Carleson curves

Abstract

The purpose of this chapter is to acquaint the reader with some simple but basic properties of Carleson curves and to provide a sufficient supply of examples. The “oscillation” of a Carleson curve Γ at a point t ∈ Γ may be measured by its Seifullayev bounds σ ⁻ _t and σ ⁺ _t as well as its spirality indices δ ⁻ _t and δ ⁺ _t The definition of the spirality indices requires the notion of the W transform and some facts from the theory of submultiplicative functions. In the spectral theory of Toeplitz and singular integral operators, the spirality indices will play a decisive role. We therefore compute the spirality indices for a sufficiently large class of concrete Carleson curves.