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

Abstract

In this paper we establish a uniform Harnack inequality for a class of degenerate equations whose prototype is Kolmogorov’s equations in R 3 : D yy u − y D z u − D t u = 0 {{\mathbf {R}}^3}:{D_{{\text {yy}}}}u - {\text {y}}{D_z}u - {D_t}u = 0 . Our approach is based on mean value formulas for solutions of the equation under consideration on the level sets of the fundamental solution.