A characterization of productBMO by commutators
A characterization of productBMO by commutators
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DOI：
10.1007/bf02392840
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发表时间：
2002-09
期刊：
影响因子：
3.7
通讯作者：
S. Ferguson;M. Lacey
中科院分区：
文献类型：
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作者：
S. Ferguson;M. Lacey
本文建立了一个换向子估计，使我们可以具体地识别a . Chang和R. Fefferman的乘积BMO空间BMO （R2+ x R2+）作为L2 （R2）上的算子空间。这个结果的单参数类比是著名的Nehari[8]定理。本文的新颖之处在于我们讨论了由双参数膨胀族控制的情况，因此空间h1和BMO具有更复杂的结构。其中R2+表示上半平面，BMO （R2+•R2+）定义为实变Hardy空间h1在积域R2+ x R2+上的对偶。有几种等效的方法来定义后一个空间，读者可以参考[5]了解各种描述。我们将对h1的生物全纯类似物更感兴趣，它可以根据R2•R2+上的生物全纯函数的边值来定义，并将在整个过程中表示为Hi （R2+•cf.[10]）。在一个变量中，空间L2 (R)分解为H2 (R)|的直和，其中H2 (R)定义为H2 （R2+）中函数的边值，H2 (R)表示H2 (R)中函数的复共轭空间。因此，空间L2 （R2）分解为四个空间H2 (R)| H2 (R)@ H2 (R) H2 (R)|和H2 (R)|其中张量积是希尔伯特空间张量积。设P~-，•分别表示L2 （r2）在第一变量和第二变量全纯/反全纯子空间上的正交投影，设Hj表示第j变量j—1,2中的一维希尔伯特变换。根据投影P+，•
In this paper we establish a commutator estimate which allows one to concretely identify the product BMO space, BMO (R2+ x R2+), of A. Chang and R. Fefferman, as an operator space on L2 (R2). The one-parameter analogue of this result is a well-known theorem of Nehari [8]. The novelty of this paper is that we discuss a situation governed by a twoparameter family of dilations, and so the spaces H 1 and BMO have a more complicated structure.Here R2+ denotes the upper half-plane and BMO (R2+• R2+) is defined to be the dual of the real-variable Hardy space H 1 on the product domain R2+ x R2+. There are several equivalent ways to define this latter space, and the reader is referred to [5] for the various characterizations. We will be more interested in the biholomorphic analogue of H 1, which can be defined in terms of the boundary values of biholomorphic functions on R 2• R2+ and will be denoted throughout by Hi (R2+• cf.[10]. In one variable, the space L2 (R) decomposes as the direct sum H2 (R)| where H2 (R) is defined as the boundary values of functions in H2 (R2+) and H2 (R) denotes the space of complex conjugate of functions in H2 (R). The space L2 (R2), therefore, decomposes as the direct sum of the four spaces H2 (R)| H2 (R)@ H2 (R), H2 (R)| and H2 (R)| where the tensor products are the Hilbert space tensor products. Let P~-,• denote the orthogonal projection of L2 (R 2) onto the holomorphic/anti-holomorphic subspaces, in the first and second variables, respectively, and let Hj denote the one-dimensional Hilbert transform in the jth variable, j--1, 2. In terms of the projections P+,•