Les domaines de Carleson. (Carleson domains)
A Carleson measure \(\mu\) in a plane domain \(\Omega\) satisfies (by definition) the inequality \[ \| \mu \| =:\sup \{r^{-1}| \mu | (D(z,r)):\quad z\in \partial \Omega,\quad r>0\}, \] D(z,r) being the open unit disc centered at z with the radius r. Suppose a Jordan domain with a rectifiable boundary enjoys the following property: \(\mu\) a Carleson measure in \(\Omega\), \(f\in H^ 1(\Omega)\Rightarrow \int | f| | d\mu | \leq C\| \mu \| \| f\|_ 1\), with a constant C not depending on f and \(\mu\), and \(H^ 1(\Omega)\) the usual Smirnov class for \(\Omega\) \((f\in H^ 1(\Omega)\Leftrightarrow (f\circ \phi)\phi '\in H^ 1({\mathbb{D}})\), where \(\phi\) is a conformal isomorphism of the unit disc \({\mathbb{D}}\) onto \(\Omega)\). Then \(\Omega\) is called a Carleson domain. This notion can be naturally extended to an arbitrary Jordan domain (with possibly non rectifiable \(\partial \Omega)\). The principal result is a complete characterization of Carleson domains in terms of the mapping \(\phi\).Namely, \(\Omega\) is Carleson iff log \(\phi\) \({}'\in BMOA({\mathbb{D}})\). This yields a lot of explicit geometric conditions ensuring carlesonity of a domain and leads to examples of non Carleson domains with rectifiable boundary. Applications to free interpolation of bounded analytic functions are given.
scientific article; zbMATH DE number 4132527