Dirichlet symbols and the nonlinear wave equation

Abstract

We study the operator symbols of Dirichlet type introduced by Hedenmalm and Shimorin (2020), in connection with a given contraction on $L^2$ of the unit disk. They are always holomorphic functions on the bidisk. Such Dirichlet symbols associated with the Grunsky operator of a univalent function on the disk or exterior disk are of particular significance. From the work of Hedenmalm and Shimorin, we know they are characterized as solutions of a certain nonlinear wave equation. We perform a local analysis of such symbols near the diagonal on the bidisk, and in so doing, we provide alternative chart coordinates for the infinite-dimensional manifolds of univalent functions of the (exterior) disk. Those coordinates allow us to characterize $\log\psi'$ for $\psi$ in the class $\Sigma$ of normalized univalent functions without explicitly touching the univalence property. Moreover, that manifold extends the universal Teichm\"uller space of Lipman Bers beyond the quasicircle boundary setting, allowing for even more fractality. The fractality of harmonic measure for the domain associated with the given univalent function can be studied in terms of the asymptotic variance introduced by McMullen (2008). The asymptotic variance captures the $L^2$ average amplitude of the nonlinearity. We here introduce the new concept of Schwarzian asymptotic variance, which measures the average amplitude of the Schwarzian derivative in place of the nonlinearity. For this new Schwarzian asymptotic variance, we find that the effective average amplitude of $(1-|z|^2)^4|\Sop(\vp)|^2$ on the disk in the hyperbolic metric sense is at most $9.07735\ldots$, considerably smaller than the maximum amplitude of $36$. Here, $\Sop(\vp)$ is the Schwarzian derivative of $\varphi\in\mathscr{S}$, and the analogous statement is valid for $\psi\in\Sigma$ as well.