Riesz's Theorem for Lumer's Hardy Spaces

TL;DR

This paper extends Riesz's theorem to Lumer's Hardy spaces on arbitrary domains, showing harmonic conjugates of functions in these spaces also belong to the space with a sharp norm bound.

Contribution

It establishes a version of Riesz's theorem for Lumer's Hardy spaces on arbitrary domains, including a precise norm estimate with the optimal constant.

Findings

Harmonic conjugates in Lumer's Hardy spaces also belong to the space.

Derived a sharp norm estimate for conjugate harmonic functions.

Extended classical Riesz's theorem to a broader class of domains.

Abstract

In this note we obtain a version of the well-known Riesz's theorem on conjugate harmonic functions for Lumer's Hardy spaces $(Lh)^2(\Omega)$ on arbitrary domains $\Omega$: If a real-valued harmonic function $U\in (Lh)^2(\Omega)$ has a harmonic conjugate $V$ on $\Omega$ (i.e., a real-valued harmonic function such that $U+ iV$ is analytic on $\Omega$), then $U+iV$ also belongs to $(Lh)^2(\Omega)$, and for the normalized conjugate we have the norm estimate $\|U+iV\|_{(Lh)^2(\Omega)}\le\sqrt{2} \|U\|_{(Lh)^2(\Omega)}$, with the best possible constant.