Multipliers of the Haar series
The system of functions \[ \displaystyle\begin{cases}\chi_0^0(t):=1,\;\chi_n^k(t):=1 &\text{when }t\in((k-1)2^{-n},(k-1/2)2^{-n}),\\ \chi_n^k(t):=-1&\text{when }t\in((k-1/2)2^{-n},k2^{-n}),\\ \chi_n^k(t)=0&\text{for all the other }t\in[0,1],\end{cases} \] where \(1\leq k\leq 2^n\) with \(n\in\{0,1,\dots\}\), is called the Haar system. Denote by \(\Omega\) the set of all indices \((n,k)\) corresponding to the Haar system. Given \((n,k)\in\Omega\), let \(\Delta_n^k:=(\frac{k-1}{2^n},\frac{k}{2^n})\) and \(A\) be a set \(K=(1,k_1,\dots,k_n)\) such that \(\Delta_0^1\supset\Delta_1^{k_1}\supset\cdots\supset\Delta_n^{k_n}\). Let \(W\) be the set of all sequences \(s:=\{s_{n,k}:(n,k)\in\Omega\}\) such that \(\|s\|_W:=\sup_{K\in A,n\in\mathbb N}\sum_{m=1}^{n}|s_{m,k_m}-s_{m-1,k_{m-1}}|<\infty.\) For any given sequence \(s:=\{s_1,s_2,\dots\}\), define a multiplier \(S\) on Haar polynomials as \[ S\left(\sum_{m}c_m\chi_m\right):=\sum_{m}s_mc_m\chi_m. \] A Banach function space \(E\) on \([0,1]\) equipped with the Lebesgue measure is called the rearrangement invariant space if the following two conditions are satisfied: (i) if \(|x(t)|\leq |y(t)|\) and \(y\in E\), then \(x\in E\) and \(\|x\|_E\leq \|y\|_E\); (ii) if \(y\in E\), then the equimeasurability of two functions \(x(t)\) and \(y(t)\) implies that \(x\in E\) and \(\|x\|_E=\|y\|_E\). In this paper, the authors first show that a multiplier \(S\) is bounded in \(L_1\) if and only if \(s\in W\); moreover, \(\|S\|_{L_1}\) is equivalent to \(\|s\|_W+\|s\|_{\ell_\infty}\). As a corollary, the authors obtain that a multiplier \(S\) is bounded in every rearrangement invariant space if and only if \(s\in W\). Let \(\varphi\) be an increasing concave function on \([0,1]\) satisfying \(\varphi(0)=0\) and \(\varphi(1)=1\). The Lorentz space \(\Lambda(\varphi)\) is defined to be the set of all functions \(x\) on \([0,1]\) such that \(\|x\|_{\Lambda(\varphi)}:=\int_0^1x^\ast(t)\,d\varphi(t)<\infty\), where \(x^\ast\) is the rearrangement of \(x\). The authors also prove that, if \(\chi_{n_i}^1\) is a subsystem of the Haar system such that \(\sup_{k\in\mathbb N}\log n_{k+1}/\log n_k<\infty\), then the orthogonal projection onto the subsystem \(\chi_{n_i}^1\) is bounded in the Lorentz space if and only if \(\sup_{k\in\mathbb N}n_k\sum_{j=k}^\infty\frac1{n_j}<\infty\).
scientific article; zbMATH DE number 6105682