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.