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arXiv 1906.01405math.FA
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$H^1$空间中的Hoeffding分解

Hoeffding decomposition in $H^1$ spaces

Maciej Rzeszut, Michał Wojciechowski

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摘要：

Bourgain和Kwapiestan的著名结果指出，投影$P_{\leq m}$到希尔伯特空间$L^2\left(Ω^\infty\right)$的子空间上，由依赖于至多$m$变量的函数所张成，在$L^p$中有界，对于$1<p<\infty$，范数为$\leq c_p^m$。我们将关注两种端点估计。我们证明了$P_{\leq m}$在$L^1\left(\mathbb{T}^\infty\right)$解析函数空间$H^1\left(\mathbb{D}^\infty\right)$上是有界的.我们还证明了$P_{\leq 2}$是有界的鞅Hardy空间与一个自然的双指标过滤，更一般地说，我们展示了一个多指标鞅Hardy空间，其中包含$H^1\left(\mathbb{D}^\infty\right)$作为一个子空间和$P_{\leq m}$是有界的。

英文摘要：

The well known result of Bourgain and Kwapień states that the projection $P_{\leq m}$ onto the subspace of the Hilbert space $L^2\left(Ω^\infty\right)$ spanned by functions dependent on at most $m$ variables is bounded in $L^p$ with norm $\leq c_p^m$ for $1<p<\infty$. We will be concerned with two kinds of endpoint estimates. We prove that $P_{\leq m}$ is bounded on the space $H^1\left(\mathbb{D}^\infty\right)$ of functions in $L^1\left(\mathbb{T}^\infty\right)$ analytic in each variable. We also prove that $P_{\leq 2}$ is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains $H^1\left(\mathbb{D}^\infty\right)$ as a subspace and $P_{\leq m}$ is bounded on it.

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