Revista Matemática Iberoamericana


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Volume 21, Issue 2, 2005, pp. 511–556
DOI: 10.4171/RMI/428

Published online: 2005-08-31

Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs

Marco Bramanti[1] and Luca Brandolini[2]

(1) Politecnico di Milano, Italy
(2) Università di Bergamo, Dalmine, Italy

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.

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

Bramanti Marco, Brandolini Luca: Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs. Rev. Mat. Iberoamericana 21 (2005), 511-556. doi: 10.4171/RMI/428