Critical exponents of the Riesz projection

Let 𝔭_d(q) denote the critical exponent of the Riesz projection from L^q(𝕋^d) to the Hardy space H^p(𝕋^d), where 𝕋 is the unit circle. We present the state-of-the-art on the conjecture that 𝔭_1(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+2d+2q-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.