On generalized M. Riesz conjugate function theorem for harmonic mappings
Anton Gjokaj, David Kalaj, Djordjije Vujadinovic
TL;DR
This work sharpens the generalized M. Riesz conjugate framework for harmonic mappings by determining the best constant A_{p,c} in the inequality || (|P_+[f]|^2 + c|P_-[f]|^2)^{p/2} ||_{L^p} ≤ A_{p,c} ||f||_{L^p} for 2 ≤ p < ∞ and c > 0, where P_+ and P_- are the analytic and co-analytic projections. The authors develop a subharmonic majorant approach, deriving an explicit sharp constant a_{p,c} = 2^{-1/2} sqrt((1+c+S) csc^2(π/p)) with S = sqrt(1+c^2+2c cos(2π/p)), and show sharpness via extremal quasiconformal harmonic mappings (γ → 1/p) as the minimizing sequence. They prove the main inequality (Theorem Kao) and its sharpness (2.2), and then establish a related result (Theorem theo123) by parametrizing c with R and splitting the proof into p ∈ [2,4] and p ≥ 4, using carefully constructed subharmonic majorants. The results extend Verbitsky’s sharp M. Riesz-type bounds and Kalaj’s harmonic-map inequalities, providing a precise, extremal-structure description with implications for generalized conjugate-function theory in harmonic analysis.
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 $f_γ$ and letting $γ\to 1/p$. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.