Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups

This article completes the authors's series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness $\sqrt{K-1}$ in the uniform norm and with order of closeness $K-1$ in the Sobolev $L^1_p$-norm for all $p<\frac C{K-1}$. We construct two examples, demonstrating the asymptotic sharpness of our results.