The topology of the class of functions representable by Carleman type formulae, duality and applications

Abstract

We set $D$ to be a simply connected domain and we consider exhaustion function spaces, $X_\infty(D)$ with the projective topology. We show that the natural topology on the topological dual of $X_\infty(D)$, $(X_\infty(D))'$, is the inductive topology. As a main application we assume that $D$ has a Jordan rectifiable boundary $\partial D$, and $M\subset \partial D$ to be an open analytic arc whose Lebesgue measure satisfies $0<m(M)<m(\partial D)$. We prove a result for the dual of ${\mathcal{N}\mathcal{H}}^1_M(D)$, which is the class of holomorphic functions in $D$ which are represented by Carleman formulae on $M\subset \partial{D}$. Furthermore we show that the Cauchy Integral associated to $f \in \near$ is an element of $\near$. Lastly, we solve an extremal problem for the dual of ${\mathcal{N}\mathcal{H}}^1_M(D)$.