We consider a class of possibly degenerate second order elliptic operators A on R^n. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Holder spaces both for elliptic equations and for parabolic Cauchy problems involving A. The considered Holder spaces are defined with respect to a possibly non-euclidean metric related to the operator A. Schauder estimates are deduced by sharp L^1 − C^{\alpha} estimates on the spatial derivatives of the associated diffusion semigroup which are of independent interest. In the proof we also use probabilistic techniques.

Global Schauder estimates for a class of degenerate Kolmogorov equations

PRIOLA, Enrico
2009-01-01

Abstract

We consider a class of possibly degenerate second order elliptic operators A on R^n. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Holder spaces both for elliptic equations and for parabolic Cauchy problems involving A. The considered Holder spaces are defined with respect to a possibly non-euclidean metric related to the operator A. Schauder estimates are deduced by sharp L^1 − C^{\alpha} estimates on the spatial derivatives of the associated diffusion semigroup which are of independent interest. In the proof we also use probabilistic techniques.
2009
194
117
153
http://arxiv.org/pdf/0705.2810v1
http://journals.impan.gov.pl/sm/
Schauder estimates; degenerate elliptic and parabolic equations; Kolmogorov equations; hypoelliptic operators; diffusion semigroups
E. Priola
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/71663
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