Asymptotic variance of the Beurling transform

TL;DR

This paper investigates the asymptotic variance of the Beurling transform, establishing bounds between 0.87913 and 1, and introduces new methods for estimating this variance using polynomial Julia sets.

Contribution

It provides the first non-trivial bounds for the asymptotic variance of the Beurling transform and introduces a novel fractal approximation scheme for its evaluation.

Findings

Proved that the asymptotic variance $ ext{Sigma}^2$ lies between 0.87913 and 1.

Constructed polynomial Julia sets demonstrating the lower bound of $ ext{Sigma}^2$.

Developed a new approximation method for $ ext{Sigma}^2$ using nearly circular polynomial Julia sets.

Abstract

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a $k$-quasicircle is at most $1+k^2$, it is natural to expect that the maximum asymptotic variance $\Sigma^2 = 1$. In this paper, we prove $0.87913 \le \Sigma^2 \le 1$. For the lower bound, we give examples of polynomial Julia sets which are $k$-quasicircles with dimensions $1+ 0.87913 \, k^2$ for $k$ small, thereby showing that $\Sigma^2 \ge 0.87913$. The key ingredient in this construction is a good estimate for the distortion $k$, which is better than the one given by a straightforward use of the $\lambda$-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $\Sigma^2$ in terms of nearly circular polynomial Julia sets.