##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## A. Volberg

#### Michigan State University

## Story of analytic capacity and problems of Painleve, Ahlfors and Vitushkin

##### Abstract:

We will tell the story of the solution of several famous problems of Painlev´e, Ahlfors and Vitushkin. \vskip .1in \noindent Essentially, the theory of nonhomogeneous Calder´on-Zygmund (CZ) operators is the topic of the lecture. The main cornerstone of the theory of CZ operators turns out not to be a cornerstone at all. Namely, one can completely get rid of homogeneity of the underlying measure. The striking application of this theory is the solution of the series of problems of Painlev´e, Ahlfors and Vitushkin on the borderline of Harmonic Analysis and Geometric Measure Theory. \vskip .1in \noindent We will show how the ideas from nonhomogeneous CZ theory interplay with Tolsa’s ideas of capacity theory with Calder´on-Zygmund kernels to give Tolsa’s solution of the famous Vitushkin conjecture of semiadditivity of analytic capacity. We also show what changes should be maid if we want to grow the dimension and to prove the semiadditivity of Lipschitz harmonic capacity in $Rn, n > 2$, where the wonderful tool of Menger-Melnikov’s curvature extensively used in $2D$ is “cruelly missing”.

Host: Bill Helton

### November 22, 2004

### 3:00 PM

### AP&M 6438

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