DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 657-664

Pertti Mattila

Title: Rectifiability, Analytic Capacity, and Singular Integrals

This is a survey of some interplay between geometric measure theory (rectifiability), complex analysis (analytic capacity) and harmonic analysis (singular integrals). Vaguely, it deals with the following three principles: \roster \item"1." The analytic capacity of a $1$-dimensional compact subset of the complex plane $\C$ is zero if and only if $E$ is purely unrectifiable. \item"2." The analytic capacity of a $1$-dimensional compact subset $E$ of $\C$ is positive if and only if the Cauchy singular integral operator is $L^2$-bounded on a large part of $E$. \item"3." Singular integrals behave nicely on an $m$-dimensional subset $E$ of $\Rn$ if and only if $E$ is in some sense rectifiable. \endroster

1991 Mathematics Subject Classification: Primary 28A75; Secondary 31A05, 42B20.

Keywords and Phrases: Analytic capacity, Cauchy integral, rectifiable set, Menger curvature.

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