The class of mappings with bounded specific oscillation, and integrability of mappings with bounded distortion on Carnot groups

This paper is the first of the author's three articles on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness $\sqrt{K-1}$ in the uniform norm and order of closeness $K-1$ in the Sobolev norm $L_p^1$ for all $p<\frac C{K-1}$.
In the present article we study integrability of mappings with bounded specific oscillation on spaces of homogeneous type. As an example, we consider mappings with bounded distortion on the Heisenberg group. We prove that a mapping with bounded distortion belongs to the Sobolev class $W^1_{p,\mathrm{loc}}$, where $p\to\infty$ as the distortion coefficient tends to 1.