On M. Riesz Conjugate Function Theorem for Harmonic Functions

Abstract

Let \(L^p(\textbf{T})\) be the Lesbegue space of complex-valued functions defined in the unit circle \(\textbf{T}=\{z: |z|=1\}\subseteq \mathbb {C}\). In this paper, we address the problem of finding the best constant in the inequality of the form:

$$ \Vert f\Vert _{L^p(\textbf{T})}\le A_{p,b} \Vert (|P_+ f|^2+b| P_{-} f|^2)^{1/2}\Vert _{L^p(\textbf{T})}. $$

Here \(p\in [1,2]\), \(b>0\), and by \(P_{-} f\) and \( P_+ f\) are denoted the co-analytic and analytic projections of a function \(f\in L^p(\textbf{T})\). The sharpness of the constant \(A_{p,b}\) follows by taking a family quasiconformal harmonic mapping \(f_c\) and letting \(c\rightarrow 1/p\). The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.