Cones, rectifiability, and singular integral operators

Abstract

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