A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields

TL;DR

This paper proves local Hölder regularity for weak solutions of divergence parabolic equations with coefficients related to Hörmander's vector fields, extending classical results to a more general setting.

Contribution

It establishes a Nash type regularity result for divergence parabolic equations associated with Hörmander's vector fields, including new inequalities and the Harnack inequality.

Findings

Proved local Hölder regularity of weak solutions.

Derived parabolic Sobolev and Poincaré inequalities for Hörmander's vector fields.

Established the Harnack inequality for weak solutions.

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 Poincar\'e 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.