Topological and Functional Properties of Some F-Algebras of Holomorphic Functions

Let Np (1<p<∞) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane. The space Np equipped with the topology given by the metric dp defined by dp(f,g)=(∫02π‍(log(1+|f∗(eiθ)-g∗(eiθ)|))p(dθ/2π))1/p, f,g∈Np, becomes an F-algebra. For each p>1, we also consider the countably normed Fréchet algebra Fp of holomorphic functions on D which is the Fréchet envelope of the space Np. Notice that the spaces Fp and Np have the same topological duals. In this paper, we give a characterization of bounded subsets of the spaces Fp and weakly bounded subsets of the spaces Np with p>1. If (Fp)∗ denotes the strong dual space of Fp and Npw∗ denotes the space Sp of complex sequences γ={γn}n satisfying the condition γn=Oexp-cn1/(p+1), equipped with the topology of uniform convergence on weakly bounded subsets of Np, then we prove that Fp∗=Npw∗ both set theoretically and topologically. We prove that for each p>1 Fp is a Montel space and that both spaces Fp and (Fp)∗ are reflexive.

Abstract licence: CC BY

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