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Gradient estimates for diffusion semigroups with singular coefficients

By Enrico Priola and Feng-Yu Wang

Abstract

AbstractUniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space–time bounded harmonic functions with respect to the underlying diffusion semigroup

Publisher: Published by Elsevier Inc.
Year: 2006
DOI identifier: 10.1016/j.jfa.2005.12.010
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