Non-divergence evolution operators modeled on HÃ¶rmander vector fields with Dini continuous coefficients

Matteo Faini

analysis of partial differential equations

Non-divergence evolution operators modeled on HÃ¶rmander vector fields with Dini continuous coefficients

arXiv:2502.19073

summary

The paper studies non‑divergence parabolic operators built from Hörmander vector fields on Carnot groups with coefficients that are only Dini continuous, constructing a fundamental solution and establishing Gaussian bounds for it and its derivatives, and using these results to solve the associated Cauchy problem.

Abstract

In this paper we analyze operators $H = \partial_{t} - \sum_{i, j} a_{ij} (x, t) X_{i} X_{j}$ , where the $X_{i}$ 's are HÃ¶rmander 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 HÃ¶lder 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, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.

Topics & keywords

#hörmander vector fields #carnot groups #non-divergence operators #dini continuity #gaussian estimates #fundamental solutionevolution operatordouble Dini continuityGaussian boundsCauchy problemparabolic PDEsubelliptic operators