Ermanno Lanconelli

On the Poincaré inequality for vector fields

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

Smoothness of Lipschitz-continuous graphs with nonvanishing Levi curvature

In (1), (2), ~=(x , y, t) denotes the point of R 3, ut is the first derivative of u with respect to t, and analogous notations are used for the other firstand second-order derivatives of u. The notion of Levi curvature for a real manifold was introduced by E.E. Levi in 1909 in order to characterize the holomorphy domains of C 2. Since then, it has played a crucial role in the geometric theory of several complex variables. In looking for the polynomial hull of a graph, Slodkowski and Tomassini implicitly introduced in 1991 the following definition of Levi curvature for Lipschitz-continuous graphs [16].

Wiener's criterion for parabolic equations with variable coefficients and its consequences

Dans un ensemble borne de R n+1 , on etudie la regularite des points limites pour le probleme de Dirichlet pour un operateur parabolique a coefficients lisses. On donne une caracterisation geometrique de ces points limites qui sont reguliers

Liouville-type theorems for real sub-Laplacians

Abstract: An inequality generalizing the classical Liouville and Harnack Theorems for real sub-Laplacians ℒ is proved. A representation formula for functions

Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities

u

Fundamental solutions for non-divergence form operators on stratified groups

for which ℒu is a polynomial is also showed. As a consequence, some conditions are given ensuring that u is a polynomial whenever ℒu is a polynomial. Finally, an application of this last result is given: if ψ is a C2 map commuting with ℒ, then any of its component is a polynomial function.

Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type

In this work the authors deal with linear second order partial differential operators of the following type

Sul problema di Dirichlet per equazioni paraboliche del secondo ordine a coefficienti discontinui

H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)

A sphere theorem for a class of Reinhardt domains with constant Levi curvature

where

Link of groups and homogeneous Hörmander operators

X_{1},X_{2},\ldots,X_{q}