Cα regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models, and prove that weak solutions are H?lder continuous, which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity condition. The new ingredients are manifested in two aspects: on the one hand, for lower-order terms, we exploit a new Sobolev inequality suitable for the Moser iteration by improving the result of Pascucci and Polidoro(2004); on the other hand, we explore the G-function from an early idea of Kruzhkov(1964) and an approximate weak Poincaré inequality for non-negative weak sub-solutions to prove the H?lder regularity.