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.

40 pages; removed the measurability assumption from Theorem 1.4, many minor improvements; to appear in Rev. Mat. Iberoam

References in corpus (7)

• Conical upper density theorems and porosity of measures
• Boundedness of the square function and rectifiability
• Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ${\mathbb{R}}^d$
• Plenty of big projections imply big pieces of Lipschitz graphs
• On upper conical density results
• Rectifiability of pointwise doubling measures in Hilbert Space
• Two examples related to conical energies

Cited by in corpus (3)

• Two examples related to conical energies
• Cone and paraboloid points of arbitrary subsets of Euclidean space
• Rectifiability; a survey