Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform

ISBN-13: 9781470422523 / Angielski

Xavier Tolsa

Opis

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

Spis treści

• Introduction
• Preliminaries
• A compactness argument
• The dyadic lattice of cells with small boundaries
• The Main Lemma
• The stopping cells for the proof of Main Lemma 5.1
• The measure $\tilde\mu$ and some estimates about its flatness}
• The measure of the cells from $\mathsf$, $\mathsf$, $\mathsf$ and $\mathsf$
• The new families of cells $\mathsf\beta$, $\mathsf$, $\mathsf$, $\mathsf$ and $\mathsf$
• The approximating curves $\Gamma^k$
• The small measure $\tilde\mu$ of the cells from $\mathsf\beta$
• The approximating measure $\nu^k$ on $\Gamma^k_$
• Square function estimates for $\nu^k$
• The good measure $\sigma^k$ on $\Gamma^k$
• The $L^2(\sigma^k)$ norm of the density of $\nu^k$ with respect to $\sigma^k$
• The end of the proof of the Main Lemma 5.1
• Proof of Theorem 1.3: Boundedness of $T_\mu$ implies boundedness of the Cauchy transform
• Some Calderon-Zygmund theory for $T_\mu$
• Proof of Theorem 1.3: Boundedness of the Cauchy transform implies boundedness of $T_\mu$
• Bibliography.