Pointwise convergence of almost periodic Fourier series and associated series of dilates

Christophe Cuny and Michel Weber

Vol. 292 (2018), No. 1, 81–101

DOI: 10.2140/pjm.2018.292.81

Abstract

Let ${\mathcal{S}}^2$ be the Stepanov space with norm $\parallel f{\parallel }_{{\mathcal{S}}^2}=\underset{x\in ℝ}{sup}{\left({\int \; <!--nolimits-->}_x^{x+1}|f\left(t\right){|}^{2}dt\right)}^{1∕2}$, ${\lambda }_n\uparrow \infty$, and let ${\left(a_n\right)}_{n\ge 1}$ satisfy Wiener’s condition $\left[b\right]{\sum }_{n\ge 1}{\left({\sum }_{k:n\le {\lambda }_k\le n+1}\left|a_k\right|\right)}^2<\infty$. We establish the following maximal inequality:

$${\parallel \underset{N\ge 1}{sup}\left|\sum _{n=1}^N{a}_n{e}^{i{\lambda }_{n}t}\right|\parallel }_{{\mathcal{S}}^2}\le C{\left(\sum _{n\ge 1}{\left(\sum _{k:n\le {\lambda }_k\le n+1}\left|a_k\right|\right)}^2\right)}^{1∕2},$$

where $C>0$ is a universal constant. Moreover, the series ${\sum }_{n\ge 1}{a}_n{e}^{it{\lambda }_n}$ converges for $\lambda$-a.e. $t\in ℝ$. We give a simple and direct proof. This contains as a special case Hedenmalm and Saksman’s result for Dirichlet series. We also obtain maximal inequalities for corresponding series of dilates. Let ${\left({\lambda }_n\right)}_{n\ge 1}$, ${\left({\mu }_n\right)}_{n\ge 1}$, be nondecreasing sequences of real numbers greater than 1. We prove the following interpolation theorem. Let $1\le p,q\le 2$ be such that $1∕p+1∕q=\frac{3}{2}$. There exists $C>0$ such that for any sequences ${\left({\alpha }_n\right)}_{n\ge 1}$ and ${\left({\beta }_n\right)}_{n\ge 1}$ of complex numbers such that ${\sum }_{n\ge 1}{\left({\sum }_{k:n\le {\lambda }_k<n+1}\left|{\alpha }_k\right|\right)}^p<\infty$ and ${\sum }_{n\ge 1}{\left({\sum }_{k:n\le {\mu }_k<n+1}\left|{\beta }_k\right|\right)}^q<\infty$, we have

$${\parallel \underset{N\ge 1}{sup}\left|\sum _{n=1}^N{\alpha }_{n}D\left({\lambda }_{n}t\right)\right|\parallel }_{{\mathcal{S}}^2}\le C{\left(\sum _{n\ge 1}{\left(\sum _{k:n\le {\lambda }_k<n+1}\left|{\alpha }_k\right|\right)}^p\right)}^{1∕p}{\left(\sum _{n\ge 1}{\left(\sum _{k:n\le {\mu }_k<n+1}\left|{\beta }_k\right|\right)}^q\right)}^{1∕q},$$

where $D\left(t\right)={\sum }_{n\ge 1}{\beta }_n{e}^{i{\mu }_{n}t}$ is defined in ${\mathcal{S}}^2$. Moreover, ${\sum }_{n\ge 1}{\alpha }_{n}D\left({\lambda }_{n}t\right)$ converges in ${\mathcal{S}}^2$ and for $\lambda$-a.e. $t\in ℝ$. We further show that if $\left\{{\lambda }_k,k\ge 1\right\}$ satisfies the condition

$$\sum _{\genfrac{}{}{0ex}{}{k\ne \ell ,k^{\prime }\ne {\ell }^{\prime }}{\left(k,\ell \right)\ne \left(k^{\prime },{\ell }^{\prime }\right)}}{\left(1-|\left({\lambda }_k-{\lambda }_{\ell }\right)-\left({\lambda }_{k^{\prime }}-{\lambda }_{{\ell }^{\prime }}\right)|\right)}_{+}^2<\infty ,$$

then the series ${\sum }_k{a}_k{e}^{i{\lambda }_{k}t}$ converges on a set of positive Lebesgue measure only if the series ${\sum }_{k=1}^{\infty }|a_k{|}^2$ converges. The above condition is in particular fulfilled when $\left\{{\lambda }_k,k\ge 1\right\}$ is a Sidon sequence.

Keywords

almost periodic function, Stepanov space, Carleson theorem, Dirichlet series, dilated function, series, almost everywhere convergence

Mathematical Subject Classification 2010

Primary: 42A75

Secondary: 42A24, 42B25

Milestones

Received: 11 August 2016

Accepted: 31 March 2017

Published: 22 September 2017