Annales Academię Scientiarum Fennicę

Mathematica

Volumen 33, 2008, 585-596

# PLANAR BEURLING TRANSFORM
AND GRUNSKY INEQUALITIES

## Håkan 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
*L*^{2}(**C**).
As the Beurling operator is also bounded on
*L*^{p}(**C**) for 1 p L^{p} setting.
Here, we consider weighted Hilbert spaces
*L*_{\theta}^{2}(**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*_{\theta}^{2}(**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*_{\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 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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