Inequality on the optimal constant of Young's convolution inequality for locally compact groups and their closed subgroups

2023/02/02 by Takashi SATOMI, Satomi, Takashi
Mathematics · #28C10 #39B62 #43A15 (Secondary) #46E30 (Primary) 22D15 #Advanced Harmonic Analysis Research #Algebraic Topology (math.AT) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Group Theory (math.GR) #Nonlinear Partial Differential Equations #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2302.01084

openalex publication_date 2023/02/02 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28

Abstract

We define the optimal constant Y ( p₁ , p₂ ; G ) of Young's convolution inequality as Y ( p₁ , p₂ ; G ) := sup \ ‖ ϕ₁ * ( ϕ₂ Δ^{1 / p₁'} ) ‖_p | ϕ₁ , ϕ₂ \colon G → ℂ , ‖ ϕ₁ ‖_p1 = ‖ ϕ₂ ‖_p2 = 1 \ for a locally compact group G and 1 ≤ p₁ , p₂ , p ≤ ∞ with 1 / p₁ + 1 / p₂ = 1 + 1 / p. Here p' is the Hölder conjugate of p, ‖ ⋅ ‖ _p is the L^p-norm on a left Haar measure, and Δ\colon G → ℝ_{> 0} is the modular function. The main result of this paper is that Y ( p₁ , p₂ ; G ) ≤ Y ( p₁ , p₂ ; H ) for any closed subgroup H ⊂ G. It follows from this inequality that Y ( p₁ , p₂ ; G ) ≤ Y ( p₁ , p₂ ; ℝ ) ^{dim G - r ( G )} for any connected Lie group G such that the center of the semisimple part is a finite group such as connected linear Lie groups and connected solvable Lie groups, where r ( G ) is the dimension of the maximal compact subgroups of G.

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