Regularity theory of quasilinear elliptic and parabolic equations in the Heisenberg group

This note provides a succinct survey of the existing literature concerning the Hölder regularity for the gradient of weak solutions of PDEs of the form $$\sum _{i=1}^{2n}{X}_i{A}_i({\mathrm{\nabla }}_{0}u)=0\text{ and }{\mathrm{\partial }}_{t}u=\sum _{i=1}^{2n}{X}_i{A}_i({\mathrm{\nabla }}_{0}u)$$ modeled on the $p$-Laplacian in a domain $\mathrm{\Omega }$ in the Heisenberg group ${\mathbb{H}}^n$, with $1\le p<\mathrm{\infty }$, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present.