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 $T \\in S'(\\mathbb R^s)$ 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 $B^q{ap} (\\mathbb R^s, \\Lambda)$, if it is restricted to the spectrum $\\Lambda$. Finally, we prove that the Sobolev-Besicovich spaces $H^{N,q}{sp} (\\mathbb R^s, \\Lambda)$ coincide if $N \\in \\mathbb N$.