The $C^{\a}$ regularity of a class hypoelliptic ultraparabolic equations

TL;DR

This paper proves that weak solutions to a class of ultraparabolic equations with measurable coefficients are continuously differentiable with a certain Hölder exponent, extending previous regularity results.

Contribution

The authors establish $C^{eta}$ regularity for weak solutions of a broader class of ultraparabolic equations with measurable coefficients, generalizing earlier homogeneous cases.

Findings

Weak solutions are $C^{eta}$ continuous.

Generalization to equations with measurable coefficients.

Simplified proof technique.

Abstract

We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form ${\ptl_t u}= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j u)+X_0 u$. The result is proved by simplifying and generalizing our earlier arguments for the $C^{\a}$ regularity of homogeneous ultraparabolic equations.