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

2025/02/27 by Cardona, Duván, Martínez, Manuel Alejandro
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2502.20575

Abstract

In this paper we prove L^p-estimates for Hörmander classes of pseudo-differential operators on the torus \mathbbTⁿ. The results are presented in the context of the global symbolic calculus of Ruzhansky and Turunen on \mathbbTⁿ×ℤⁿ by using the discrete Fourier analysis on the torus which extends the usual (ρ,δ)-Hörmander classes on \mathbbTⁿ. The main results extend Álvarez and Hounie's method for ℝⁿ to the torus, and Fefferman's L^p-boundedness theorem in the toroidal setting allowing the condition δ≥ρ. When δ≤ ρ, our results recover the available estimates in the literature.

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