Pichorides定理的多参数推广
Multi-parameter extensions of a theorem of Pichorides
摘要：
将Pichorides和Zygmund的工作推广到$d$ -维的情形，我们证明了解析Hardy空间$H^p_A(\mathbb{T}^d)$的单位球上的Littlewood-Paley平方函数的$L^p$ -范数的上确界像$(p-1)^{-d}$一样爆破为$p\to 1^+$ .进一步，我们在$H^1_A(\mathbb{T}^d)$上得到了平方函数的$L\log^d L$ -估计.也得到了Pichorides定理的欧氏变体。
英文摘要：
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.
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