We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.
We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal Lévy process with the characteristic exponent ψ satisfying some scaling condition. We derive sharp estimates of the potential measure and capacity of balls, and further, under the assumption that ψ satisfies the lower scaling condition, sharp estimates of the potential kernel of the underlying process. This allows us to establish the Krylov-Safonov type estimate, which is the key ingredient in the approach of Bass and Levin, that we follow.
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