Vol. 292, No. 1, 2018

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

Christophe Cuny and Michel Weber

Vol. 292 (2018), No. 1, 81–101
Abstract

Let ${\mathsc{S}}^{2}$ be the Stepanov space with norm $\parallel f{\parallel }_{{\mathsc{S}}^{2}}=\underset{x\in ℝ}{sup}{\left({\int }_{x}^{x+1}|f\left(t\right){|}^{2}\phantom{\rule{0.3em}{0ex}}dt\right)}^{1∕2}$, ${\lambda }_{n}↑\infty$, and let ${\left({a}_{n}\right)}_{n\ge 1}$ satisfy Wiener’s condition $\left[b\right]{\sum }_{n\ge 1}{\left({\sum }_{k:n\le {\lambda }_{k}\le n+1}|{a}_{k}|\right)}^{2}<\infty$. We establish the following maximal inequality:

 ${\parallel \underset{N\ge 1}{sup}|\sum _{n=1}^{N}{a}_{n}{e}^{i{\lambda }_{n}t}|\phantom{\rule{0.3em}{0ex}}\parallel }_{{\mathsc{S}}^{2}}\le C{\left(\sum _{n\ge 1}{\left(\sum _{k:n\le {\lambda }_{k}\le n+1}|{a}_{k}|\right)}^{2}\right)}^{1∕2},$

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} and ${\sum }_{n\ge 1}{\left({\sum }_{k:n\le {\mu }_{k}, we have

${\parallel \underset{N\ge 1}{sup}|\sum _{n=1}^{N}{\alpha }_{n}D\left({\lambda }_{n}t\right)|\parallel }_{{\mathsc{S}}^{2}}\le C{\left(\sum _{n\ge 1}{\left(\sum _{k:n\le {\lambda }_{k}

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

 $\sum _{\genfrac{}{}{0}{}{k\ne \ell ,\phantom{\rule{0.3em}{0ex}}{k}^{\prime }\ne {\ell }^{\prime }}{\left(k,\ell \right)\ne \left({k}^{\prime },{\ell }^{\prime }\right)}}{\left(1-|\left({\lambda }_{k}-{\lambda }_{\ell }\right)-\left({\lambda }_{{k}^{\prime }}-{\lambda }_{{\ell }^{\prime }}\right)|\right)}_{+}^{2}<\infty ,$

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