0000000000814651
AUTHOR
Luca Capogna
Conformality and $Q$-harmonicity in sub-Riemannian manifolds
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
Conformal equivalence of visual metrics in pseudoconvex domains
We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.
Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian
Abstract We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype ∂ t u = − ∑ i = 1 m X i ⁎ ( | X u | p − 2 X i u ) where p ⩾ 2 , X = ( X 1 , … , X m ) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and X i ⁎ denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M ; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincare inequality involving X and μ. Our results extend the recent work in [16] , [36] , to a more general setting including the model cases of (1)…