Comparison of measures on the maximal ideal space of \(H^{\infty}(W)\) with applications to the Dirichlet problem (Q791112)

Let \({\mathbb{C}}^ n\), \(n\geq 1\), denote the n-dimensional complex Euclidean space. Let W be any subdomain of \({\mathbb{C}}^ n\) which admits either nonconstant bounded analytic functions, or non-constant harmonic functions. Let \(H^{\infty}(W)\) denote the commutative Banach algebra of all bounded analytic functions on W, endowed with the uniform norm. Let \(HB_ R(W)\) denote the order complete Banach lattice of all real-valued bounded harmonic functions in \(W(\subseteq {\mathbb{R}}^{2n})\) with the sup- norm topology. Finally, let \(M(H^{\infty}(W))\) stand for the maximal ideal space of \(H^{\infty}(W)\). This paper deals with the Dirichlet problem on the Shilov boundary S of \(M(H^{\infty}(W))\). More specifically, the author investigates a positive linear map from \(C_ R(S)\) into \(HB_ R(W)\) which acts on Re \(H^{\infty}(W)\) as the identity map. Since \(HB_ R(W)\) is an order complete Banach lattice, such a map always exists. The author's primary aim is to study the maximal solutions of the Dirichlet problem on S using the Choquet order relation. The notion of maximality is defined in terms of the boundary measures introduced by \textit{E. M. Alfsen} [Compact Convex Sets and Boundary Integrals (1971; Zbl 0209.426)]. The paper concludes with a result concerning a closed subalgebra of \(H^{\infty}(W)\).