On Hollenbeck-Verbitsky conjecture for $4/3 < p < 2$ and reverse Riesz-type inequalities for $2<p<4$

Abstract

Let \(P_+\) be the Riesz's projection operator and let \(P_-=I-P_+.\) We find the best upper estimates of the expression \(\left\lVert \left( \left\lvert P_+f \right\rvert ^s + \left\lvert P_-f \right\rvert ^s \right) ^{1/s} \right\rVert _p \) in terms of Lebesgue p-norm of the function \(f \in L^p(\mathbf{T})\) for \(p \in (4/3,2)\) and \(0 < s \leq \frac{p}{p-1},\) thus extending results from \cite{Melentijevic_2022} and \cite{Melentijevic_2023}, where the mentioned range is not considered. Also, we find the best lower estimates of the same quantities for \(p \in (2,4)\) and \(s \geq \frac{p}{p-1},\) thus extending results from \cite{melentijevic-reverse-2025}.