Abstract

Given a general Hörmander’s operator $P={\sum }_{j=1}^m{X}_j^2+Y+b$ in an open set $\Omega \subset {ℝ}^n$, where $Y,X_1,\dots ,X_m$ are smooth real vector fields in $\Omega$, $b\in C^{\infty }\left(\Omega \right)$, and given also an open, relatively compact set ${\Omega }_0$ with ${\overline{\Omega }}_0\subset \Omega$, and $s\in ℝ$, $s\ge 1$, such that the coefficients of $P$ are in $G^s\left({\Omega }_0\right)$ and $P$ satisfies a $\frac{1}{p}$-Sobolev estimate in ${\Omega }_0$, our aim is to establish local estimates reflecting local domination of ordinary derivatives by powers of $P$, in ${\Omega }_0$. As an application, we give a direct proof of the $G^{2ps}\left({\Omega }_0\right)$-regularity of any $G^s\left({\Omega }_0\right)$-vector of $P$.

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Mathematical Subject Classification 2010

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