On some regularity problems for kinetic-type equations

In this thesis we study some regularity problems for kinetic-type partial differential equations. These equations are characterized by the fact that their second order part is fully degenerate, but the presence of a first order operator restores good properties for the solution. In the first part of the thesis we consider a class of Backward Kolmogorov equations with rough coefficients, namely coefficients that are measurable in time and Hölder continuous in space. We prove optimal regularity for the fundamental solution and Schauder estimates for the Cauchy problem. In the second part we study boundary regularity for a kinetic Fokker-Planck equation with constant coefficients. We also prove a Nash inequality for kinetic Sobolev spaces.