Remainder terms of a nonlocal Sobolev inequality1

2023/05/26 by Shengbing Deng, Xingliang Tian, Deng, Shengbing +5 · 2 citations
Engineering · Mathematics · #35J20 #35P30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2305.16857

openalex publication_date 2023/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we study a nonlocal version of the Sobolev inequality ∫_ℝ^N|∇ u|² dx ≥ S_HLS(∫_ℝ^N(|x|^-α ∗ u^2_α^∗)u^2_α^∗ dx)^\frac12_α^∗, ∀ u∈ D^1,2(ℝ^N), where S_HLS is the best constant, ∗ denotes the standard convolution and D^1,2(ℝ^N) denotes the classical Sobolev space with respect to the norm ‖u‖_D^1,2(ℝ^N)=‖∇ u‖_L²(ℝ^N). By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak L^(N)/(N-2)-norm of above inequality for all 0<α

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