Rectifiable measures, square functions involving densities, and the Cauchy transform

TL;DR

This paper characterizes rectifiable measures and sets using square functions involving densities and explores the boundedness of the Cauchy transform in relation to these square functions, providing new criteria for rectifiability and boundedness.

Contribution

It provides a new characterization of rectifiable measures and sets via square functions and establishes a criterion for the boundedness of the Cauchy transform based on these functions.

Findings

Rectifiability characterized by finiteness of a specific square function.

Boundedness of the Cauchy transform in L^2() linked to a square function condition.

Equivalent conditions for rectifiability and Cauchy transform boundedness established.

Abstract

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad \int_0^1\left|\frac{\mu(B(x,r))}{r} - \frac{\mu(B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty$$ for $\mu$-a.e. $x\in\mathbb R^d$, then $\mu$ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set $E\subset\mathbb R^d$ with finite $1$-dimensional Hausdorff measure $H^1$ is rectifiable if and only $$\int_0^1\left|\frac{H^1(E\cap B(x,r))}{r} - \frac{H^1(E\cap B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty \quad\mbox{ for $H^1$-a.e. $x\in E$.}$$ The second result of the paper deals with the relationship between a similar square function in the complex plane and the Cauchy transform $C_\mu f(z) = \int \frac1{z-\xi}\,f(\xi)\,d\mu(\xi)$. Suppose that $\mu$ has linear growth, that is, $\mu(B(z,r))\leq c\,r$ for all $z\in\mathbb C$ and all $r>0$. It is proved that $C_\mu$ is bounded in $L^2(\mu)$ if and only if $$ \int_{z\in Q}\int_0^\infty\left|\frac{\mu(Q\cap B(z,r))}{r} - \frac{\mu(Q\cap B(z,2r))}{2r}\right|^2\,\frac{dr}r\,d\mu(z)\leq c\,\mu(Q) \quad\mbox{ for every square $Q\subset\mathbb C$.} $$