Nonlocal Quasilinear Parabolic Equations in Heisenberg Group: Local Boundedness with an Optimal Tail

TL;DR

This work establishes local boundedness for a nonlocal, quasilinear parabolic equation on the Heisenberg group under an optimal tail condition. By combining time regularization, a refined Caccioppoli inequality that incorporates the tail into level sets, and a De Giorgi-type iteration, the authors obtain a quantitative local bound in terms of local energy data and Tail$(u_+)$. They also develop interpolation inequalities and an extension theorem for fractional Sobolev spaces on the Heisenberg group, enriching the regularity toolkit in this non-Euclidean setting. The results advance the regularity theory for parabolic nonlocal equations in sub-Riemannian geometries and connect tail behavior with interior boundedness in a sharp way.

Abstract

We prove local boundedness for a quasilinear parabolic equation on the Heisenberg group \[ \partial_t u(ξ,t) + \text{p.v.}\int_{\mathbb{H}^N} \frac{|u(ξ,t)-u(η,t)|^{p-2}(u(ξ,t)-u(η,t))}{|η^{-1}\circ ξ|^{Q+sp}} \,dη= 0, \] with optimal regularity assumption on the tail term. We also prove interpolation inequalities and an extension theorem for fractional Sobolev spaces on the Heisenberg group.

Theorems & Definitions (52)

• Lemma 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Lemma 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• proof