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

About this Title

Xavier Tolsa, ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Catalonia, and Dept. of Mathematics and BGSMath, Universitat Autònoma de Barcelona, 01893 Bellaterra, Catalonia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 245, Number 1158
ISBNs: 978-1-4704-2252-3 (print); 978-1-4704-3605-6 (online)
DOI: https://doi.org/10.1090/memo/1158
Published electronically: July 15, 2016
Keywords: Rectifiability, square function, density, Cauchy transform
MSC: Primary 28A75, 42B20; Secondary 28A78

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Table of Contents

Chapters

• 1. Introduction
• 2. Preliminaries
• 3. A compactness argument
• 4. The dyadic lattice of cells with small boundaries
• 5. The Main Lemma
• 6. The stopping cells for the proof of Main Lemma
• 7. The measure $\tilde \mu$ and some estimates about its flatness
• 8. The measure of the cells from ${\mathsf {BCF}}$, ${\mathsf {LD}}$, ${\mathsf {BS}\Delta }$ and ${\mathsf {BCG}}$
• 9. The new families of cells ${{\mathsf {BS}\beta }}$, ${\mathsf {NTerm}}$, ${\mathsf {NGood}}$, ${\mathsf {NQgood}}$ and ${\mathsf {NReg}}$
• 10. The approximating curves $\Gamma ^k$
• 11. The small measure $\tilde \mu$ of the cells from ${{\mathsf {BS}\beta }}$
• 12. The approximating measure $\nu ^k$ on $\Gamma ^k_{ex}$
• 13. Square function estimates for $\nu ^k$
• 14. The good measure $\sigma ^k$ on $\Gamma ^k$
• 15. The $L^2(\sigma ^k)$ norm of the density of $\nu ^k$ with respect to $\sigma ^k$
• 16. The end of the proof of the Main Lemma
• 17. Proof of Theorem : Boundedness of $T_\mu$ implies boundedness of the Cauchy transform
• 18. Some Calderón-Zygmund theory for $T_\mu$
• 19. Proof of Theorem : Boundedness of the Cauchy transform implies boundedness of $T_\mu$

Abstract

This monograph 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 \begin{equation*} \limsup _{r\to 0} \frac {\mu (B(x,r))}{r}>0\quad \mathrm {and}\quad \int _0^1\left |\frac {\mu (B(x,r))}{r} - \frac {\mu (B(x,2r))}{2r}\right |^2\,\frac {dr}r< \infty \end{equation*} 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 $\mathcal {H}^1$ is rectifiable if and only if \[ \int _0^1\left |\frac {\mathcal {H}^1(E\cap B(x,r))}{r} - \frac {\mathcal {H}^1(E\cap B(x,2r))}{2r}\right |^2\,\frac {dr}r< \infty \qquad \mathrm {for}\; \mathcal {H}^1\mathrm {-a.e.}\ x\in E.\]

The second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform $\mathcal {C}_\mu f(z) = \int \frac 1{z-\xi }\,f(\xi )\,d\mu (\xi )$. Assuming that $\mu$ has linear growth, it is proved that $\mathcal {C}_\mu$ is bounded in $L^2(\mu )$ if and only if \[ \displaystyle \int _{z\in Q}\int _0^\infty \left |\frac {\mu (Q\cap B(z,r))}{r} - \frac {\mu (Q\cap B(z,2r))}{2r}\right |^2\,\frac {dr}r\,d\mu (z)\leq c\,\mu (Q)\] for every square $Q\subset \mathbb {C}$.