Boundedness of pseudo-differential operators on the torus revisited. II

TL;DR

This paper extends the analysis of pseudo-differential operators on the torus, establishing new boundedness results in Hardy and Lebesgue spaces for classes of these operators, using global symbolic analysis techniques.

Contribution

It proves $H^p$-$L^p$ and $H^p$-estimates for H"ormander classes on the torus, extending prior Euclidean results to the periodic setting with discrete Fourier analysis.

Findings

Established $H^p$-$L^p$ boundedness for pseudo-differential operators on $ orus^n$

Extended Euclidean boundedness results to the torus case

Utilized global symbolic analysis with discrete Fourier methods

Abstract

In this paper we continue our program of revisiting the new aspects about the boundedness properties of pseudo-differential operators on the torus. Here we prove $H^p$-$L^p$ and $H^p$-estimates for H\"ormander classes of pseudo-differential operators on the torus $\mathbb{T}^n$ for $p\leq 1$. The results are presented in the context of the global symbolic analysis developed by Ruzhansky and Turunen on $\mathbb{T}^n \times \mathbb{Z}^n$ by using the discrete Fourier analysis, which extends the $(\rho, \delta)$-H\"ormander classes on $\mathbb{T}^n$ defined by local coordinate systems. These results extend those proved by \'Alvarez and Hounie for the Euclidean case, considering even the case $\rho\leq\delta$.