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


About this Title

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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Table of Contents


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 .

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