Title: A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields,
Abstract: In this paper we consider the divergence parabolic equation with bounded and
measurable coefficients related to Hormander's vector fields and establish a
Nash type result, i.e., the local Holder regularity for weak solutions. After
deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of
Hormander's vector fields and a De Giorgi type Lemma, the Holder 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 Holder 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.