Annales Academić Scientiarum Fennicć
Mathematica
Volumen 33, 2008, 585-596

PLANAR BEURLING TRANSFORM AND GRUNSKY INEQUALITIES

Hĺkan Hedenmalm

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

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 L^p(C) for 1 < p < +\infty (with so far unknown norm), an 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_\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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