Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants

TL;DR

This paper proves the optimal asymptotic stability of Sobolev inequalities on the Heisenberg group using a rearrangement-free approach via the CR Yamabe flow, with implications for related inequalities.

Contribution

It establishes the first optimal stability results for Sobolev inequalities on the Heisenberg group without rearrangement techniques, using the CR Yamabe flow.

Findings

Optimal local stability of Sobolev inequality on the CR sphere.

Dimension-dependent constants in stability estimates.

Extension to Hardy-Littlewood-Sobolev inequality stability.

Abstract

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.