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

TL;DR

This work studies sharp reverse Riesz-type inequalities for analytic and co-analytic projections on the circle, quantifying how the $L^p$ norm of a function is controlled by the $L^p$ norm of a combined modulus of $P_+f$ and $P_-f$. The authors develop a concerted approach based on plurisubharmonic minorants, a family of nearly-extremal test functions, and a detailed stationary-point analysis to reduce the problem to boundary cases, yielding exact constants in key regimes: for $1<p\le 2$ with $s\ge p$, the optimal constant is $B_{p,s}=2^{1-1/s}\sin(\pi/(2p))$; for $p\ge 4$ with $s\ge p/(p-1)$, it is $B_{p,s}=2^{1-1/s}\cos(\pi/(2p))$; and for $s\in(0,1]$, the constant is $B_{p,s}=1$. The results extend previous $s=2$ cases and provide a comprehensive sharp characterization across broad parameter ranges with potential impact on harmonic analysis and related inequalities for Hardy spaces and Riesz 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 $p':=\min\{p,\frac{p}{p-1}\}.$

Theorems & Definitions (7)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• proof : Proof of the First Part of Theorem 1.1
• proof : Proof of the Second Part of Theorem 1.1
• Lemma 6.1
• proof