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

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.

38 Pages