Multi-parameter extensions of a theorem of Pichorides

Bakas, Odysseas; Rodríguez-López, Salvador; Sola, Alan

doi:10.1090/proc/14251

Multi-parameter extensions of a theorem of Pichorides
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by Odysseas Bakas, Salvador Rodríguez-López and Alan A. Sola PDF

Proc. Amer. Math. Soc. 147 (2019), 1081-1095 Request permission

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’ theorem are also obtained.

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