Singular Integral Equations Boundary Problems Of Function Theory And Their Application To Mathematical Physics N I Muskhelishvili 🔔

is bounded on Hölder spaces and ( L^p ) ((1<p<\infty)). Find a sectionally analytic function ( \Phi(z) ) (vanishing at infinity as ( O(1/z) ) for the “exterior” problem) satisfying on ( \Gamma ):

defines two analytic functions: ( \Phi^+(z) ) inside, ( \Phi^-(z) ) outside. Their boundary values on ( \Gamma ) satisfy is bounded on Hölder spaces and ( L^p

[ \Phi^\pm(t_0) = \pm \frac12 \phi(t_0) + \frac12\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt, ] \textP.V. \int_\Gamma \frac\phi(t)t-t_0