Topological and Functional Properties of SomeF-Algebras of Holomorphic Functions

Abstract

LetNp  (1<p<∞)be the Privalov class of holomorphic functions on the open unit diskDin the complex plane. The spaceNpequipped with the topology given by the metricdpdefined bydp(f,g)=(∫02π‍(log(1+|f∗(eiθ)-g∗(eiθ)|))p(dθ/2π))1/p,f,g∈Np, becomes anF-algebra. For eachp>1, we also consider the countably normed Fréchet algebraFpof holomorphic functions onDwhich is the Fréchet envelope of the spaceNp. Notice that the spacesFpandNphave the same topological duals. In this paper, we give a characterization of bounded subsets of the spacesFpand weakly bounded subsets of the spacesNpwithp>1. If(Fp)∗denotes the strong dual space ofFpandNpw∗denotes the spaceSpof complex sequencesγ={γn}nsatisfying the conditionγn=Oexp-cn1/(p+1), equipped with the topology of uniform convergence on weakly bounded subsets ofNp, then we prove thatFp∗=Npw∗both set theoretically and topologically. We prove that for eachp>1  Fpis a Montel space and that both spacesFpand(Fp)∗are reflexive.