A note on the boundedness of Riesz transform for some subelliptic operators
A note on the boundedness of Riesz transform for some subelliptic operators
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DOI：
10.1093/imrn/rnr271
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发表时间：
2011-05
期刊：
影响因子：
--
通讯作者：
Fabrice Baudoin;N. Garofalo
中科院分区：
文献类型：
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作者：
Fabrice Baudoin;N. Garofalo
令 $\M$ 为光滑连通非紧流形，具有光滑测度 $\mu$ 和光滑局部亚椭圆扩散算子 $L$，满足 $L1=0$，且相对于 $\mu$ 对称。我们证明，如果 $L$ 在非负曲率参数 $\rho_1$ 下满足下面 \eqref{CD} 中的广义曲率不等式，则对于每个 $p>1$，Riesz 变换在 $L^p (\bM)$ 内有界，即 \[\| \sqrt{\Gamma((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM), \] 其中 $\Gamma$ 是与 $L$ 关联的 \textit{carr\'e du champ}。我们的结果特别适用于水平 Tanaka-Webster Ricci 曲率非负的所有 Sasakian 流形、所有具有第二步的卡诺群以及 Riemannian 流形上主丛的宽子类（Ricci 曲率非负）。
Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. We show that if $L$ satisfies, with a non negative curvature parameter $\rho_1$, the generalized curvature inequality in \eqref{CD} below, then the Riesz transform is bounded in $L^p (\bM)$ for every $p>1$, that is \[\| \sqrt{\Gamma((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM), \] where $\Gamma$ is the \textit{carr\'e du champ} associated to $L$. Our results apply in particular to all Sasakian manifolds whose horizontal Tanaka-Webster Ricci curvature is nonnegative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.