Positive-homogeneous operators, heat kernel estimates and the Legendre-Fenchel transform

TL;DR

This paper studies positive-homogeneous differential operators and their heat kernels, highlighting the role of the Legendre-Fenchel transform in deriving sharp off-diagonal heat kernel estimates.

Contribution

It introduces a general class of positive-homogeneous operators, extends heat kernel estimates using the Legendre-Fenchel transform, and constructs fundamental solutions for variable-coefficient cases.

Findings

Heat kernels arise as limits of convolution powers on lattices.

Off-diagonal heat kernel estimates are expressed via Legendre-Fenchel transform.

Results are sharp in many cases and extend classical theories.

Abstract

We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These so-called positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. More generally, we introduce a class of variable-coefficient operators, each of which is uniformly comparable to a positive-homogeneous operator, and we study the corresponding Cauchy problem for the heat equation. Under the assumption that such an operator has H\"{o}lder continuous coefficients, we construct a fundamental solution to its heat equation by the method of E. E. Levi, adapted to parabolic systems by A. Friedman and S. D. Eidelman. Though our results in this direction are implied by the long-known results of S. D. Eidelman for $2\vec{b}$-parabolic systems, our focus is to highlight the role played by the Legendre-Fenchel transform in heat kernel estimates. Specifically, we show that the fundamental solution satisfies an off-diagonal estimate, i.e., a heat kernel estimate, written in terms of the Legendre-Fenchel transform of the operator's principal symbol--an estimate which is seen to be sharp in many cases.