Critical Exponents of the Riesz Projection

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