Planar Beurling transform and Grunsky inequalities
2008 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 33, no 2, 585-596 p.Article in journal (Refereed) Published
In recent work with Baranov, it was explained how to view the classical Grunsky inequalities in terms of an operator identity, involving a transferred Beurling operator induced by the conformal mapping. The main property used is the fact that the Beurling operator is unitary on L-2(C). As the Beurling operator is also bounded oil L-p(C) for 1 < p < infinity (with so far unknown norm), all analogous operator identity was found which produces a generalization of the Grunsky inequalities to the L-p setting. Here, we consider weighted Hilbert spaces L-theta(2)(C) with weight, vertical bar z vertical bar(2 theta), for 0 <= theta <= 1, and find that the Beurling operator perturbed by adding a Cauchy-type operator acts unitarily on L-0(2) (C). After transferring to the unit disk D with the conformal mapping, we find a generalization of the Grunsky inequalities ill the setting of the space L-theta(2) (D); this generalization seems to be essentially known, but the formulation is new. As a special case, the generalization of the Grunsky inequalities contains the Prawitz theorem used in a recent paper with Shirnorin. We also mention an application to quasiconformal maps.
Place, publisher, year, edition, pages
2008. Vol. 33, no 2, 585-596 p.
Beurling transform, Grunsky inequalities
IdentifiersURN: urn:nbn:se:kth:diva-17692ISI: 000257661000014ScopusID: 2-s2.0-59949105682OAI: oai:DiVA.org:kth-17692DiVA: diva2:335737
QC 201005252010-08-052010-08-05Bibliographically approved