Critical exponents of the Riesz projection

Abstract

Let ${\mathfrak{p}}_d(q)$ denote the critical exponent of the Riesz projection from $L^q({\mathbb{T}}^d)$ to the Hardy space $H^p({\mathbb{T}}^d)$, where $\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that ${\mathfrak{p}}_1(q)=4(1-1/q)$ for $1\le q\le \infty$ and prove that it holds in the endpoint case $q=1$. We then extend the conjecture to \[\mathfrak{p}_d(q) = 2+\cfrac{2}{d+\cfrac{2}{q-2}}\] for $d\ge 1$ and $\frac{2d}{d+1}\le q\le \infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the conjecture holds in the endpoint case $q=4/3$.

References in corpus (3)

• Hollenbeck-Verbitsky conjecture on best constant inequalities for analytic and co-analytic projections
• Point evaluation in Paley--Wiener spaces
• Contractive inequalities between Dirichlet and Hardy spaces