Global pseudo-differential operators on the Lie group G= (-1,1)ⁿ

2022/09/20 by Duván Cardona, Roland Duduchava, Cardona, Duván +5
Mathematics · Medicine · Physics and Astronomy · #Advanced Topics in Algebra #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Waves and Solitons #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.2209.09751

openalex publication_date 2022/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work we characterise the Hörmander classes \symbClassOnmρδ\group,\textnormalHör on the open manifold \group = (-1,1)ⁿ. We show that by endowing the open manifold \group = (-1,1)ⁿ with a group structure, the corresponding global Fourier analysis on the group allows one to define a global notion of symbol on the phase space \group × \Rⁿ. Then, the class of pseudo-differential operators associated to the global Hörmander classes \symbClassOnmρδ\group × \Rⁿ recovers the Hörmander classes \symbClassOnmρδ\group,\textnormalloc defined by local coordinate systems. The analytic and qualitative properties of the classes \symbClassOnmρδ\group × \Rⁿ are presented in terms of the corresponding global symbols. In particular, L^p-Fefferman type estimates and Calderón-Vaillancourt theorems are analysed, as well as the spectral properties of the operators.

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