Cones, rectifiability, and singular integral operators
Damian Dąbrowski · Revista Matemática Iberoamericana · 2021
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.