Cones, rectifiability, and singular integral operators

TL;DR

This paper introduces conical energies to analyze Radon measures, providing new characterizations of rectifiability and conditions for the boundedness of singular integral operators with smooth kernels.

Contribution

It defines conical energies to characterize rectifiability and big pieces of Lipschitz graphs, and establishes a sufficient condition for singular integral boundedness under polynomial growth.

Findings

Conical energies effectively characterize rectifiability.

Polynomial growth measures lead to bounded singular integrals.

New criteria connect geometric measure theory with harmonic analysis.

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.