Global Schauder estimates for a class of degenerate Kolmogorov equations

Enrico Priola ¹

¹ Dipartimento di Matematica Università di Torino via Carlo Alberto 10 10123 Torino, Italy

Studia Mathematica, Tome 194 (2009) no. 2, pp. 117-153

Résumé

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.

Détail

DOI : 10.4064/sm194-2-2

Keywords: consider class possibly degenerate second order elliptic operators cal mathbb class includes hypoelliptic ornstein uhlenbeck type operators having additional first order term unbounded coefficients establish global schauder estimates lder spaces elliptic equations parabolic cauchy problems involving cal lder spaces question defined respect possibly non euclidean metric related operator cal schauder estimates deduced sharp infty c theta estimates spatial derivatives associated diffusion semigroup which independent interest proof probabilistic techniques