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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 ℂn+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.

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Correspondence to Andrew Raich.

Additional information

Communicated by Karlheinz Gröchenig.

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

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Boggess, A., Raich, A. Heat Kernels, Smoothness Estimates, and Exponential Decay. J Fourier Anal Appl 19, 180–224 (2013). https://doi.org/10.1007/s00041-012-9249-y

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  • DOI: https://doi.org/10.1007/s00041-012-9249-y

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