, Volume 19, Issue 1, pp 180-224
Date: 09 Nov 2012

Heat Kernels, Smoothness Estimates, and Exponential Decay

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Abstract

In this article, we characterize functions whose Fourier transforms have exponential decay. We characterize such functions by showing that they satisfy a family of estimates that we call quantitative smoothness estimates (QSE). Using the QSE, we establish Gaussian decay in the “bad direction” for the □ b -heat kernel on polynomial models in ℂ n+1. On the transform side, the problem becomes establishing QSE on a heat kernel associated to the weighted \(\bar{\partial}\) -operator on L 2(ℂ). The bounds are established with Duhamel’s formula and careful estimation. In ℂ2, we can prove both on and off-diagonal decay for the □ b -heat kernel on polynomial models.

Communicated by Karlheinz Gröchenig.
A. Raich is partially supported by NSF grant DMS-0855822.