JournalszaaVol. 17, No. 4pp. 917–935

Fourier Multipliers for Besicovitch Spaces

  • R. Grande

    Università di Roma La Sapienza, Italy
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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 ΛRs\Lambda \subset \mathbb R^s, a distribution TS(Rs)T \in S'(\mathbb R^s) is almost periodic if it can be represented as λΛaλeiλx\sum_{\lambda \in \Lambda} a_{\lambda} e^{i \lambda x}, where the sequence (aλ)λΛ(a_{\lambda})_{\lambda \in \Lambda} is tempered. The main result states that any Fourier multipliers for Lq(Rs)L^q(\mathbb R^s) of the Michlin-Hörmander type is also a Fourier multiplier for the Besicovich spaces Bapq(Rs,Λ)B^q_{ap} (\mathbb R^s, \Lambda), if it is restricted to the spectrum Λ\Lambda. Finally, we prove that the Sobolev-Besicovich spaces HspN,q(Rs,Λ)H^{N,q}_{sp} (\mathbb R^s, \Lambda) coincide if NNN \in \mathbb N.

Cite this article

R. Grande, Fourier Multipliers for Besicovitch Spaces. Z. Anal. Anwend. 17 (1998), no. 4, pp. 917–935

DOI 10.4171/ZAA/859