On generalized M. Riesz conjugate function theorem for harmonic mappings

TL;DR

This paper extends the M. Riesz conjugate function theorem to harmonic mappings, establishing the best constant in a related inequality for projections of functions on the unit circle, with implications for quasiconformal harmonic mappings.

Contribution

It provides a sharp inequality involving harmonic projections and determines the optimal constant, extending classical theorems to a broader harmonic setting.

Findings

Derived the best constant in the inequality for harmonic projections.

Extended the M. Riesz conjugate function theorem to harmonic mappings.

Connected the sharp constant to 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: $$\|(|P_+ f|^2+c| P_{-} f|^2)^{1/2}\|_{L^p(\mathbf{T})}\le A_{p,c} \|f\|_{L^p(\mathbf{T})}.$$ Here $2\le p<\infty$, $c>0$, and by $P_{-} f$ and $ P_+ f$ are denoted co-analytic and analytic projection of a function $f\in L^p(\mathbf{T})$. The sharpness of the constant $A_{p,c}$ follows by taking a family quasiconformal harmonic mapping $f_\gamma$ and letting $\gamma\to 1/p$. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.