Resumen de Cones, rectifiability, and singular integral operators

Damian Dabrowski

• Let \muμ be a Radon measure on \mathbb{R}^dR d . We define and study conical energies \mathcal{E}_{\mu,p}(x,V,\alpha)E μ,p (x,V,α), which quantify the portion of \muμ lying in the cone with vertex x\in\mathbb{R}^dx∈R d , direction V\in G(d,d-n)V∈G(d,d−n), and aperture \alpha\in (0,1)α∈(0,1). We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that \muμ has polynomial growth, we give a sufficient condition for L^2(\mu)L 2 (μ)-boundedness of singular integral operators with smooth odd kernels of convolution type.