Multi-parameter extensions of a theorem of Pichorides

TL;DR

This paper extends Pichorides's theorem to higher dimensions, showing how the supremum of Littlewood-Paley square function norms behaves near p=1 and providing new estimates in multidimensional Hardy spaces.

Contribution

It generalizes Pichorides's theorem to the d-dimensional setting and establishes the blow-up rate of square function norms as p approaches 1 from above.

Findings

Supremum of L^p norms blows up like (p-1)^{-d} as p→1+.

Established an L log^d L estimate for square functions on H^1_A(𝕋^d).

Derived Euclidean variants of Pichorides's theorem.

Abstract

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.