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

Abstract EN

Let N p ( 1 < p < ∞ ) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane.

The space N p equipped with the topology given by the metric d p defined by d p ( f , g ) = ( ∫ 0 2 π ( l o g ( 1 + | f ∗ ( e i θ ) - g ∗ ( e i θ ) | ) ) p ( d θ / 2 π ) ) 1 / p , f , g ∈ N p , becomes an F -algebra.

For each p > 1 , we also consider the countably normed Fréchet algebra F p of holomorphic functions on D which is the Fréchet envelope of the space N p .

Notice that the spaces F p and N p have the same topological duals.

In this paper, we give a characterization of bounded subsets of the spaces F p and weakly bounded subsets of the spaces N p with p > 1 .

If ( F p ) ∗ denotes the strong dual space of F p and N p w ∗ denotes the space S p of complex sequences γ = { γ n } n satisfying the condition γ n = O e x p - c n 1 / ( p + 1 ) , equipped with the topology of uniform convergence on weakly bounded subsets of N p , then we prove that F p ∗ = N p w ∗ both set theoretically and topologically.

We prove that for each p > 1 F p is a Montel space and that both spaces F p and ( F p ) ∗ are reflexive.