A Theory of Hardy Spaces Associated to the Herz Spaces
A Theory of Hardy Spaces Associated to the Herz Spaces
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DOI：
10.1112/plms/s3-69.3.605
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发表时间：
1994-11
影响因子：
1.8
通讯作者：
J. García-cuerva;M. Herrero
中科院分区：
文献类型：
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作者：
J. García-cuerva;M. Herrero
在[5]中，Chen 和 Lau 介绍了与 Beurling 代数 Ap 相关的 Hardy 空间。 J. Garcfa-Cuerva [9] 进一步发展了他们的理论。代数 Ap 是由 A. Buerling [2] 结合谱合成引入的。它们是 L1 的卷积子代数的嵌套系统，其并集为 L1。 Feichtinger [6] 为 Ap 提供了等效范数，使得扩展 [9] 成为可能。关联的 Hardy 空间 HAP 是一个嵌套的空间系统，其并集是普通的 Hardy 空间//'。 HAP 的原子分解与 Hl 的不同之处在于，原子必须以 0 为中心，并且每个原子的大小由 Lp 范数给出，Lp 范数通常由其支持度测量的倒数控制。这种将 HAP 视为一种//'在某一点的观点特别有吸引力，并为哈代空间理论的一般性质提供了一些线索。考虑到这一背景，尝试将理论扩展到 q< 1 的 Hq 似乎是很自然的事情。这种扩展是本文的主题，也是第二作者博士论文的一部分。事实证明，现在扮演 Beurling 代数 Ap 角色的空间 Apq 或 Apq 之前已由 C. Herz [11] 用不同的符号引入。我们的符号适应最适合我们目标的 Feichtinger 规范。在第 1 节中，我们介绍了空间 Apq 和 Apq 以及它们的对偶 Bp_q 和 Bp-q，并简要研究了那些对于哈代空间理论的发展非常重要的性质。在第 2 节中，我们将 Hardy 空间 HApq 和 HApq 定义为调和分布空间，其非切线泊松极大函数分别属于 Apq 或 Apq。我们获得了几个等效的特征，包括原子分解，这些特征被收集在定理 2.14 中。本节包括对这些空间的复杂插值的描述。这个问题比 Banach 空间的问题更困难，我们按照 Calderon 和 Torchinsky [4] 引入的方法来解决它。最后，第 3 节专门讨论了 Kp= s2 和 0<< 7^ 1 时 HApq 的 Littlewood-Paley 表征。通过这种方式，我们能够完成 Lu 和 Yang [13] 先前获得的结果。
In [5] Chen and Lau introduced Hardy spaces associated to the Beurling algebras Ap. Their theory was further developed by J. Garcfa-Cuerva [9]. The algebras Ap were introduced by A. Buerling [2] in connection with spectral synthesis. They are a nested system of convolution subalgebras of L1 whose union is L1. Feichtinger [6] provided the equivalent norm for the Ap which made the extension [9] possible. The associated Hardy spaces HAP are a nested system of spaces whose union is the ordinary Hardy space//'. The atomic decomposition for HAP differs from that of Hl in that atoms have to be centred at 0 and the size of each atom is given by an Lp-norm controlled as usual by the reciprocal of the measure of its support. This view of HAP as a kind of//'at a point is particularly appealing and casts some light on the general nature of Hardy space theory. Given this background, it seems to be natural to try to extend the theory to Hq for q< 1. This extension is the subject of the present paper, which is part of the Ph. D. thesis of the second author. It turns out that the spaces Apq or Apq, which now play the role of the Beurling algebras Ap, had previously been introduced by C. Herz [11] with different notation. Our notation is adapted to the Feichtinger norms which are the most appropriate for our aims. In § 1 we introduce the spaces Apq and Apq and their duals Bp_q and Bp-q and briefly study those properties which will be important for the development of Hardy space theory. In § 2 we define the Hardy spaces HApq and HApq as spaces of tempered distributions whose non-tangential Poisson maximal function belongs to Apq or Apq respectively. We obtain several equivalent characterizations including an atomic decomposition, which are collected in Theorem 2.14. This section includes a description of the complex interpolation for these spaces. This problem is more difficult than that for Banach spaces and we solve it along the lines of the method introduced by Calderon and Torchinsky [4]. Finally § 3 is devoted to the Littlewood-Paley characterization of HApq for Kp= s2 and 0<< 7^ 1. In this way we are able to complete previous results obtained by Lu and Yang [13].