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# memo_has_moved_text();Rectifiable measures, square functions involving densities, and the Cauchy transform

Xavier Tolsa

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

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. A compactness argument
• Chapter 4. The dyadic lattice of cells with small boundaries
• Chapter 5. The Main Lemma
• Chapter 6. The stopping cells for the proofof Main Lemma 5.1
• Chapter 7. The measure $\tilde \mu$ and some estimatesabout its flatness
• Chapter 8. The measure of the cells from $\BCF$, $\LD$, $\BSD$and $\BCG$
• Chapter 9. The new families of cells $\bsb$, $\nterm$, $\ngood$, $\nqgood$ and $\nreg$
• Chapter 10. The approximating curves $\Gamma ^k$
• Chapter 11. The small measure $\tilde \mu$ of the cells from $\bsb$
• Chapter 12. The approximating measure $\nu ^k$ on $\Gamma ^k_ex$
• Chapter 13. Square function estimates for $\nu ^k$
• Chapter 14. The good measure $\sigma ^k$ on $\Gamma ^k$
• Chapter 15. The $L^2(\sigma ^k)$ norm of the density of $\nu ^k$ with respect to $\sigma ^k$
• Chapter 16. The end of the proof of the Main Lemma 5.1
• Chapter 17. Proof of Theorem 1.3: Boundedness of $T_\mu$ implies boundedness of the Cauchy transform
• Chapter 18. Some Calderón-Zygmund theory for $T_\mu$
• Chapter 19. Proof of Theorem 1.3: 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 is a Radon measure in satisfyingfor -a.e. , then is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set with finite -dimensional Hausdorff measure is rectifiable if and only ifH^1x2EThe second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform . Assuming that has linear growth, it is proved that is bounded in if and only iffor every square .