Семинар за анализу, 11. новембар 2021.

• 05. Новембар, 2021
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Наредни састанак Семинара биће одржан онлајн у четвртак, 11. новембра 2021. са почетком у 15 часова преко Zoom платформе.

Предавач: Oleg Ivrii, Универзитет у Тел Авиву

Наслов предавања: ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM

Апстракт:
Many questions in geometric function theory can be reduced to how the harmonic measure is distributed on the boundary of a simply-connected domain. One can alternatively encode this information in analytic terms using the integral means spectrum. For a regular fractal such as a hyperbolic Julia set or a Fuchsian group, the asymptotic variance describes the second derivative of the integral means spectrum at the origin and appears in the Makarov’s law of the iterated logarithm. In the infinitesimal setting, the work of McMullen relates the asymptotic variance to dimensions of quasicircles and the Weil-Petersson metric.

In view of a theorem of Smirnov, which states that the dimension of a k-quasicircle is at most 1+k^2, we conjectured that the maximal asymptotic variance of the Beurling transform was 1 and showed that the optimal value was between 0.87913 and 1. For the lower bound, we found polynomial Julia sets of dimension 1 + 0.87913 k^2 for k small. The key ingredient was to find a good estimate on the distortion k. After our paper was published, Håkan Hedemalm surprised us by showing that the optimal value was less 1, which suggested that Smirnov’s bound was not sharp after all. This suspicion was later confirmed using fractal approximation techniques.

This is joint work with Kari Astala, István Prause and Antti Perälä.

Линк за приступ предавању:
https://us02web.zoom.us/j/6636215428?pwd=bllJSTUvZ0E5MjhkMkxYa3RYYzZpQT09
Meeting ID: 663 621 5428
Passcode: 7h6KR1