$H^1$ and dyadic $H^1$
$H^1$ and dyadic $H^1$
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发表时间：
2008-09
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通讯作者：
S. Treil
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作者：
S. Treil
本文给出了一个简单的证明：函数的并矢H^1 $-范数在所有并矢格上的平均给出了一个等价的H^1 $-范数。我们提出的证明工程单参数和多参数哈代空间。这种类型的结果是已知的。第一个结果（对于单参数哈代空间）属于Burgess Davis（1980）。同时，通过对偶性，这些结果等价于Garnett-Jones（1982）对单参数情形证明的“BMO from dyadic BMO”命题和Piper-Ward（2008）对双参数情形证明的“BMO from dyadic BMO”命题.虽然本文将这些结果推广到多参数设置，但这不是其主要目标。本文的目的是提出一种方法，导致一个简单的证明，它适用于单参数和多参数的情况下。的主要思想治疗平方函数作为卡尔德龙-Zygmind运营商是一个司空见惯的调和分析的主要观察，该文件的基础上，是一个可以治疗随机并矢平方函数这种方式。然后，利用Hilbert空间值环境下Calderon-Zygmind算子的标准和众所周知的结果证明了这一切。作为额外的奖励，我们得到了多参数情况下包含$\text{BMO}\subset \text {BMO}_d$，$H^1_d \subset H^1$的简单证明（对偶等价）。注意，与单参数情形不同，一般情况下的包含远非微不足道。
In this paper we give a simple proof of the fact that the average over all dyadic lattices of the dyadic $H^1$-norm of a function gives an equivalent $H^1$-norm. The proof we present works for both one-parameter and multi-parameter Hardy spaces. The results of such type are known. The first result (for one-parameter Hardy spces) belongs to Burgess Davis (1980). Also, by duality, such results are equivalent to the "BMO from dyadic BMO" statements proved by Garnett-Jones(1982} for one parameter case, and by Pipher-Ward (2008) for two-parameter case. While the paper generalizes these results to the multi-parameter setting, this is not its main goal. The purpose of the paper is to present an approach leading to a simple proof, which works in both one-parameter and multi-parameter cases. The main idea of treating square function as a Calderon--Zygmind operator is a commonplace in harmonic analysis; the main observation, on which the paper is based, is that one can treat the random dyadic square function this way. After that, all is proved by using the standard and well-known results about Calderon--Zygmind operators in the Hilbert-space-valued setting. As an added bonus, we get a simple proof of the (equivalent by duality) inclusion $\text{BMO}\subset \text{BMO}_d$, $H^1_d \subset H^1$ in the multi-parameter case. Note, that unlike the one-parameter case, the inclusions in the general situation are far from trivial.