A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.