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arXiv · 2502.20575

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

Duván Cardona·Manuel Alejandro Martínez
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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 ($ρ,δ$)-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 $δ\geqρ$. When $δ\leq ρ$, our results recover the available estimates in the literature.

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Duván Cardona, Manuel Alejandro Martínez. 2025-02-27. Estimates for pseudo-differential operators on the torus revisited. I. https://doi.org/10.1016/j.jmaa.2025.129959

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