Revista Matemática Iberoamericana


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Published online first: 2021-08-10
DOI: 10.4171/RMI/1301

Cones, rectifiability, and singular integral operators

Damian Dąbrowski[1]

(1) Universitat Autònoma de Barcelona, Spain

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.

Keywords: Rectifiability, cone, singular integral operators, conical density, big pieces of Lipschitz graphs

Dąbrowski Damian: Cones, rectifiability, and singular integral operators. Rev. Mat. Iberoam. Electronically published on August 10, 2021. doi: 10.4171/RMI/1301 (to appear in print)