Espaces de Hardy et domains de Denjoy. (Hardy spaces and Denjoy domains) (Q1824062)

Let \(\Omega\) be a Denjoy domain in the plane, that is, a domain of the form \(\Omega ={\mathbb{C}}\setminus K\), where \(K\subseteq {\mathbb{R}}\) is a compact set of positive Lebesgue measure. Let \(H^ 1(\Omega)\) be the Hardy space of all functions analytic in \(\Omega\) satisfying \(\lim_{| z| \to \infty}zF(z)=0\) and \[ \| F\|_{H^ 1(\Omega)}:=\sup_{\delta >0}\int_{\Gamma_{\delta}}| F(z)| | dz| <\infty, \] where \(\Gamma_{\delta}\) is the boundary of the union of all squares centered on K with length \(\delta\). The author studies the problem for which compact sets \(K\subseteq {\mathbb{R}}\) there exists a constant \(C>0\) such that \[ \int_{L\cap \Omega}| F(z)| | dz| \leq C\| F\|_{H^ 1(\Omega)},\quad \Omega ={\mathbb{C}}\setminus K, \] for every \(F\in H^ 1(\Omega)\) and every line L in \({\mathbb{C}}\). A positive answer is given for the so-called homogeneous sets. This notion was introduced by \textit{L. Carleson} [Harmonic analysis, Conf. in Honor A. Zygmund, Chicago 1981, Vol. 2, 349-372 (1983; Zbl 0527.46048)] in relation with the corona theorem for Denjoy domains. An n- dimensional version of the results is briefly sketched in the last section.