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

TL;DR

This paper establishes precise bounds for Riesz projection inequalities in various function spaces, improving previous results and confirming conjectures, with applications to conjugate harmonic functions and Hardy spaces.

Contribution

The paper provides exact estimates for Riesz projection inequalities for a broad range of parameters, advancing prior work and confirming a conjecture on the operator’s behavior.

Findings

Improved bounds for $p \\geq 2$ and $0<s\\leq p$ in Riesz projection inequalities.

Confirmation of Hollenbeck and Verbitsky's conjecture in certain cases.

Extension of Riesz-type theorems to various function spaces, including Hardy spaces.

Abstract

\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}