Heat kernel estimates and lower bound of eigenvalues.

Shiu-Yuen Cheng; Peter Li

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Heat kernel estimates for critical fractional diffusion operators

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We construct the heat kernel of the 1/2-order Laplacian perturbed by a first-order gradient term in Hölder spaces and a zero-order potential term in a generalized Kato class, and obtain sharp two-sided estimates as well as a gradient estimate of the heat kernel, where the proof of the lower bound is based on a probabilistic approach.