Boundedness of pseudo-differential operators on the torus revisited. II
Duván Cardona, Manuel Alejandro Martínez
TL;DR
This work establishes $H^p$-$L^p$ and $H^p$-boundedness for toroidal pseudo-differential operators in Hörmander classes on the torus for $p\le1$, using the global toroidal calculus of $\mathbb{T}^n\times\mathbb{Z}^n$ and discrete Fourier analysis. It develops kernel estimates for operator-valued kernels, implements a dyadic annulus decomposition, and employs an $H^p$-molecule framework together with a $T^*(\overline{e})=0$ (vector-valued $T(1)$-type) condition to obtain $H^p$-boundedness on $\mathbb{T}^n$. The main results give explicit bounds: for $T\in\Psi^m_{\rho,\delta}(\mathbb{T}^n\times\mathbb{Z}^n)$ with $m\le -\beta-n\lambda$ and $(1-\rho)\frac{n}{2}\le\beta<\frac{n}{2}$, $T$ maps $H^p(\mathbb{T}^n)$ to $L^p(\mathbb{T}^n)$ for $1\ge p\ge p_0$ (and to $H^p(\mathbb{T}^n)$ when $T^*(1)=0$) with $\displaystyle \frac{1}{p_0}=\frac{1}{2}+\frac{\beta(\omega/\alpha+n/2)}{n(\omega/\alpha-\omega+\beta)}$; these results extend Euclidean bounds to the torus and accommodate $\rho\le\delta$ cases. The framework integrates Ruzhansky–Turunen calculus, discrete Fourier analysis, and atomic/molecular $H^p$-theory to broaden boundedness results for toroidal operators.
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 $p\leq 1$. 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 $(ρ, δ)$-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 $ρ\leqδ$.