Calculus of Variations and Geometric Measure Theory

M. Bramanti - G. Cupini - E. Lanconelli - E. Priola

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

created by cupini on 03 Sep 2012
modified on 09 May 2014


Published Paper

Inserted: 3 sep 2012
Last Updated: 9 may 2014

Journal: Math. Nachr.
Volume: 286
Pages: 1087 - 1101
Year: 2013
Doi: 10.1002/mana.201200189


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.

Keywords: Ornstein-Uhlenbeck operators, hypoelliptic operators, singular integrals, Global $L^{p}$ estimates, Nondoubling spaces