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

Abstract

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In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to H\u00f6rmander's vector fields and establish a Nash type result, i.e., the local H\u00f6lder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincar\u00e9 inequality of H\u00f6rmander's vector fields and a De Giorgi type Lemma, the H\u00f6lder 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 H\u00f6lder 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.<\/p>"