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

Abstract

\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_{-}=I-P_{+}$. We consider estimates of the expression $\Vert (|P_{+}f{|}^s+|P_{-}f{|}^s{)}^{\frac{1}{s}}{\Vert }_{L^p(\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f\in L^p(\mathbf{T})$. We find the accurate estimates for $p\ge 2$ and $0<s\le p$, 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}

27 pages; main results of this paper are considerably improved, some of them are shortened; part with reverse estimates is removed