$L^p$-boundedness and $L^p$-nuclearity of multilinear pseudo-differential operators on $\mathbb{Z}^n$ and the torus $\mathbb{T}^n$

TL;DR

This paper systematically studies the boundedness and nuclearity of multilinear pseudo-differential operators on $ abla^n$ and $ abla^n$ spaces, extending known Fourier multiplier theorems and exploring applications like the Kato-Ponce inequality.

Contribution

It extends multilinear Fourier multiplier theorems to periodic and discrete pseudo-differential operators and classifies their $s$-nuclearity properties on Lebesgue spaces.

Findings

Established analogues of multilinear Fourier multiplier theorems for periodic and discrete operators.

Classified $s$-nuclearity of multilinear pseudo-differential operators on Lebesgue spaces.

Applied results to periodic Kato-Ponce inequality and nuclearity of Fourier integral operators.

Abstract

In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the $s$-nuclearity, $0<s \leq 1,$ of periodic and discrete pseudo-differential operators. To accomplish this, we classify those $s$-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on $\sigma$-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the $s$-nuclearity of multilinear Bessel potentials as well as the $s$-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.