On M. Riesz Conjugate Function Theorem for Harmonic Functions

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