Håkan Hedenmalm:Bloch functions and asymptotic tail variance

ABSTRACT: We prove a sharp exponential square integrability theorem for Bergman projections of bounded functions on the unit disk. The theorem is parallel to Beurling's theorem from 1933, but it does not imply Beurling's theorem and Beurling's theorem does not imply it. The theorem has applications to the quasiconformal integral means spectrum, and gives asymptotically as \(k\to 0\) the conjectured bounds. It also gives an estimate for the dimension of quasicircles: \(D(k)\leq 1+k^2+O(k^3)\) which is barely weaker than Smirnov's theorem (Acta Math 2010).