Heat Kernels, Smoothness Estimates and Exponential Decay

TL;DR

This paper proves Gaussian decay for the heat kernel on polynomial models in complex space, using Fourier transform techniques and smoothness estimates to establish exponential decay.

Contribution

It introduces a novel approach combining Fourier transform and smoothness estimates to analyze heat kernel decay on polynomial models.

Findings

Gaussian decay for the Box_b-heat kernel established

Exponential decay achieved via partial Fourier transform

Quantitative smoothness estimates derived for the heat kernel

Abstract

In this article, we establish Gaussian decay for the Box_b-heat kernel on polynomial models in C^2. Our technique attains the exponential decay via a partial Fourier transform. On the transform side, the problem becomes finding quantitative smoothness estimates on a heat kernel associated to the weighted dbar-operator on L^2(C). The bounds are established with Duhamel's formula and careful estimation.