The Grunsky operator and the Schatten ideals
Theorem 1. Let \(g\) be a conformal map of the unit disc to the interior of a Jordan curve. The Grunsky operator lies in the \(p\)th Schatten ideal \(\gamma_p\) \((p\geq 1)\) of operators on the Dirichlet space if and only if log \(g'\in B_p\), where \(B_p\) stands for the Besov space. Theorem 2. Let \(J\) be a quasicircle with welding \(h\) and let \(V_h\) be the composition operator \(f\rightarrow f\circ h\). Then the commutator \([V_h, H]\) of \(V_h\) with the Hilbert transform \(H\) lies in \(\gamma_p\) if and only if log \(g'\in B_p\), where \(g\) is the conformal map to the interior.
scientific article; zbMATH DE number 1548295