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This is a Quantitative Study study.

October 16, 2025Open Access

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

Key Points

Gaussian estimates were shown for operators modeled on Hormander's vector fields, enhancing understanding in mathematical analysis.
The fundamental solution constructed for the operator H satisfies conditions that ensure suitable Gaussian estimates.
An existence result for the Cauchy problem is proved under specific Dini-type conditions applied to the source.
The operator involves a symmetric matrix with Dini continuous coefficients, which play a critical role in the analysis.

Abstract

In this paper we construct a fundamental solution for operators of the form H = aᵢj (x, t) Xᵢ Xⱼ - d/dt (having adopted Einstein's convention on repeated indexes) and we show that the latter satisfies suitable Gaussian estimates. Here the Xᵢ are H\"ormander's vector fields generating a Carnot group and A = (aᵢj) 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.

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Cite This Study

Matteo Faini (Wed,) studied this question.

synapsesocial.com/papers/68f0f51d8dd8ea469b1d6ebb https://doi.org/https://doi.org/10.48550/arxiv.2502.19073

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