Global Schauder estimates for a class of degenerate Kolmogorov equations

Volume 194 / 2009

Enrico Priola Studia Mathematica 194 (2009), 117-153 MSC: 35K65, 35B65, 35J70, 47D07, 60J35. DOI: 10.4064/sm194-2-2


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


  • Enrico PriolaDipartimento di Matematica
    Università di Torino
    via Carlo Alberto 10
    10123 Torino, Italy

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