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We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp estimates on the spatial derivatives of the associated diffusion semigroup which are of independent interest. In the proof we also use probabilistic techniques.
Enrico Priola. "Global Schauder estimates for a class of degenerate Kolmogorov equations." Studia Mathematica 194.2 (2009): 117-153. <http://eudml.org/doc/284768>.
@article{EnricoPriola2009, abstract = {We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp $L^\{∞\}-C^\{θ\}$ estimates on the spatial derivatives of the associated diffusion semigroup which are of independent interest. In the proof we also use probabilistic techniques.}, author = {Enrico Priola}, journal = {Studia Mathematica}, keywords = {diffusion semigroups; Ornstein-Uhlenbeck type operators; unbounded coefficients; probabilistic techniques}, language = {eng}, number = {2}, pages = {117-153}, title = {Global Schauder estimates for a class of degenerate Kolmogorov equations}, url = {http://eudml.org/doc/284768}, volume = {194}, year = {2009}, }
TY - JOUR AU - Enrico Priola TI - Global Schauder estimates for a class of degenerate Kolmogorov equations JO - Studia Mathematica PY - 2009 VL - 194 IS - 2 SP - 117 EP - 153 AB - We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp $L^{∞}-C^{θ}$ estimates on the spatial derivatives of the associated diffusion semigroup which are of independent interest. In the proof we also use probabilistic techniques. LA - eng KW - diffusion semigroups; Ornstein-Uhlenbeck type operators; unbounded coefficients; probabilistic techniques UR - http://eudml.org/doc/284768 ER -