首页 | 本学科首页   官方微博 | 高级检索  
    按 检索      


coefficients
Authors:Marco Bramanti  Luca Brandolini
Institution:Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy ; Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende, 87036 Rende (CS), Italy
Abstract:Let $X_1,X_2,\ldots,X_q$ be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain $\Omega\subset\mathbb{R}^n$ ($n>q$). We consider the differential operator

\begin{equation*}\mathcal{L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*}

where the coefficients $a_{ij}(x)$ are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

\begin{equation*}\mu|\xi|^2\leq\sum _{i,j=1}^qa_{ij}(x)\xi _i\xi _j\leq\mu^{-1}|\xi|^2 \end{equation*}

for a.e. $x\in\Omega$, every $\xi\in\mathbb{R}^q$, some constant $\mu$. Moreover, we assume that the coefficients $a_{ij}$ belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields $X_1,X_2,\ldots,X_q$. We prove the following local $\mathcal{L}^p$-estimate:

\begin{equation*}\left\|X_iX_jf\right\|_{\mathcal{L}^p(\Omega')}\leq c\left\{\left\|\mathcal{L}f\right\|_{\mathcal{L}^p(\Omega)}+\left\|f\right \|_{\mathcal{L}^p(\Omega)}\right\} \end{equation*}

for every $\Omega'\subset\subset\Omega$, $1<p<\infty$. We also prove the local Hölder continuity for solutions to $\mathcal{L}f=g$ for any $g\in\mathcal{L}^p$ with $p$ large enough. Finally, we prove $\mathcal{L}^p$-estimates for higher order derivatives of $f$, whenever $g$ and the coefficients $a_{ij}$ are more regular.

Keywords:Hypoelliptic operators  discontinuous coefficients
点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
点击此处可从《Transactions of the American Mathematical Society》下载免费的PDF全文
设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号