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September/October 2013 $BMO$ estimates for nonvariational operators with discontinuous coefficients structured on Hörmander's vector fields on Carnot groups
Marco Bramanti, Maria Stella Fanciullo
Adv. Differential Equations 18(9/10): 955-1004 (September/October 2013). DOI: 10.57262/ade/1372777765

Abstract

We consider the class of operators \[ Lu=\sum_{i,j=1}^{q}a_{ij}(x)X_{i}X_{j}u, \] where $X_{1},X_{2},\dots,X_{q}$ are homogeneous left-invariant Hörmander's vector fields on $\mathbb{R}^{N}$ with respect to a structure of Carnot group, $q\leq N,$ the matrix $\{ a_{ij}\} $ is symmetric and uniformly positive on $\mathbb{R}^{q},$ the coefficients $a_{ij} $ belong to $L^{\infty}\cap VLMO_{loc}( \Omega) $ ("vanishing logarithmic mean oscillation") with respect to the distance induced by the vector fields (in particular, they can be discontinuous), and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$. We prove local estimates in $BMO_{loc}\cap L^{p}$ of the following kind: \begin{align*} & \Vert X_{i}X_{j}u\Vert _{BMO_{loc}^{p}( \Omega^{\prime }) }+\Vert X_{i}u\Vert _{BMO_{loc}^{p}( \Omega ^{\prime}) } \\ & \leq c\big\{ \Vert Lu\Vert _{BMO_{loc}^{p}( \Omega) }+\Vert u\Vert _{BMO_{loc}^{p}( \Omega) }\big\} \end{align*} for any $\Omega^{\prime}\Subset\Omega$, $1 < p < \infty$. Even in the uniformly elliptic case $X_{i}=\partial_{x_{i}}$, $q=N$ our estimates improve the known results.

Citation

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Marco Bramanti. Maria Stella Fanciullo. "$BMO$ estimates for nonvariational operators with discontinuous coefficients structured on Hörmander's vector fields on Carnot groups." Adv. Differential Equations 18 (9/10) 955 - 1004, September/October 2013. https://doi.org/10.57262/ade/1372777765

Information

Published: September/October 2013
First available in Project Euclid: 2 July 2013

zbMATH: 1292.35064
MathSciNet: MR3100057
Digital Object Identifier: 10.57262/ade/1372777765

Subjects:
Primary: 35B45 , 35H10 , 42B20 , 43A80

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.18 • No. 9/10 • September/October 2013
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