Critical exponents of the Riesz projection

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 ${\mathfrak{p}}_1(q)=4(1-1\mathrm{/}q)$ for $1\le q\le \mathrm{\infty }$ and prove that it holds in the endpoint case $q=1$. We then extend the conjecture to $${\mathfrak{p}}_d(q)=2+{\displaystyle \frac{2}{d+{\displaystyle \frac{2}{q-2}}}}$$ for $d\ge 1$ and $\frac{2d}{d+1}\le q\le \mathrm{\infty }$ 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 $q=4\mathrm{/}3$.