Multi-parameter extensions of a theorem of Pichorides

Extending work of Pichorides and Zygmund to the $d$-dimensional setting, we show that the supremum of $L^p$-norms of the Littlewood-Paley square function over the unit ball of the analytic Hardy spaces $H^p_A(\mathbb{T}^d)$ blows up like $(p-1)^{-d}$ as $p\to 1^+$. Furthermore, we obtain an $L\log^d L$-estimate for square functions on $H^1_A(\mathbb{T}^d)$. Euclidean variants of Pichorides's theorem are also obtained.