Analysis of the Carleson-Vasyunin Condition

Abstract

The Carleson condition (C)imposed on the Blaschke product B=Π_λ∈σb_λ (σ⊂D) with simple zeros

$$\mathop {\inf }\limits_{\lambda \in \sigma } \left| {{B_\lambda }\left( \lambda \right)} \right| > 0,$$

(C)

was analyzed in great detail in Lecture VII. The power of this analysis, which helps us to visualize rather completely the geometric nature of Carleson sets, depends to a large extent on the local character of condition (C) — it is expressed exclusively in terms of the local parameters of B, more exactly, the values of its derivative ∣B′(λ)∣(=(1−∣λ∣²)⁻¹∣B_λ(λ)∣,λ=σ)on the spectrum σ(B)=σ. Not less important is the circumstance that condition (C) by definition is expressed in terms of the representing measure µ^B of B and a fixed kernel on the product σ × σ:

$$\mathop {\inf }\limits_{\lambda \in \sigma } 2\int_{D\backslash \left\{ \lambda \right\}} {\frac{{\log \left| {\frac{{\zeta - \lambda }}{{1 - \bar \zeta \lambda }}} \right|}}{{1 - {{\left| \zeta \right|}^2}}}} d{u^B}\left( \zeta \right)> - \infty $$

.