case of Orlicz spaces L M , unconditionality of this system is also equivalent to the fact that a certain exponential Orlicz space embeds into L M .

On unconditionality of fractional Rademacher chaos in symmetric spaces

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

“…Indeed, by (8) and in view of Bernstein's inequality (see for example, [35], Ch. 1, § 6, formula (42), or [2], Proposition 1.2), we have, for any t ∈ [0, 1] and λ > 0,…”

“…where ϕ X is the fundamental function of X. Using successively condition (4), embedding (7), the embedding L ∞ ⊂ X, estimate (10), and the inequality α 0 ⩽ δ, we obtain…”

“…Specifically, it was shown that this system is equivalent in X to the canonical basis in ℓ 2 if and only if X contains the separable part of the Orlicz space Exp L generated by the function N 1 (u) = e u − 1. Moreover, both these properties were found to be equivalent to the formally weaker (than the equivalence to the canonical basis in ℓ 2 ) property of unconditionality of the basic sequence {r j1 (t) • r j2 (t)} j1>j2 in X (see [8]). Note that the Rademacher system itself is an unconditional (and even symmetric with constant 1) basic sequence in any symmetric space (see for example, Proposition 2.2 in [2]).…”

“…A characterization of the symmetric spaces X in which the sequence {r j } ∞ j=1 is equivalent to the canonical basis in ℓ 2 was given by Rodin and Semenov in [6], who proved that this equivalence holds if and only if X contains the separable part of the Orlicz space Exp L 2 generated by the function N 2 (u) = e u 2 −1. In [7], a similar question was studied for the system {r j1 (t) • r j2 (t)} j1>j2 of products of Rademacher functions, which is usually called the second-order Rademacher chaos. Specifically, it was shown that this system is equivalent in X to the canonical basis in ℓ 2 if and only if X contains the separable part of the Orlicz space Exp L generated by the function N 1 (u) = e u − 1.…”