Fourier Multipliers for Besicovitch Spaces

Abstract

In this paper a generalization of some results from Fourier analysis on periodic function spaces to the almost periodic case is given. We consider almost periodic distributions which constitute a subclass of tempered distributions. Under suitable conditions on the spectrum $\Lambda \subset {\mathbb{R}}^s$, a distribution $T\in S^{\prime }({\mathbb{R}}^s)$ is almost periodic if it can be represented as ${\sum }_{\lambda \in \Lambda }{a}_{\lambda }{e}^{i\lambda x}$, where the sequence $(a_{\lambda }{)}_{\lambda \in \Lambda }$ is tempered. The main result states that any Fourier multipliers for $L^q({\mathbb{R}}^s)$ of the Michlin-Hörmander type is also a Fourier multiplier for the Besicovich spaces $B_{ap}^q({\mathbb{R}}^s,\Lambda )$, if it is restricted to the spectrum $\Lambda$. Finally, we prove that the Sobolev-Besicovich spaces $H_{sp}^{N,q}({\mathbb{R}}^s,\Lambda )$ coincide if $N\in \mathbb{N}$.