Characterizations for quasiconformal weighted Bergman spaces

Abstract

We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces $A_\alpha^p(\mathbb D)$. For $p>0$ and $\alpha>-1$, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivative. These include quasiconformal analogues of characterizations established by Karafyllia and Karamanlis for conformal mappings. Moreover, for $-1<\alpha\leq 0$, we obtain an additional equivalent characterization in terms of the Hausdorff measure of level sets. The power $\alpha$ is sharp: for every $\alpha>0$, we construct a bounded $K_\alpha$-quasiconformal mapping for which the weighted level-set length integral with exponent $\alpha+2$ diverges.