ON THE ANALYTICITY OF THE
CAUCHY INTEGRAL IN SCHAUDER
SPACES

MASSIMO LANZA DE CRISTOFORIS
AND
LUCA PRECISO

Abstract:

As is well known, if the contour of integration and the density function belong to a suitable Schauder space, the Cauchy integral belongs to the same Schauder space. We analyze, in this Schauder space setting, the dependence of the Cauchy integral upon its contour and its density function, which we think of as functional variables, and we prove a result of complex analyticity for such dependence. We prove our statement by constructing a functional equation which involves the Cauchy integral, the contour of integration and the density function and by applying to such functional equation the implicit function theorem in its formulation for nonlinear maps between Banach spaces.