Limits on an extension of Carleson's \(\bar\partial\)-theorem
The author studies a theorem due essentially to Carleson about solving the \(\overline\partial\)-equation on the unit disk. He shows that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, his main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. He exhibits a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to \(\infty\). Since Carleson's \(\overline\partial\)-theorem depends on the notion of a Carleson measure, the author also discusses Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.
scientific article; zbMATH DE number 930668
scientific article