We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0, 1] random variables stays to one side of a given line.

We study real Smirnov functions and investigate a certain ∗-closed subalgebra of the Smirnov class N+ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space Hp ∩ Hp. This leads to a natural analog of the Riesz projection on a certain quotient space of Lp for p ∈ (0, 1). We also study a Herglotz-like integral transform for singular measures on ...

We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala's conjecture that the Hausdorff dimension of a $k$-quasicircle is