Bloch functions and asymptotic tail variance

Abstract

Let P denote the Bergman projection on the unit disk D,Pμ(z):=∫Dμ(w)(1−zw¯)2dA(w),z∈D, where dA is normalized area measure. We prove that if |μ(z)|≤1 on D, then the integralIμ(a,r):=∫02πexp⁡{ar4|Pμ(reiθ)|2log⁡11−r2}dθ2π,0<r<1, has the bound Iμ(a,r)≤C(a):=10(1−a)−3/2 for 0<a<1, irrespective of the choice of the function μ. Moreover, for a>1, no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection Pμ (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0<a<π264=0.1542… . We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k-quasiconformal extension, for 0<k<1. The estimate reads, for t∈C and 0<k<1,B(k,t)≤{14k2|t|2(1+7k)2,for|t|≤2k(1+7k)2,k|t|−1(1+7k)2,for|t|≥2k(1+7k)2, which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with |t|≤2/k, we should have B(k,t)=14k2t2.