Stein-Weiss inequality on non-compact symmetric spaces

2023/10/30 by Vishvesh Kumar, Michael Ruzhansky, Kumar, Vishvesh +3 · 1 citation
Mathematics · #22E30 #43A85 #43A90 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2310.19412

openalex publication_date 2023/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Δ be the Laplace-Beltrami operator on a non-compact symmetric space of any rank, and denote the bottom of its L²-spectrum as -|ρ|². In this paper, we provide a comprehensive characterization of both the sufficient and necessary conditions ensuring the validity of the Stein-Weiss inequality for the entire family of operators \lbrace(-Δ+b)^-\fracσ2\rbrace_{σ≥0, b≥-|ρ|²}. As an application, some weighted functional inequalities, such as Heisenberg's uncertainty principle, Gagliardo-Nirenberg's interpolation inequality, Pitt's inequality, etc., become available in this context. In particular, their sets of admissible indices are larger than those in the Euclidean setting.

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• Shifted Pitt and uncertainty inequalities on Riemannian symmetric spaces of noncompact type

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