Riesz's Theorem for Lumer's Hardy Spaces

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 $\Vert U+iV{\Vert }_{(Lh{)}^2(\Omega )}\le \sqrt{2}\Vert U{\Vert }_{(Lh{)}^2(\Omega )}$, with the best possible constant.

The American Mathematical Monthly, to appear. arXiv admin note: substantial text overlap with arXiv:1912.08944