Best Constants in Inequalities Involving Analytic and Co-Analytic Projections and Riesz’S Theorem in Various Function Spaces

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