The trifecta of Hilbert spaces on Unit Disc

Abstract

The Hilbert spaces are common. But the direct connection between them is rare. The aim of this paper is to establish a direct relation among the three Hilbert spaces Hardy, Bergman and Dirichlet without defining any of the Hilbert space in weighted sense. In order to accomplish this goal, this paper develops the Littlewood-Paley type Identities for Bergman and Dirichlet space. After defining these identities, the vision of connecting all the three Hilbert spaces via a direct connection is achieved.