On generalized M. Riesz conjugate function theorem for harmonic mappings

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.