Best constants in reverse Riesz-type inequalities for analytic and co-analytic projections

TL;DR

This paper determines the exact constants in reverse Riesz-type inequalities involving analytic and co-analytic projections on the unit circle, for specific ranges of p and s, advancing understanding of these inequalities.

Contribution

It establishes sharp constants for reverse Riesz-type inequalities involving projections, for certain p and s ranges, which was previously unresolved.

Findings

Sharp constants B_{p,s} are found for p in (1,2] and p ≥ 4.

The inequalities hold with these sharp constants for s in [p', +∞).

The results extend the understanding of reverse inequalities for analytic projections.

Abstract

Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider the inequalities of the following form $$ \|f\|_{L^p(\mathbb{T})}\leq B_{p,s}\|( |P_ + f | ^s + |P_- f |^s) ^{\frac 1s}\|_{L^p (\mathbb{T})} $$ and prove them with sharp constant $B_{p,s}$ for $s \in [p',+\infty)$ and $1<p\leq 2$ and $p\geq 4,$ where $p':=\min\{p,\frac{p}{p-1}\}.$