Annales Academię Scientiarum Fennicę
Volumen 33, 2008, 585-596


Håkan Hedenmalm

The Royal Institute of Technology, Department of Mathematics
S-100 44 Stockholm, Sweden; haakanh 'at'

Abstract. 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 L2(C). As the Beurling operator is also bounded on Lp(C) for 1 p Lp setting. Here, we consider weighted Hilbert spaces L\theta2(C)$ with weight |z|2\theta, for 0 \le \theta \le 1, and find that the Beurling operator perturbed by adding a Cauchy-type operator acts unitarily on L\theta2(C). After transferring to the unit disk D with the conformal mapping, we find a generalization of the Grunsky inequalities in the setting of the space L\theta2(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 Shimorin. We also mention an application to quasiconformal maps.

2000 Mathematics Subject Classification: Primary 30C55, 30C60; Secondary 42B10, 42B20, 46E22.

Key words: Beurling transform, Grunsky inequalities.

Reference to this article: H. Hedenmalm: Planar Beurling transform and Grunsky inequalities. Ann. Acad. Sci. Fenn. Math. 33 (2008), 585-596.

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