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 $H_A^p({\mathbb{T}}^d)$ blows up like $(p-1{)}^{-d}$ as $p\to 1^{+}$. Furthermore, we obtain an $L{\log }^{d}L$-estimate for square functions on $H_A^1({\mathbb{T}}^d)$. Euclidean variants of Pichorides’s theorem are also obtained.