Global L p estimates for degenerate Ornstein‐Uhlenbeck operators with variable coefficients

We 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\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \mathcal {A}\equiv \sum _{i,j=1}^{p_{0}}a_{ij}(x) \partial _{x_{i}x_{j}}^{2}+\sum _{i,j=1}^{N}b_{ij}x_{i}\partial _{x_{j}} $$ \end{document} where ( a ij ) is symmetric uniformly positive definite on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{p_{0}}$\end{document} ( p 0 ≤ N ), with uniformly continuous and bounded entries, and ( b ij ) 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 a ij ( x 0 ) is hypoelliptic. For this class of operators we prove global L p estimates (1 < p < ∞) of the kind:\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \big \Vert \partial _{x_{i}x_{j}}^{2}u\big \Vert _{L^{p}\left( \mathbb {R} ^{N}\right) }\le c\left\lbrace \left\Vert \mathcal {A}u\right\Vert _{L^{p}\left( \mathbb {R}^{N}\right) }+\left\Vert u\right\Vert _{L^{p}\left( \mathbb {R} ^{N}\right) }\right\rbrace \quad \mbox{for}\quad i,j=1,2,\ldots ,p_{0}. $$ \end{document} We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \big \Vert \partial _{x_{i}x_{j}}^{2}u\big \Vert _{L^{p}\left( S_{T}\right) }\le c\left\lbrace \left\Vert Lu\right\Vert _{L^{p}\left( S_{T}\right) }+\left\Vert u\right\Vert _{L^{p}\left( S_{T}\right) }\right\rbrace \quad \mbox{for}\quad i,j=1,2,\ldots ,p_{0}, $$ \end{document} for any \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$u\in C_{0}^{\infty }\!\left( S_{T}\right) ,$\end{document} where S T 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\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ L\equiv \sum _{i,j=1}^{p_{0}}a_{ij}(x,t) \partial _{x_{i}x_{j}} ^{2}+\sum _{i,j=1}^{N}b_{ij}x_{i}\partial _{x_{j}}-\partial _{t} $$ \end{document} with uniformly continuous and bounded a ij 's.