Matteo Faini | Searcharxiv

arXiv2026

Global Schauder estimates for nondivergence stationary operators modeled on homogeneous H\"ormander vector fields

In this paper we prove global regularity results and Schauder estimates for non-divergence stationary operators of the form L=\sum_{i,j=1}^m a_{ij}(x) X_i X_j, where X_1, ..., X_m are homogeneous (but not necessarily left-invariant) H\"ormander vector fields in R^n (n>m), and [a_{ij}(x)] is a symmetric uniformly positive-definite matrix with H\"older-continuous entries w.r.t. the control distance induced by the vector fields X_1, ..., X_m.

math.AP↗

arXiv2025

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

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 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, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.

math.AP↗