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coefficients
Authors:Marco Bramanti   Luca Brandolini
Affiliation: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 $Omegasubsetmathbb{R}^n$ ($n>q$). 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|^2leqsum _{i,j=1}^qa_{ij}(x)xi _ixi _jleqmu^{-1}|xi|^2 end{equation*}

for a.e. $xinOmega$, every $xiinmathbb{R}^q$, 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_jfright|_{mathcal{L}^p(Omega')}leq cleft{left|mathcal{L}fright|_{mathcal{L}^p(Omega)}+left|fright |_{mathcal{L}^p(Omega)}right} end{equation*}

for every $Omega'subsetsubsetOmega$, $1<p<infty$. We also prove the local Hölder continuity for solutions to $mathcal{L}f=g$ for any $ginmathcal{L}^p$ 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
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