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Zeitschrift für Analysis und ihre Anwendungen

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Volume 17, Issue 4, 1998, pp. 917–935
DOI: 10.4171/ZAA/859

Published online: 1998-12-31

Fourier Multipliers for Besicovitch Spaces

R. Grande[1]

(1) Università di Roma La Sapienza, Italy

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 $N \in \mathbb N$.

Keywords: Almost periodic functions, distributions, multipliers

Grande R.: Fourier Multipliers for Besicovitch Spaces. Z. Anal. Anwend. 17 (1998), 917-935. doi: 10.4171/ZAA/859