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

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, $0<s\le 1,$ 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.

40 pages, This version is a revised version based on reviewer's comments. Final version appeared in JFAA

References in corpus (3)

• Schatten classes on compact manifolds: Kernel conditions
• The essential spectrum of Schrödinger operators on lattices
• Kernel and symbol criteria for Schatten classes and $r$-nuclearity on compact manifolds

Cited by in corpus (2)

• Global solutions to Stokes-Magneto equations with fractional dissipations
• Sharp quasi-invariance threshold for the cubic Szegő equation