# Davies' method for heat-kernel estimates: An extension to the semi-elliptic setting

## TL;DR

This paper extends Davies' heat-kernel estimates to a class of anisotropic, semi-elliptic operators with variable coefficients, providing Gaussian bounds based on the Legendre-Fenchel transform of the principal symbol.

## Contribution

It develops a general theory for anisotropic, positive-homogeneous operators with variable coefficients, extending Davies' approach to a broader semi-elliptic setting.

## Key findings

- Established off-diagonal Gaussian estimates for heat kernels
- Extended Davies' method to semi-elliptic, anisotropic operators
- Connected heat kernel behavior to the Legendre-Fenchel transform of the symbol

## Abstract

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in different directions. The heat kernels associated to these so-called positive-homogeneous operators are seen to arise naturally as the limits of convolution powers of complex-valued measures, just as the classical heat kernel appears in the central limit theorem. Building on the functional-analytic approach developed by E. B. Davies for higher-order uniformly elliptic operators with measurable coefficients, we formulate a general theory for (anisotropic) self-adjoint variable-coefficient operators, each comparable to a positive-homogeneous operator, and study their associated heat kernels. Specifically, under three abstract hypotheses, we show that the heat kernels satisfy off-diagonal (Gaussian type) estimates involving the Legendre-Fenchel transform of the operator's principle symbol. Our results extend those of E. B. Davies and G. Barbatis and partially extend results of A. F. M. ter Elst and D. Robinson.