Non-divergence evolution operators modeled on Hörmander's vector fields with Dini continuous coefficients
Matteo Faini
TL;DR
The paper tackles the construction of a fundamental solution for the non-divergence, degenerate parabolic operator $H$ driven by Hörmander vector fields on a Carnot group, with coefficient matrix $A$ that is symmetric, uniformly positive definite and double Dini continuous. It extends the parabolic parametrix method by freezing coefficients and building a correction kernel via a Neumann-series framework, yielding a fundamental solution $\Gamma$ that satisfies Gaussian-type bounds and regularity estimates. The authors establish the continuity and Dini-based regularity of the auxiliary kernel $\mu$ and of the integral term $J$, ensuring $H\Gamma=0$ away from the diagonal and enabling robust Gaussian controls for derivatives of $\Gamma$. They then derive well-posedness results for the Cauchy problem Hu=f with initial data, in both homogeneous and nonhomogeneous settings, under Dini-type regularity on the source and appropriate growth, thereby broadening the class of admissible coefficients beyond Hölder or time-measurable cases. Overall, the work provides a rigorous fundamental solution theory and well-posedness framework for a broad family of degenerate parabolic equations on Carnot groups, with potential applications to sub-Riemannian heat kernels and related regularity theory.
Abstract
In this paper we construct a fundamental solution for operators of the form H = a_ij(x,t) X_i X_j - d/dt (having adopted Einstein's convention on repeated indexes) and we show that the latter satisfies suitable Gaussian estimates. Here the X_i are Hörmander's vector fields generating a Carnot group and A = (a_ij) is a symmetric and uniformly positive-definite matrix with bounded double Dini continuous entries. As a consequence of this procedure we also prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.