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

TL;DR

This paper establishes $L^p$-estimates for H"ormander classes of pseudo-differential operators on the torus, extending classical results from Euclidean spaces to the toroidal setting using global symbolic calculus.

Contribution

It extends Fefferman's $L^p$-boundedness theorem and the method of Alvarez and Hounie to the torus, broadening the scope of pseudo-differential operator analysis.

Findings

Proves $L^p$-estimates for H"ormander classes on $ orus^n$

Extends Fefferman's theorem to the toroidal setting with $ ho, ho$ conditions

Recovers existing estimates when $ ho eq ho$

Abstract

In this paper we prove $L^p$-estimates for H\"ormander classes of pseudo-differential operators on the torus $\mathbb{T}^n$. The results are presented in the context of the global symbolic calculus of Ruzhansky and Turunen on $\mathbb{T}^n\times\mathbb{Z}^n$ by using the discrete Fourier analysis on the torus which extends the usual ($\rho,\delta$)-H\"ormander classes on $\mathbb{T}^n$. The main results extend \'Alvarez and Hounie's method for $\mathbb{R}^n$ to the torus, and Fefferman's $L^p$-boundedness theorem in the toroidal setting allowing the condition $\delta\geq\rho$. When $\delta \leq \rho$, our results recover the available estimates in the literature.