Wiener amalgams and pointwise summability of Fourier transforms and Fourier series

Abstract

This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces $W(L_p,\ell_q)({\mathbb R}^d)$, Herz spaces and weighted versions of Feichtinger's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the $\theta$-means $\sigma_T^\theta f$ can be estimated pointwise by the Hardy–Littlewood maximal function. From this it follows that $\sigma_T^\theta f \,{\to}\, f$ a.e. for all $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$, hence $f\in L_p({\mathbb R}^d)$ for any $1\leq p\leq \infty$. Moreover, $\sigma_T^\theta f(x)$ converges to $f(x)$ at each Lebesgue point of $f\in L_1({\mathbb R}^d)$ (resp. $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$) if and only if the Fourier transform of $\theta$ is in a suitable Herz space. In case $\theta$ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for $\theta$ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.