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Rademacher functions in symmetric spaces. (English. Russian original) Zbl 1229.46020
J. Math. Sci., New York 169, No. 6, 725-886 (2010); translation from Sovrem. Mat. Fundam. Napravl. 32, 3-161 (2009).
This is a memoir on the Rademacher functions \(r_n(t)\) on the interval \(I=(0,1)\), their 2-fold products (Rademacher chaos), their closed linear spans in symmetric (rearrangement invariant) Banach spaces, their multiplicator spaces, the equivalence in distribution of a sequence of random variables (r.v.) to \((r_n)\), and other related topics. In the memoir, results are presented in which the author has taken a part, and there are related results of other authors. The Rademacher functions are considered as elements of a symmetric function space on \(I\) as well as a sequence of independent Bernoulli r.v. A symmetric space \(X\) is a natural frame for the Rademacher system because it generates a symmetric sequence space in \(X\). The classical Khintchin inequality can be considered as a starting point of the memoir. The other starting point is V. A. Rodin’s and E. M. Semyonov’s paper [Analysis Math. 1, 207–222 (1975; Zbl 0315.46031)], where detailed investigations of the Rademacher system in symmetric spaces were performed first. Both interpolation theory and probabilistic methods are the main tools of proof in the memoir.
Let us present the contents of the memoir and some simply formulated results. Chapter 0 contains some necessary general definitions and results. In particular, the following notation is used: \(a=(a_n)\) is a sequence of real numbers, \(c\) is a positive constant; \(\kappa_a(p)\) is the Peetre \(\mathcal{K}\)-functional constructed on the Banach couple \((\ell_1,\ell_2)\). Let us denote \(R_a(t)=\sum_na_nr_n(t)\). The subspace \(\mathcal{R}(X)\) consists of all functions \(x\in X\), represented as a.e. convergent series \(x(t)=R_a(t)\). Given a symmetric space \(X\) on \(I\), let \(X(I\times I)\) be the corresponding symmetric space on the square, consisting of functions \(x(s,t)\) whose non-increasing rearrangement \(x^*\) is in \(X\). The symbol \(\mathrm{Rad}X\) denotes the set of all functions \(x\in X(I\times I)\) represented in the form of a.e. converging series \(x(s,t)=\sum_nx_n(s)r_n(t)\), \(x_n\in X\). Chapter 1 contains classical facts on the Rademacher system such as the Khintchin inequality.
In Chapter 2 we have newer results which have not been considered in the monograph literature so far. In particular, we have the following essential improvement of the Khintchin inequality and of the estimations of tails.
Hitczenko theorem. {There exists a constant \(c\) such that \(c\kappa_a(\sqrt{p})\leq\|R_a\|_p\leq \kappa_a(\sqrt{p})\) for all \(p\geq 1\) and all \(a\in \ell_2\).}
Montgomery-Smith theorem. {There exists a constant \(c\) such that for all \(p\geq 1\) and all \(a\in \ell_2\) the measure \(m\{R_a>\kappa_a(p)\}\leq e^{-p^2/2}\) and \(m\{R_a>c^{-1}\kappa_a(p)\}\geq c^{-1}e^{-cp^2/2}\).}
Chapter 3 contains descriptions of symmetric spaces in which the Khintchin theorem holds and in which the subspace \(\mathcal{R}(X)\) is complemented. These are spaces which are “far” from \(L_{\infty}\). The closure \(G\) of \(L_{\infty}\) in the Orlicz space generated by the function \(e^{t^2}-1\) and its associated (Köthe dual) \(G'\) play a central role here.
Rodin-Semenov and Lindenstrauss-Tzafriri theorem. 1. {The sequence \((r_n)\) is equivalent in \(X\) to the standard basis of \(\ell_2\) iff \(G\subset X\).} 2. {The subspace \(\mathcal{R}(X)\subset X\) is complemented iff \(G\subset X\subset G'\).}
On the other hand, \((r_n)\) in \(X\) is equivalent to the standard basis of \(\ell_1\) iff \(X=L_{\infty}\) (Rodin and Semenov).
The remaining chapters contain the author’s results or results in which the author has taken a part. Chapter 4 continues the Rodin and Semenov investigations of the Rademacher system for spaces “close” to \(L_{\infty}\). A typical result is (\(\asymp\) denotes the equivalence of basic sequences):
Theorem 4.2. Let the symmetric spaces \(X_0,X_1\) be interpolation spaces between \(L_{\infty}\) and \(G\). If \((r_n)_{X_0}\asymp (r_n)_{X_1}\), then \(X_0=X_1\) (with equivalent norms).
Chapter 5 contains the connection of the Rademacher functions and cones of step functions. The first results in this direction were obtained by Rodin and Semenov in 1975 where a weaker version of the following Theorem 5.1 was proved. For a symmetric space, \(X\) let \(q(X)\) denote the symmetric space of sequences \(a\) such that \(\|a\|_{q(X)}:= \|\sum_na_n^*\chi_{(0,2^{-n+1}]}\|_X<\infty\), where \((a_n^*)\) is the non-increasing rearrangement of the sequence \((a_n)\). Then we always have that \(\|a\|_{q(X)}\leq 4\|R_a\|_X\) and we may ask about the reverse estimate. To obtain an answer, the author considers the following sublinear operator:
\[ \mathcal{L}_2x(t)=\left(\frac{1}{\varphi(t)}\int_0^tx(s)^2d\varphi(s)\right)^{1/2}, \]
where \(\varphi(s)=\log_2^{-1}(2/s)\), \(0<s\leq 1\).
