Complete convergence of the Hilbert transform

Abstract

Suppose that $\{a_j\}\in {\ell }^1$, and suppose that for any sequence $(t_n)$ of integers there exits a constant $C_1>0$ such that $$\sharp\left\{k\in\mathbb{Z}:\sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n-t_n} \!\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{a_{k+i}}{i}\right|>λ\right\}\\ \leq C_1\sharp\left\{k\in\mathbb{Z}:\sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n} \!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{a_{k+i}}{i}\right|>λ\right\},$$ for all $\lambda >0$, where ${\mathcal{B}}_n=\{-n,-(n-1),-(n-2),\dots ,n-2,n-1,n\}$. Then there is a constant $C_2>0$ which does not depend on the sequence $\{a_j\}$ such that $$\sum_{n=1}^\infty\sharp\left\{k\in\mathbb{Z}:\left|\sum_{i=-n}^{n} \!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{a_{k+i}}{i}\right|>λ\right\}\leq\frac{C_2}λ\sum_{i=-\infty}^{\infty}|a_i|$$ for all $\lambda >0$. Let $(X,\mathcal{B},\mu )$ be a measure space, $\tau :X\to X$ an invertible measure-preserving transformation, and suppose that $f\in L^1(X)$ such that for any sequence $(t_n)$ of integers there exists a constant $C_1>0$ such that $$μ\left\{ x: \sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n-t_n}\!\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{f(τ^ix)}{i}\right| >λ\right\}\leq C_1μ\left\{x: \sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n}\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{f(τ^i x)}{i}\right|>λ\right\} $$ for all $\lambda >0$, where ${\mathcal{B}}_n=\{-n,-(n-1),-(n-2),\dots ,n-2,n-1,n\}$. Then there exists a constant $C_2>0$ which does not depend on $f$ such that $$\sum_{n=1}^\inftyμ\left\{x:\left|\sum_{i=-n}^{n}\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$} \;\frac{f(τ^ix)}{i}\right|>λ\right\}\leq\frac{C_2}λ\|f\|_1$$ for all $\lambda >0$.