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 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 $\rho\leq\delta$.