Regularity for quasilinear equations and $1-$ quasiconformal maps in Carnot groups

Quasiregular maps and the conductivity equation in the Heisenberg group

“…In particular, such maps are smooth. This was shown for C 4 maps by Korányi and Reimann [11] and later by Capogna [4] without the regularity assumption.…”

Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

“…Therefore, we also need to improve the integrability of difference quotients. We return to estimate (3)(4)(5)(6)(7)(8)(9)(10), and apply the subelliptic Sobolev embedding theorem as follows:…”

“…In the Heisenberg group setting, up to now the conditions (1)(2)(3)(4) and (1)(2)(3)(4)(5) have been considered with q = p, i.e. in the standard p-growth case.…”

“…Hölder regularity results for the gradient of solutions of quasi-linear equations were given by Capogna [4,5], using a technique involving fractional pseudo-differential operators; this was also generalized to partial regularity for nonlinear systems by Capogna and Garofalo [7] (see also [17]). Regularity results focusing on the vertical derivative were developed by Marchi [39,40] and Domokos [8]: In particular, T u ∈ L p loc was established for solutions of equations with p-growth, under the condition that 2 ≤ p < 4, employing an iterative method based on fractional difference quotients.…”

Capacities in metric spaces

“…There are also proofs of Hölder continuity for some Carnot groups, see e.g. [2,3]. (5) Using the results and techniques of [30], it should be possible to prove continuity of extremal functions for a wide class of axiomatic Sobolev space (perhaps assuming that D is local and µ is doubling) …”