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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$L^p$ estimates for nonvariational hypoelliptic operators with $VMO$ coefficients
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by Marco Bramanti and Luca Brandolini PDF
Trans. Amer. Math. Soc. 352 (2000), 781-822 Request permission

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
  • 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
  • Received by editor(s): February 4, 1998
  • Published electronically: September 21, 1999
  • © Copyright 1999 American Mathematical Society
  • 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
  • MathSciNet review: 1608289