$L_p\rightarrow L_q$ boundedness of Fourier multipliers

TL;DR

This paper establishes sufficient and necessary conditions for the boundedness of Fourier multipliers from Lp to Lq spaces, including criteria for M-generalized monotone functions, advancing understanding of Fourier analysis.

Contribution

It provides new Hormander and Lizorkin type theorems and criteria for M-generalized monotone functions, extending the theory of Fourier multiplier boundedness.

Findings

Derived sufficient conditions for boundedness

Established necessary conditions for Fourier multipliers

Provided criteria for M-generalized monotone functions

Abstract

We investigate the $L_p \mapsto L_q$ boundedness of the Fourier multipliers. We obtain sufficient conditions, namely, we derive Hormander and Lizorkin type theorems. We also obtain the necessary conditions. For $M$-generalized monotone functions, we obtain a criteria for boundedness of the corresponding Fourier multipliers.