Theorem 5.1. {Let \(X\) be an interpolation space between \(L_{\infty}\) and the Orlicz space \(L_N\) generated by the function \(N(u)=e^u-1\). The inequality \(\|R_a\|_X\leq c\|a\|_{q(X)}\) holds with \(c=c(X)\) iff the operator \(\mathcal{L}_2\) is bounded in \(X\).}
In Chapter 6, the systems \((r_i(t)r_j(t))_{1\leq i<j<\infty}\) and \((r_i(s)r_j(t))_{i,j=1}^{\infty}\) in symmetric spaces on \(I\) and \(I\times I\), respectively, are studied. The main question here is: Which results on the usual Rademacher functions can be transferred to these 2-fold products?
Theorem 6.4. {The system \((r_i(s)r_j(t))\) is equivalent in a symmetric space \(X(I\times I)\) to the standard basis of \(\ell_2\) iff \(H\subset X\), where \(H\) is the closure of \(L_{\infty}\) in the above-mentioned Orlicz space \(L_N\).}
This result is used to prove the following one.
Theorem 6.3. {The system \((r_i(t)r_j(t))\) is unconditional in \(X\) iff it is equivalent to the standard basis of \(\ell_2\) iff \(H\subset X\).}
In addition, it is shown that, for spaces that are “close” to \(L_{\infty}\), the considered (basic) sequences, in contrast to the ordinary Rademacher system, are not unconditional.
In Chapter 7, the subspace \(\mathrm{Rad}\, X\) of \(X(I\times I)\) is considered. Theorems 7.1, 7.2. {Let \(X\) have an order semi-continuous norm.} 1. {The system \((r_i(s)r_j(t))\) in \(X(I\times I)\) is equivalent to the standard basis of \(\ell_2\) iff the lower Boyd index \(\alpha_X>0\).} 2. {The subspace \(\mathrm{Rad}\,X\subset X(I\times I)\) is complemented iff \(0<\alpha_X\leq\beta_X<1\).}
In the next two chapters only \(L_p\) spaces are considered, but interpolation methods still play an important role. Here the equivalence problem of a system of r.v. and the Rademacher system is studied. Two sequences \((f_n)\) and \((g_n)\) of r.v. are equivalent in distribution if there is \(c>0\) such that for all finite sums and \(z>0\)
\[ c^{-1}\mathbb{P}\{|\sum_{n\geq 1}a_nf_n|>cz\}\leq \mathbb{P}\{|\sum_{n\geq 1}a_ng_n|>z\}\leq c\mathbb{P}\{|\sum_{n\geq 1}a_nf_n|>c^{-1}z\}. \]
Theorem 8.2. {A sequence \((f_n)\) of r.v. is equivalent to \((r_n)\) in distribution iff \((f_n)_p\asymp(r_n)_p\), where the equivalence constant does not depend on \(p\geq 1\).}
The main aim of Chapter 9 is to find conditions under which a sequence of r.v. contains a subsequence equivalent in distribution to the Rademacher system.
Theorem 9.1. {A sequence \((f_n)\) of r.v. contains a subsequence equivalent in distribution to \((r_n)\) iff there is a uniformly bounded subsequence \((g_n)\subset (f_n)\) which converges weakly to \(0\) in \(L_2\) and for which \(\inf_n\|g_n\|_2>0\).}
The author also gives the following quantitative version of this theorem.
Theorem 9.6. {Let the functions \((f_n)_1^N\) on \(I\) be orthonormal and \(|f_n(t)|\leq d\). Then there exists a subsequence \((f_{n_i})_1^s\) with \(s\geq \frac{1}{7}\log_2N\) and \(c=c(d)\) such that
\[ c^{-1}m\{|\sum_1^sa_ir_i|>cz\}\leq m\{|\sum_1^sa_if_{n_i}|>z\}\leq cm\{|\sum_1^sa_ir_i|>c^{-1}z\}, \]
for all \((a_i)\) and \(z>0\).}
Chapter 10 is devoted to the study of multiplicator spaces generated by the Rademacher system. The Rademacher multiplicator space \(\Lambda(\mathcal{R},X)\) of a symmetric space \(X\) consists of those measurable functions \(x\) on \(I\) such that \(x\cdot y\in X\) for every \(y\in \mathcal{R}(X)\), equipped with the natural norm. Obviously, \(L_{\infty}\subset \Lambda(\mathcal{R},X)\). Not every multiplicator space is symmetric; \(\mathrm{Sym}(\mathcal{R},X)\) denotes the largest symmetric space embedded into \(\Lambda(\mathcal{R},X)\).
Theorem 10.5. {The following conditions for \(X\) are equivalent. \((i)\) \(\Lambda(\mathcal{R},X)=L_{\infty}\); \((ii)\) \(\mathrm{Sym}(\mathcal{R},X)=L_{\infty}\); \((iii)\) \(\ln^{1/2}(e/t)\not\subset X^{\circ}\), where \(X^{\circ}\) is the closure of \(L_{\infty}\) in \(X\).}
The memoir is well written, but we cannot say the same about the translation. The reviewer had to consult the Russian original to understand some parts of the memoir. Many illustrative examples of Orlicz, Lorentz and Marcinkiewicz spaces are given. Each chapter ends with historical comments and references related to its contents. Formally, all necessary definitions and auxiliary results are presented. However, e.g. [J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 97. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0403.46022)] will be useful for the better understanding of symmetric spaces and the theory of interpolation of operators.

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B70 Interpolation between normed linear spaces
46B09 Probabilistic methods in Banach space theory
46B06 Asymptotic theory of Banach spaces
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