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 $V\in G(d,dn)$, 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.14432
 Bibcode:
 2020arXiv200614432D
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 28A75 (Primary) 28A78;
 42B20 (Secondary)
 EPrint:
 40 pages