Estimates for pseudo-differential operators on the torus revisited. III

TL;DR

This paper establishes boundedness and continuity properties of toroidal pseudo-differential operators with symbols in Hörmander classes, extending Euclidean results to the torus and weighted spaces using discrete Fourier analysis.

Contribution

It provides new pointwise estimates and boundedness theorems for pseudo-differential operators on the torus, including weighted Lebesgue, Sobolev, and Besov spaces, extending prior Euclidean results.

Findings

Proves pointwise estimates using Fefferman-Stein and Hardy-Littlewood maximal functions.

Establishes boundedness on weighted Lebesgue spaces $L^p(w)$ for toroidal pseudo-differential operators.

Includes continuity results on Sobolev and Besov spaces on the torus.

Abstract

This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the H\"ormander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the global symbolic analysis defined on $\mathbb{T}^n\times \mathbb{Z}^n$ as developed by Ruzhansky and Turunen by using discrete Fourier analysis, and extend those of Park and Tomita available in the Euclidean case. Moreover, we include continuity results on Sobolev spaces $W^s_p$ and on Besov spaces $B^s_{p,q}$ on the torus. Our techniques are taken from Park and Tomita \cite{park-tomita} and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature.