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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$L^{\lowercase{p}}$ estimates for nonvariational hypoelliptic operators with $VMO$ coefficients


Authors: Marco Bramanti and Luca Brandolini
Journal: Trans. Amer. Math. Soc. 352 (2000), 781-822
MSC (1991): Primary 35H05; Secondary 35B45, 35R05, 42B20
Published electronically: September 21, 1999
MathSciNet review: 1608289
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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.


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Additional Information

Marco Bramanti
Affiliation: Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy
Email: marbra@mate.polimi.it

Luca Brandolini
Affiliation: Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende, 87036 Rende (CS), Italy

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02318-1
PII: S 0002-9947(99)02318-1
Keywords: Hypoelliptic operators, discontinuous coefficients
Received by editor(s): February 4, 1998
Published electronically: September 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society