L^p-bounds in Safarov pseudo-differential calculus on manifolds with bounded geometry

2024/03/20 by Santiago Gómez Cobos, Michael Ruzhansky, Cobos, Santiago Gómez +1 · 2 citations
Computer Science · Engineering · Mathematics · #35S05 (Secondary) #58J40 (Primary) 53B20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2403.13920

openalex publication_date 2024/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a smooth complete Riemannian manifold with bounded geometry (M,g) and a linear connection ∇ on it (not necessarily a metric one), we prove the L^p-boundedness of operators belonging to the global pseudo-differential classes Ψ_{ρ, δ}^m(Ω^κ, ∇, τ) constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: ρ>1/3 and ∇ symmetric; and ∇ flat with any values of ρ and δ. Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some L^p-L^q boundedness. Different examples and applications are presented.

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• Functional calculus for Safarov pseudo-differential operators
• Estimates for pseudo-differential operators on the torus revisited. III

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