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
E. Priola
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
