Stability of Hardy Littlewood Sobolev Inequality under Bubbling

2021/09/26 by Aryan, Shrey · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.12610

Abstract

In this note we will generalize the results deduced in arXiv:1905.08203 and arXiv:2103.15360 to fractional Sobolev spaces. In particular we will show that for s∈ (0,1), n>2s and ν∈ ℕ there exists constants δ= δ(n,s,ν)>0 and C=C(n,s,ν)>0 such that for any function u∈ H^s(ℝⁿ) satisfying, ‖ u-∑_i=1^ν U_i‖_H^s ≤ δ where U₁, U₂,⋯ U_ν is a δ-interacting family of Talenti bubbles, there exists a family of Talenti bubbles U₁, U₂,⋯ U_ν such that ‖ u-∑_i=1^ν U_i‖_H^s ≤ C\Γ · amp; \text if 2s · lt; n · lt; 6s,
Γ|log Γ|^(1)/(2) · amp; \text if n=6s,
Γ^(p)/(2) · amp; \text if n · gt; 6s. for Γ=‖Δu+u|u|^p-1‖_H^-s and p=2^*-1=(n+2s)/(n-2s).

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