Hilbert Space of Complex-Valued Harmonic Functions in the Unit Disc

TL;DR

This paper extends the Hilbert space framework from analytic functions to complex-valued harmonic functions in the unit disc, showing many analogous properties and results hold in this larger space.

Contribution

It introduces and analyzes a Hilbert space of complex-valued harmonic functions, demonstrating that many properties of analytic function spaces also apply to harmonic functions.

Findings

Functions exhibit analogous norm and growth properties

Reproducing kernels are established for the harmonic space

Many classical results extend to the harmonic function space

Abstract

We investigate an extended version of Hilbert space of analytic functions called Hilbert space of complex-valued harmonic functions. It is found that functions in Hilbert space of complex-valued harmonic functions exhibit many properties analogous to its analytic counter part such as complex-valued harmonic function analogous of norm, equivalent norms, reproducing kernels, growth estimates and Littlewood-Paley Identity Theorem. In conclusion we prove that many results in Hilbert space of analytic functions also hold in larger Hilbert space of complex-valued harmonic functions.