Bloch functions and asymptotic tail variance

TL;DR

This paper establishes an optimal exponential integrability result for Bergman projections, linking it to the asymptotic tail variance in Bloch space, with applications to quasiconformal Teichmüller theory and integral means spectrum estimates.

Contribution

It provides the first optimal exponential square integrability theorem for Bergman projections of bounded functions, connecting tail variance to Bloch space seminorms and applying it to quasiconformal mappings.

Findings

Asymptotic tail variance of bounded functions is at most 1.

Derived an upper bound for the integral means spectrum of certain quasiconformal maps.

Connected tail variance with Bloch space seminorms and Teichmüller theory applications.

Abstract

We obtain an optimal exponential square integrability theorem for the Bergman projection of a function bounded by 1 in modulus. This is interpreted as the statement that the asymptotic tail variance of such a function is at most 1. The asymptotic tail variance defines a seminorm on the Bloch space. We apply the main result to quasiconformal Teichm\"uller theory, and obtain an estimate of the integral means spectrum of k-quasiconformal mappings that are conformal in the exterior disk: $B(k,t)\le\frac14k^2|t|^2(1+7k)^2$. This is conjectured asymptotically sharp as k tends to 0, by Prause and Smirnov (2011).