2025/07/17 by Lu Chen, Guozhen Lu, Chen, Lu +5 · 2 citations
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2507.12725
openalex publication_date 2025/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are concerned with the optimal asymptotic lower bound for the stability of Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of Sobolev inequality on the CR sphere through bispherical harmonics and complicated orthogonality technique ( see Lemma 3.1). The loss of rearrangement inequality in the CR setting makes it impossible to use any rearrangement flow technique (either differential rearrangement flow or integral rearrangement flow) to derive the optimal stability of Sobolev inequality on the CR sphere from corresponding optimal local stability. To circumvent this, we will use the CR Yamabe flow to establish the optimal stability of Sobolev inequality on the Heisenberg group with the dimension-dependent constants (see Theorem 1.1). As an application, we also establish the optimal stability of the Hardy-Littlewood-Sobolev (HLS) inequality for special conformal index with the dimension-dependent constants (see Theorem 1.3). Our approach is rearrangement-free and can be used to study the optimal stability problem for fractional Sobolev inequality or HLS inequality on the Heisenberg group once the corresponding continuous flow is established.