On M. Riesz conjugate function theorem for harmonic functions

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.