Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$L^p$ 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
DOI: https://doi.org/10.1090/S0002-9947-99-02318-1
Published electronically: September 21, 1999
MathSciNet review: 1608289
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35H05, 35B45, 35R05, 42B20

Retrieve articles in all journals with MSC (1991): 35H05, 35B45, 35R05, 42B20


Additional Information

Marco Bramanti
Affiliation: Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy
MR Author ID: 289427
Email: marbra@mate.polimi.it

Luca Brandolini
Affiliation: Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende, 87036 Rende (CS), Italy
MR Author ID: 294667
ORCID: 0000-0002-9670-9051

Keywords: Hypoelliptic operators, discontinuous coefficients
Received by editor(s): February 4, 1998
Published electronically: September 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society