Best constants in inequalities involving analytic and co-analytic projections and Riesz theorem for various function spaces

\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$. Interestingly, for this range of $s$ there holds the appropriate vector-valued inequality with the same constant. Also, we obtain the right asymptotic of the constants for large $s$. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. As a consequence of inequalities we have proved in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. In particular, slightly general version of Stout's theorem for Lumer Hardy spaces is obtained by a new approach. \end{abstract}