Best constants in reverse Riesz-type inequalities for analytic and co-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 $p':=\min\{p,\frac{p}{p-1}\}.$