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 $$\Vert f{\Vert }_{L^p(\mathbb{T})}\le B_{p,s}\Vert (|P_{+}f{|}^s+|P_{-}f{|}^s{)}^{\frac{1}{s}}{\Vert }_{L^p(\mathbb{T})}$$ and prove them with sharp constant $B_{p,s}$ for $s\in [p^{\prime },+\infty )$ and $1<p\le 2$ and $p\ge 4,$ where $p^{\prime }:=\min \{p,\frac{p}{p-1}\}.$

21 pages; this version has been accepted for publication in Journal of Mathematical Analysis and Applications