EXTENSIONS OF HARDY SPACES AND THEIR USE IN ANALYSIS
EXTENSIONS OF HARDY SPACES AND THEIR USE IN ANALYSIS
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DOI：
10.1090/s0002-9904-1977-14325-5
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发表时间：
1977-01-01
影响因子：
1.3
通讯作者：
WEISS, G
中科院分区：
文献类型：
--
作者：
COIFMAN, RR;WEISS, G
1. 介绍。众所周知，函数理论在傅里叶级数的经典理论中占有重要的地位。正因为如此，在调和分析中，Hp空间得到了广泛的研究。当p >1时，If和Hp本质上是相同的；然而，当p< 1时，空间Hp{%更适合傅立叶级数理论中出现的问题。我们将研究p< 1时Hp的一些性质，并描述最近表征这些空间的方法。这些特征使我们能够将它们的定义扩展到一个非常普遍的环境，这将使我们能够统一对经典谐波分析的许多扩展的研究。Rn上的Hp空间理论最近从C. Fefferman和EM . Stein bbb的工作中得到了重要的推动。他们的工作导致了许多涉及卷积算子尖锐估计的应用。目前还不清楚R ‘ ’的微分结构在获得这些结果中起了多大的作用。我们的目的是从这个理论中分离出一些测量理论和几何性质，这些性质使我们能够以统一的形式获得许多这些应用以及谐波分析中的其他结果。我们将不处理那些涉及与我们的目的无关的Hp空间的问题。涉及谐波分析和Hp空间的一般参考文献有[23]，[64],[27],[62]，[57]和[55]。我们开发的主要工具是Calderón-Zygmund将函数分解为“好”和“坏”*部分的扩展和细化。这个工具出现在定理A的证明中，有点技术性。把它包括在这里，是为了使我们发展的理论在本质上是自成一体的。在一些例子中，我们给出了这一理论的应用，这些应用需要的材料在这里没有给出。不过，我们确实会提供必要的参考资料。从这个意义上说，我们希望这个论述对普通读者来说是可以理解的。在开始我们的演讲之前，我们要感谢我们的同事a . Baernstein， Y. Meyer， R. Rochberg和EM Stein，他们阅读了大部分手稿并提出了许多有用的建议。假设f是平面上单位圆盘的周长T上的实值可积函数（我们通常用[-7T， TT]表示）。假设ƒ
1. Introduction. It is well known that the theory of functions plays an important role in the classical theory of Fourier series. Because of this certain function spaces, the Hp spaces, have been studied extensively in harmonic analysis. When p> 1, If and Hp are essentially the same; however, when p< 1 the space Hp {% much better adapted to problems arising in the theory of Fourier series. We shall examine some of the properties of Hp for p< 1 and describe ways in which these spaces have been characterized recently. These characterizations enable us to extend their definition to a very general setting that will allow us to unify the study of many extensions of classical harmonic analysis.The theory of Hp spaces on Rn has recently received an important impetus from the work of C. Fefferman and EM Stein [29]. Their work resulted in many applications involving sharp estimates for convolution operators. It is not immediately apparent how much of a role the differential structure of R" plays in obtaining these results. Our purpose is to isolate from this theory some of the measure theoretic and geometric properties that enable us to obtain in a unified form many of these applications as well as other results in harmonic analysis. We shall not deal with those questions involving Hp spaces that are not relevant to our purpose. Some general references involving harmonic analysis and Hp spaces are [23],[64],[27],[62],[57] and [55]. The main tool in our development is an extension and a refinement of the Calderón-Zygmund decomposition of a function into a" good" and" bad'* part. This tool is presented in the proof of Theorem A and is of a somewhat technical nature. It is included here in order to make the presentation of the theory we develop essentially self-contained. In some examples we give applications of this theory that require material not presented here. We do, however, give the necessary references. In this sense, we hope that this exposition is accessible to a general audience* Before beginning our presentation we would like to thank our colleagues A. Baernstein, Y. Meyer, R. Rochberg and EM Stein who read a large part of this manuscript and made many useful suggestions. Suppose ƒ is a real-valued integrable function on T, the perimeter of the unit disc in the plane (which we identify in the usual way with [-7T, TT)). Suppose ƒ