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July, 2005 Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs
Marco Bramanti, Luca Brandolini
Rev. Mat. Iberoamericana 21(2): 511-556 (July, 2005).

Abstract

Let us consider the class of ``nonvariational uniformly hypoelliptic operators'': $$ Lu\equiv\sum_{i,j=1}^{q}a_{ij} (x) X_{i} X_{j} u $$ where: $X_1,X_2,\ldots,X_q$ is a system of H\"ormander vector fields in $\mathbb{R}^{n}$ ($n>q$), $\{a_{ij}\}$ is a $q\times q$ uniformly elliptic matrix, and the functions $a_{ij} (x)$ are continuous, with a suitable control on the modulus of continuity. We prove that: $$ \| X_{i} X_{j} u \|_{BMO(\Omega^{\prime})} \leq c \left\{\left\|Lu\right\|_{BMO(\Omega)} + \left\| u\right\|_{BMO(\Omega)} \right\} $$ for domains $\Omega^{\prime}\subset\subset\Omega$ that are regular in a suitable sense. Moreover, the space $BMO$ in the above estimate can be replaced with a scale of spaces of the kind studied by Spanne. To get this estimate, several results are proved, regarding singular and fractional integrals on general spaces of homogeneous type, in relation with function spaces of $BMO$ type.

Citation

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Marco Bramanti. Luca Brandolini. "Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs." Rev. Mat. Iberoamericana 21 (2) 511 - 556, July, 2005.

Information

Published: July, 2005
First available in Project Euclid: 11 August 2005

zbMATH: 1082.35060
MathSciNet: MR2174915

Subjects:
Primary: 35H10
Secondary: 42B20 , 43A85

Keywords: BMO , hypoelliptic operators , singular integrals , spaces of homogeneous type

Rights: Copyright © 2005 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.21 • No. 2 • July, 2005
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