Heat Kernels, Smoothness Estimates, and Exponential Decay

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 ∂̄-operator on L²(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.