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 ):
with given Hölder-continuous ( G(t) \neq 0 ) and ( g(t) ). The of the problem is
with ( a(t), b(t) ) Hölder continuous. The key is to set
Title: Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics Author: N. I. Muskhelishvili (also spelled Muskhelishvili) Original Russian Publication: 1946 (frequently revised) English Translation: 1953 (P. Noordhoff, Groningen; later Dover reprints)
[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(t)t-z , dt ]
