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

Abstract

In this paper we prove $L^p$-estimates for Hörmander 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 ($\ddot{I},\hat{I}´$)-Hörmander classes on ${\mathbb{T}}^n$. The main results extend Álvarez 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 $\hat{I}´\ge \ddot{I}$. When $\hat{I}´\le \ddot{I}$, our results recover the available estimates in the literature.

23 pages