JournalsrmiVol. 21, No. 2pp. 511–556

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

  • Marco Bramanti

    Politecnico di Milano, Italy
  • Luca Brandolini

    Università di Bergamo, Dalmine, Italy
Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs cover
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Abstract

Let us consider the class of ``nonvariational uniformly hypoelliptic operators'':

Lui,j=1qaij(x)XiXjuLu\equiv\sum_{i,j=1}^{q}a_{ij} (x) X_{i} X_{j} u

where: X1,X2,,XqX_1,X_2,\ldots,X_q is a system of H\"ormander vector fields in Rn\mathbb{R}^{n} (n>qn>q), {aij}\{a_{ij}\} is a q×qq\times q uniformly elliptic matrix, and the functions aij(x)a_{ij} (x) are continuous, with a suitable control on the modulus of continuity. We prove that:

XiXjuBMO(Ω)c{LuBMO(Ω)+uBMO(Ω)}\| X_{i} X_{j} u \|_{BMO(\Omega^{\prime})} \leq c \left\{\left\|Lu\right\|_{BMO(\Omega)} + \left\| u\right\|_{BMO(\Omega)} \right\}

for domains ΩΩ\Omega^{\prime}\subset\subset\Omega that are regular in a suitable sense. Moreover, the space BMOBMO 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 BMOBMO type.

Cite this article

Marco Bramanti, Luca Brandolini, Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs. Rev. Mat. Iberoam. 21 (2005), no. 2, pp. 511–556

DOI 10.4171/RMI/428