Critical exponents of the Riesz projection

Let pd(q) denote the critical exponent of the Riesz projection from Lq(Tᵈ) to the Hardy space Hᵖ(Tᵈ), where T is the unit circle. We present the state-of-the-art on the conjecture that p₁(q) = 4(1-1/q) for 1 ≤ q ≤ ∞ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \[p_d(q) = 2+{2}{d+{2}{q-2}}\] for d≥1 and 2d/d+1 ≤ q ≤ ∞ 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 $q = 4/3$.