On a real hypersurface in of class we consider a local CR structure by choosing complex vector fields in the complex tangent space. Their real and imaginary parts span a -dimensional subspace of the real tangent space, which has dimension If the Levi matrix of is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...
We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a...
We provide a structure theorem for Carnot-Carathéodory balls defined by a family of
Lipschitz continuous vector fields. From this result a proof of Poincaré inequality
follows.
In the euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous –convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for...
In this paper a -regularity result for the strong viscosity solutions to the prescribed Levi-curvature equation is announced. As an application, starting from a result by Z. Slodkowski and G. Tomassini, the -solvability of the Dirichlet problem related to the same equation is showed.
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