Published Paper

Inserted: 19 nov 2008
Last Updated: 15 oct 2010

Journal: Math. Z.
Volume: 266
Pages: 789-816
Year: 2010

Abstract:

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[ \mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} +\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}} \] where $\left( a_{ij}\right) ,\left( b_{ij}\right) $ are constant matrices, $\left( a_{ij}\right) $ is symmetric positive definite on $\mathbb{R} ^{p_{0}}$ ($p_{0}\leq N$), and $\left( b_{ij}\right) $ is such that $\mathcal{A}$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind: \[ \left\Vert \partial_{x_{i}x_{j}}^{2}u\right\Vert _{L^{p}\left( \mathbb{R} ^{N}\right) }\leq c\left\{ \left\Vert \mathcal{A}u\right\Vert _{L^{p}\left( \mathbb{R}^{N}\right) }+\left\Vert u\right\Vert _{L^{p}\left( \mathbb{R} ^{N}\right) }\right\} \text{ for }i,j=1,2,...,p_{0} \] and corresponding weak (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue $L^{p}$ spaces, for complete H\"{o}rmander's operators \[ \sum X_{i}^{2}+X_{0}, \] proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own: \[ \left\Vert \partial_{x_{i}x_{j}}^{2}u\right\Vert _{L^{p}\left( S\right) }\leq c\left\Vert Lu\right\Vert _{L^{p}\left( S\right) } \] for any $u\in C_{0}^{\infty}\left( S\right) ,$ where $S$ is the strip $\mathbb{R}^{N}\times\left[ -1,1\right] $ and $L$ is the Kolmogorov-Fokker-Planck operator $\mathcal{A}-\partial_{t}.$ To get this estimate we crucially use the left translation invariance of $L$ on a Lie group $\mathcal{K}$ in $\mathbb{R}^{N+1}$ and some results on singular integrals on nonhomogeneous spaces recently proved by the first author.

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

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