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

Xavier Tolsa, ICREA, Barcelona, Spain, and Universitat Autonoma de Barcelona, Spain.

• IntroductionPreliminariesA compactness argumentThe dyadic lattice of cells with small boundariesThe Main LemmaThe stopping cells for the proof of Main Lemma 5.1The measure $\tilde\mu$ and some estimates about its flatness}The measure of the cells from $\mathsf{BCF}$, $\mathsf{LD}$, $\mathsf{BSD}$ and $\mathsf{BCG}$The new families of cells $\mathsf{BS}\beta$, $\mathsf{NTerm}$, $\mathsf{NGood}$, $\mathsf{NQgood}$ and $\mathsf{NReg}$The approximating curves $\Gamma^k$The small measure $\tilde\mu$ of the cells from $\mathsf{BS}\beta$The approximating measure $\nu^k$ on $\Gamma^k_{ex}$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 5.1Proof of Theorem 1.3: Boundedness of $T_\mu$ implies boundedness of the Cauchy transformSome Calderon-Zygmund theory for $T_\mu$Proof of Theorem 1.3: Boundedness of the Cauchy transform implies boundedness of $T_\mu$Bibliography.