A Topological Characterization of a Class of Integral Operators
Charles Loewner · Annals of Mathematics · 1948
x = x(t) (2) y = y(t) may be interpreted as the parametric representation of a closed oriented curve in the x-y-plane. Any curve obtained in this way will be said to be 'generated by the kernel function k(t).' Various problems in analysis and geometry (study of smoothing operators, of the behavior of analytic functions on the boundary and finally of umbilical points) led the author to the following question: Which kernel functions k(t) generate only curves of non-negative circulation? A complete answer is given in this paper. It states: THE PRINCIPAL THEOREM: An Li-integrable function k(t) generates only curves of non-negative circulation if and only if, after a possible change of its values in a set of measure zero which has no influence on the operator (1), it is analytic in the open interval 0 < t < 2ir and its derivative can there be represented by a LaplaceStieltjes integral