Estimates of the derivative of a Cauchy-type integral with meromorphic density and their applications
Let \(\Gamma\) be a closed simple rectifiable curve on the complex plane. For \(f\in L^ 1(\Gamma)\) the Cauchy integrals \(K^{\pm}_{\Gamma}\) are defined by \[ (K^{\pm}_{\Gamma}f)(z)=(1/2\pi i)\int_{\Gamma}(f(\xi)/(\xi -z))d\xi,\quad z\in \Gamma^{\pm}, \] \(\Gamma\) \({}^{\pm}\) being the interior and exterior of \(\Gamma\). Given \(\lambda =\{b_ 1,...,b_ n\}\) in \(\Gamma^+\), \({\mathcal R}(\lambda)\) denotes the set of rational functions with poles (with counted multiplicities) in \(\lambda\). The author investigates the norms \(N_ p(\lambda,\Gamma^+)=\sup_{f\in M_ p(\lambda,\Gamma^+),\| f\|_{L^ p(\Gamma)}\leq 1}\quad \| K^-_{\Gamma}\|_{L^ p(\Gamma)},\) where \(M_ p(\lambda,\Gamma^+)=E_ p(\Gamma^+)+{\mathcal R}(\lambda),\) \(E_ p\) being the Smirnov class. Estimates of \(N_ p(\lambda,\Gamma^+)\) for \(p=1\), \(\infty\) are obtained. In particular, in case \(\Gamma\) is the unit circle it is proved that \(\sup_{\lambda}N_{\infty}(\lambda,\Gamma^+)=2n.\) Applications to the inverse theorems of rational approximation are found.
scientific article; zbMATH DE number 3882704