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

Author: Xavier Tolsa; American Mathematical Society, Providence, Rhode Island : American Mathematical Society, 2017. ©2016 Memoirs of the American Mathematical Society, no. 1158. eBook : Document : EnglishView all editions and formats "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 if Ĥ1x2E The 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 ." --Publisher website.  Read more... (not yet rated) 0 with reviews - Be the first.

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Genre/Form: Electronic books Document, Internet resource Internet Resource, Computer File Xavier Tolsa; American Mathematical Society, Find more information about: Xavier Tolsa 9781470436056 1470436051 965547797 “Volume 245, Number 1158 (third of 6 numbers), January 2017” 1 online resource (v, 130 pages) : illustrations. 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 $T_\mu$ Memoirs of the American Mathematical Society, no. 1158. Xavier Tolsa.

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Volume 245, number 1158 (third of 6 numbers), January 2017.  Read more...

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