Non-divergence evolution operators modeled on H\"ormander vector fields with Dini continuous coefficients

TL;DR

This paper develops fundamental solutions and Gaussian estimates for a class of parabolic operators with Hörmander vector fields and Dini continuous coefficients, extending classical results to weaker regularity conditions.

Contribution

It constructs fundamental solutions and establishes Gaussian bounds for operators with Dini continuous coefficients, a weaker regularity assumption than H"older continuity.

Findings

Established two-sided Gaussian estimates for fundamental solutions.

Proved upper Gaussian bounds for derivatives of the fundamental solution.

Demonstrated existence of solutions to the Cauchy problem under Dini conditions.

Abstract

In this paper we analyze operators H = a^{ij}(x,t) X_i X_j - d/dt (having adopted Einstein's convention on repeated indexes), where the X_i's are H\"ormander vector fields generating a Carnot group and A = [a_{ij}] is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than H\"older continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight 2. As a consequence of the whole procedure, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.