On a generalized M. Riesz conjugate function theorem for harmonic mappings

Anton Gjokaj, David Kalaj and Djordjije Vujadinović

Vol. 345 (2026), No. 2, 279–321

DOI: 10.2140/pjm.2026.345.279

Abstract

Let $L^p(\mathbb{𝕋})$ be the Lebesgue space of complex-valued functions defined on the unit circle $\mathbb{𝕋}=\{z:|z|=1\}\subseteq ℂ$. We determine the best constant in

$$\parallel {(|P_{+}f{|}^2+c|P_{-}f{|}^2)}^{1∕2}{\parallel }_{L^p(\mathbb{𝕋})}\le a_{p,c}\parallel f{\parallel }_{L^p(\mathbb{𝕋})}.$$

Here $2\le p<\infty$, $c>0$, and $P_{+}f$ and $P_{-}f$ denote the analytic and coanalytic projections of $f\in L^p(\mathbb{𝕋})$, respectively. The sharpness of the constant $a_{p,c}$ follows from an explicit extremizing family. For $c\ne 1$ the family is quasiconformal harmonic, whereas for $c=1$ it is real harmonic. The result extends a sharp version of the M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.

Keywords

subharmonic functions, harmonic mappings

Mathematical Subject Classification

Primary: 31A05

Milestones

Received: 11 November 2025

Revised: 31 May 2026

Accepted: 3 August 2026

Published: 26 October 2026