Global Heat Kernels for Parabolic Homogeneous H\"ormander Operators

TL;DR

This paper establishes the existence and key properties of a global heat kernel for a class of parabolic operators defined by Hörmander vector fields satisfying homogeneity, using a lifting technique and Carnot group structures.

Contribution

It introduces a novel approach to construct and analyze the global heat kernel for Hörmander operators with homogeneity, extending previous local results to a global setting.

Findings

Proves existence of the global heat kernel $\Gamma$

Establishes homogeneity, symmetry, and decay properties of $\Gamma$

Provides integral representations for derivatives of $\Gamma$

Abstract

The aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel $\Gamma$ for the parabolic operators $\mathcal{H}=\sum_{j=1}^m X_j^2-\partial_t$, where $X_1,\ldots,X_m$ are smooth vector fields on $\mathbb{R}^n$ satisfying H\"ormander'snrank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of $\Gamma$ is based on a (algebraic) global lifting technique, together with a representation of $\Gamma$ in terms of the integral (performed over the lifting variables) of the Heat kernel for the Heat operator associated with a suitable sub-Laplacian on a homogeneous Carnot group. Among the features of $\Gamma$ we prove: homogeneity and symmetry properties; summability properties; its vanishing at infinity; the uniqueness of the bounded solutions of the related Cauchy problem; reproduction and density properties; an integral representation for the higher-order derivatives.