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∞(D) with the projective topology (see §1). We show that the natural topology on the topological dual of X∞(D), (X∞(D))′, is the inductive topology. As a main application we assume that D has a Jordan rectifiable boundary ∂D, and M ⊂ ∂D to be an open analytic arc whose Lebesque measure satisfies 0 < m(M) < m(∂D). We prove a result for the dual of NHM (D), which is the class of holomorphic functions in D which are represented by Carleman formulae on M ⊂ ∂D. Furthermore we show that the Cauchy Integral associated to f ∈ NHM (D) is an element of NH 1 M (D). Lastly, we solve an extremal problem for the dual of NHM (D). 1. Projective and Inductive topologies We set D to be a bounded simply connected domain of the complex plane C and {Di}i=1, i = 1, 2 . . . to be an increasing sequence of bounded simply connected domains (i.e. Di ⊂ Di+1, i = 1, 2 . . .) such that D = ⋃ iDi with Di ⊂ D, i = 1, 2, . . ., ∂Di → ∂D in the sense that {∂Di}i eventually surrounds each compact subdomain of D. Such a sequence of domains {Di}i is called an exhaustion of D. Furthermore we set X(Di) to be a function space on Di with topology Ti. For simplicity in the symbolism write Xi for X(Di), and assume that each Xi carries the topology Ti for all i ∈ N. In the following we construct the projective limit associated with Xi and we provide it with the projective topology. For all i ≤ j, fi ∈ Xi, i, j = 1, 2, . . ., define the connecting maps (1.1) μij : Xj → Xi, i ≤ j, such that μij(fj) = fi is the restriction of fj on Di. In addition note that for all i, j, k = 1, 2 . . ., i ≤ j ≤ k, holds μik = μij ◦ μjk. We consider X∞ to be the subspace of ∏ iXi whose elements f = (f1, f2, . . .) satisfy the relation fi = μij(fj) for all i ≤ j. X∞ is called the projective limit of the family {Xi}i with respect to the mappings μij and is denoted by X∞ = lim ←− μij(Xi, Ti). We set μi to be the restriction to X∞ of the projection map pi of ∏ iXi onto Xi, i = 1, 2 . . ., and we give X∞ the projective topology T∞ with respect to the family {(Xi, Ti), μi}i. That is the coarsest topology on X∞ for which each of the mappings μi : X∞ → (Xi, Ti), i ∈ N, is continuous. An element fi ∈ Xi is called a representative of f ∈ X∞, if μi(f) = fi, i ∈ N. Note that each element of X∞ has a unique representative in each Xi, but that an element of Xi does not necessarily 1991 Mathematics Subject Classification. Primary:46A13, 30E20; Secondary:30D55, 30E25.