Pietro Zamboni

HARNACK INEQUALITY AND REGULARITY FOR DEGENERATE QUASILINEAR ELLIPTIC EQUATIONS

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong A∞ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove C1,α local estimates for solutions of a degenerate equation in non divergence form.

A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields

In this note we prove a Fefferman-Poincare type inequality in spaces with metric induced by Carnot-Caratheodory vector fields.

A Potential Theoretic Inequality

In this paper is proved a weighted inequality for Riesz potential similar to the classical one by D. Adams. Here the gain of integrability is not always algebraic, as in the classical case, but depends on the growth properties of a certain function measuring some local potential of the weight.

Sum Operators and Fefferman–Phong Inequalities

We define Stummel-Kato type classes in a quasimetric homogeneous setting using sum operators introduced in (J. Fourier Anal. Appl. 9(5), 511–540, 2003) by Franchi, Perez and Wheeden. Then we prove an embedding inequality of Fefferman–Phong type. As an application we give a unique continuation result for non negative solutions of some subelliptic equations.