On M. Riesz conjugate function theorem for harmonic functions

TL;DR

This paper investigates the optimal constant in a specific inequality involving harmonic functions on the unit circle, extending classical results like M. Riesz's conjugate function theorem with new sharp bounds.

Contribution

It derives the best constant in a harmonic function inequality, extending classical theorems and providing sharp estimates for harmonic and holomorphic functions.

Findings

Identifies the optimal constant A_{p,b} in the inequality.

Extends the M. Riesz conjugate function theorem to a broader setting.

Provides conditions under which the equality is attained by quasiconformal harmonic mappings.

Abstract

Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. In this paper, we address the problem of finding the best constant in the inequality of the form: $$\|f\|_{L^p(\mathbf{T})}\le A_{p,b} \|(|P_+ f|^2+b| P_{-} f|^2)^{1/2}\|_{L^p(\mathbf{T})}.$$ Here $p\in[1,2]$, $b>0$, and by $P_{-} f$ and $ P_+ f$ are denoted co-analytic and analytic projection of a function $f\in L^p(\mathbf{T})$. The equality is "attained" for a quasiconformal harmonic mapping. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.