Bloch functions and asymptotic tail variance

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,01, 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