Global Lp estimates for degenerate Ornstein‐Uhlenbeck operators with variable coefficients

Abstract

AbstractWe consider a class of degenerate Ornstein‐Uhlenbeck operators in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{N}\!$\end{document}, of the kind where (aij) is symmetric uniformly positive definite on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{p_{0}}$\end{document} (p0 ≤ N), with uniformly continuous and bounded entries, and (bij) is a constant matrix such that the frozen operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {A}_{x_{0}}$\end{document} corresponding to aij(x0) is hypoelliptic. For this class of operators we prove global Lp estimates (1 < p < ∞) of the kind: We obtain the previous estimates as a byproduct of the following one, which is of interest in its own: for any \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$u\in C_{0}^{\infty }\!\left( S_{T}\right) ,$\end{document} where ST is the strip \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{N}\times \left[-T,T\right]$\end{document}, T small, and L is the Kolmogorov‐Fokker‐Planck operator with uniformly continuous and bounded aij's.