Functions in All H p Spaces, p < ∞

Source abstract

Abstract Let Ĥ denote the class of functions analytic in | z | < 1 which are in every H p class, 0 < p < ∞. The class Ĥ strictly contains H ∞ and consists of those functions that are ‘almost in H ∞ ’ in the sense of integration. L. Hansen and W. Hayman have given simple geometric conditions for a function to belong to Ĥ . The purpose of this note is to show that Hansen and Hayman's conditions are far from necessary. Using techniques from normal functions, gap series, characterizations of BMOA, subordination, Bloch functions, and VMOA, six completely different examples of functions in Ĥ are given which ‘fill the plane’.