Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

Marco Bramanti, Giovanni Cupini, Ermanno Lanconelli, Enrico Priola

Published 2012-09-03Version 1

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leq c{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} for i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leq c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any $u\in C_{0}^{\infty}(S_{T}),$ where $S_{T}$ is the strip $\mathbb{R}^{N}\times[-T,T]$, $T$ small, and $L$ is the Kolmogorov-Fokker-Planck operator% [L\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x,t) \partial_{x_{i}x_{j}}% ^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}-\partial_{t}%] with uniformly continuous and bounded $a_{ij}$'s.