**2.2 The case of** *ν* **≥ 1**

We fix a real parameter *ν* such that 2*ν*>1 and we consider the following weighted logarithmic potential transform

$$\mathcal{L}\_{\nu}[f](x) = \int\_{\mathbb{D}} f(\xi) \log \left( \frac{1}{|\xi - x|} \right) \left( 1 - \xi \overline{\xi} \right)^{2\nu - 2} d\mu(\xi), \tag{10}$$

defined on the space *L*2,*<sup>ν</sup>* ð Þ complex-valued measurable functions are <sup>1</sup> � *ξξ* � �2*ν*�<sup>2</sup> *<sup>d</sup>μ ξ*ð Þ-square integrable on . As a convolution of *<sup>L</sup>*2,*<sup>ν</sup>* -functions with the compactly supported measure ð Þ <sup>1</sup>�*<sup>ξ</sup>* <sup>2</sup>*ν*�<sup>2</sup> *<sup>ξ</sup>* <sup>1</sup>D*dμ ξ*ð Þ <sup>L</sup>*<sup>ν</sup>* : *<sup>L</sup>*2,*<sup>ν</sup>* ð Þ! *<sup>L</sup>*2,*<sup>ν</sup>* ð Þ is obviously bounded. Moreover, it is not hard to show that L*<sup>ν</sup>* is in fact compact [4]. This raises a question concerning the spectral picture of L*ν*.
