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

### Global L^p estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

#### Abstract

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind $\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\left( x\right) \partial _{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}$ where $\left( a_{ij}\right)$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $\left( b_{ij}\right)$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}\left( x_{0}\right)$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates (\$1
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2013
M. Bramanti; G. Cupini; E. Lanconelli; E.Priola
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/133095
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