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

Abstract

Let ${\mathcal{S}}^2$ be the Stepanov space and let ${\lambda }_n\uparrow \infty$. Let $(a_n{)}_{n\ge 1}$ be satisfying Wiener's condition $A:={\sum }_{n\ge 1}({\sum }_{k:n\le {\lambda }_k\le n+1}|a_k|{)}^2<\infty$. We prove that $\Vert {\sup }_{N\ge 1}\left|{\sum }_{n=1}^N{a}_n{\mathrm{e}}^{i{\lambda }_{n}t}\right|{\Vert }_{{\mathcal{S}}^2}\le C{A}^{1/2}$ where $C>0$ denotes a universal constant. Moreover, the series ${\sum }_{n\ge 1}{a}_n{\mathrm{e}}^{it{\lambda }_n}$ converges for $\lambda$-a.e. $t\in \mathbb{R}$. This contains as a special case Hedenmalm and Saksman result for Dirichlet series. We also obtain maximal inequalities for corresponding series of dilates. Let $1\le p,q\le 2$ be such that $1/p+1/q=3/2$. Then for any sequence $({\alpha }_n{)}_{n\ge 1}$ and $({\beta }_n{)}_{n\ge 1}$ of complex numbers such that $K:={\sum }_{n\ge 1}({\sum }_{k:n\le {\lambda }_k<n+1}|{\alpha }_k|{)}^p<\infty$ and $L:={\sum }_{n\ge 1}({\sum }_{k:n\le {\mu }_k<n+1}|{\beta }_k|{)}^q<\infty$, we have $$\Vert \underset{N\ge 1}{\sup }|\sum _{n=1}^N{\alpha }_{n}D({\lambda }_{n}t)|{\Vert }_{{\mathcal{S}}^2}\le C{K}^{1/p}{L}^{1/q}$$ where $D(t)={\sum }_{n\ge 1}{\beta }_n{\mathrm{e}}^{i{\mu }_{n}t}$ is defined in ${\mathcal{S}}^2$. Moreover, the series ${\sum }_{n\ge 1}{\alpha }_{n}D({\lambda }_{n}t)$ converges in ${\mathcal{S}}^2$ and for $\lambda$-a.e. $t\in \mathbb{R}$. We further show that if $\{{\lambda }_k,k\ge 1\}$ satisfies the following condition $$\sum _{\genfrac{}{}{0ex}{}{k\ne \ell ,k^{\prime }\ne {\ell }^{\prime }}{(k,\ell )\ne (k^{\prime },{\ell }^{\prime })}}(1-|({\lambda }_k-{\lambda }_{\ell })-({\lambda }_{k^{\prime }}-{\lambda }_{{\ell }^{\prime }})|{)}_{+}^2<\infty ,$$ then the series ${\sum }_k{a}_k{\mathrm{e}}^{i{\lambda }_{k}t}$ 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 $\{{\lambda }_k,k\ge 1\}$ is a Sidon sequence.

arXiv comment15 pages