Boundary values of analytic functions in the Banach space \(P^ t(\sigma)\) on crescents
A crescent \(\Omega\) is a simply connected domain bounded by two Jordan curves. Let \(\sigma\) be the harmonic measure of \(\Omega\) and let \(P^ t (\sigma)\) be the closure of \({\mathcal P}\) in \(L^ t (\sigma)\) \((t \in [1, \infty))\). In this paper we assume that \(abpe P^ t(\sigma) = D\), the open unit disc in the plane. Let \(f \in P^ t (\sigma)\), we show that \(\widehat f\), the analytic extension of \(f\) on \(abpe P^ t (\sigma)\), has nontangent limits almost everywhere on \(\partial D\). Our goal is to answer the following natural question: If \(f \in P^ t (\sigma)\) and \(f | \partial D\) is bounded, is \(\widehat f\) a bounded function on \(D\)? We first construct a crescent \(\Omega\) and a function \(f \in P^ t (\sigma)\) such that \(\widehat f\) has bounded boundary values (in nontangential limit sense) on \(\partial D\), but \(\widehat f\) is an unbounded analytic function on \(D\). We then answer the question positively for a large class of crescents. Using the last result, we conclude this paper by showing that if \(\tau\) is a positive measure on \(D\), then the operators \(S_{\sigma + \tau}\) and \(S_ \sigma\) are similar if and only if \(\tau\) is a Carleson measure on \(D\).
scientific article; zbMATH DE number 766575