Some New Fourier Multiplier Results of Lizorkin and Hörmander Types

Abstract

This paper is devoted to the study of Fourier series and Fourier transform multipliers and contains introduction, which put some new results into a general frame. In the following Sections several further examples and results are presented and discussed. In Section 2 we present some important results (including the most early papers we know) concerning Fourier series multipliers of particular interest for the investigations. The corresponding result for Fourier transform multipliers can be found in Section 3. In Section 4 we give some applications and in Section 5 we describe shortly the main results: A generalization and sharpening of the Lizorkin theorem concerning Fourier transform multipliers between \(L_p\) and \(L_q\). The Fourier series multipliers in the case with a regular system, which is rather general. A generalization and sharpening of the Lizorkin type theorem concerning Fourier series multipliers between \(L_p\) and \(L_q\) in this general case. A generalization of the Hormander multiplier theorem for two dimensional Fourier series to the case with a general regular system .