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

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 Hörmander classes of pseudo-differential operators on the torus ${\mathbb{T}}^n$ for $p\le 1$. 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 $(\rho ,\delta )$-Hörmander classes on ${\mathbb{T}}^n$ defined by local coordinate systems. These results extend those proved by Álvarez and Hounie for the Euclidean case, considering even the case $\rho \le \delta$.

22 pages