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- W2952203837 abstract "We consider a class of degenerate Ornstein-Uhlenbeck operators in $mathbb{R}^{N}$, of the kind [ mathcal{A}equivsum_{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}uVert_{L^{p}(mathbb{R}% ^{N})}leq c{Vert mathcal{A}uVert_{L^{p}(mathbb{R}^{N})}+Vert uVert_{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 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:% [ Vert partial_{x_{i}x_{j}}^{2}uVert_{L^{p}(S)}leq cVert LuVert_{L^{p}(S)}% ] for any $uin C_{0}^{infty}(S) ,$ where $S$ is the strip $mathbb{R}^{N}times[ -1,1] $ and $L$ is the Kolmogorov-Fokker-Planck operator $mathcal{A}-partial_{t}.$" @default.
- W2952203837 created "2019-06-27" @default.
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- W2952203837 date "2008-07-25" @default.
- W2952203837 modified "2023-09-27" @default.
- W2952203837 title "Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators" @default.
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