Cones, rectifiability, and singular integral operators

Abstract

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