The Smirnov class on compact groups with ordered duals (Q2736866)

Let \(K\) be a compact abelian group whose dual group \(\Gamma\) is a dense subgroup of the real line, endowed with the discrete topology. The space \(H^p(K)\), \(p>0\), is the closure of the trigonometric polynomials in the metric of \(L^p(K)\). The Smirnov class \({\mathcal A}(K)\) consists of the null function and of all functions \(F\), whose modulus \(w=|F|\) satisfies \(-\infty<\int_{-\infty}^{+\infty}\log w(x+e_t){{dt}\over{t^2+1}}<\infty\) (where \(e_t\) is the element of \(K\) defined by \(e_t(\lambda)=e^{t\lambda}\), \(\lambda \in\Gamma\)) and such that whenever \(g\) is in \(H^2(K)\) and \(Fg\) is in \(L^2(K)\), then also \(Fg\) belongs to \(H^2(K)\). The author proves some properties of \({\mathcal A}(K)\), for instance that \({\mathcal A}(K)\) is closed under addition and multiplication and that \({\mathcal A}(K)\cap L^p(K)=H^p(K)\) and he gives also a characterization of the real elements of \({\mathcal A}(K)\) as quotients of certain functions. The proofs in this paper rely partially on results in \textit{H. Helson} [Analycity on compact groups, in: Algebras in Analysis, Proc. instr. Conf. Birmingham 1973, 1-62 (1975; Zbl 0321.43009)].