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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 operatorswe prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:%[|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leqc{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} fori,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of thefollowing one, which is of interest in its own:%[|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leqc{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any $u\inC_{0}^{\infty}(S_{T}),$ where $S_{T}$ is the strip$\mathbb{R}^{N}\times[-T,T]$, $T$ small, and $L$ is theKolmogorov-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 uniformlycontinuous and bounded $a_{ij}$'s.

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