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

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dc.creatorBramanti, M.
dc.creatorCupini, G.
dc.creatorLanconelli, E.
dc.creatorPriola, E.
dc.date2008-07-25
dc.date.accessioned2026-07-07T09:52:52Z
dc.date.available2026-07-07T09:52:52Z
dc.descriptionWe 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 $(a_{ij}) ,(b_{ij}) $ are constant matrices, $(a_{ij}) $ is symmetric positive definite on $\mathbb{R} ^{p_{0}}$ ($p_{0}\leq N$), and $(b_{ij}) $ is such that $\mathcal{A}$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% \[ \Vert \partial_{x_{i}x_{j}}^{2}u\Vert_{L^{p}(\mathbb{R}% ^{N})}\leq c\{\Vert \mathcal{A}u\Vert_{L^{p}(\mathbb{R}^{N})}+\Vert u\Vert_{L^{p}(\mathbb{R}% ^{N})}\} \text{for}i,j=1,2,...,p_{0}% \] and corresponding weak (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue $L^{p}$ spaces, for complete Hörmander'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:% \[ \Vert \partial_{x_{i}x_{j}}^{2}u\Vert_{L^{p}(S)}\leq c\Vert Lu\Vert_{L^{p}(S)}% \] for any $u\in C_{0}^{\infty}(S) ,$ where $S$ is the strip $\mathbb{R}^{N}\times[ -1,1] $ and $L$ is the Kolmogorov-Fokker-Planck operator $\mathcal{A}-\partial_{t}.$
dc.identifierhttps://arxiv.org/abs/0807.4020
dc.identifierhttp://arxiv.org/abs/0807.4020
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/165747
dc.subjectAnalysis of PDEs
dc.subject35H10 (Prrimary) ;5, 35K70, 42B20 (Secondary)
dc.titleGlobal $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators
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