Vector ₂ weights and a Hardy-Littlewood maximal function
Vector ₂ weights and a Hardy-Littlewood maximal function
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矢量 2 权重和 Hardy-Littlewood 极大值函数
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发表时间：
2001
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通讯作者：
M. Goldberg
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作者：
M. Christ;M. Goldberg
本文对有限维希尔伯特空间中取值的函数，引入了Hardy-Littlewood极大函数的一个类似物。它被证明是L2有界相对于在类A2的Treil的重量，从而延长Muckenhoupt定理从标量到向量的情况下。奇异积分算子的一个基本章节是加权范数理论，它提供了这样的算子的非负函数w的一个充分必要条件，以及Hardy-Littlewood极大函数M在L中关于测度w（x）dx有界的充分必要条件。见[3]、[6]、[12]。最近，这个理论的某些方面首先由Treil和Volberg [17，18]扩展，然后由其他作者扩展到在有限维希尔伯特空间中取值的函数，权重在相应的埃尔米特形式空间中取值。这些扩展依赖于新的想法，与早期作者在标量情况下采用的想法完全不同，并为标量理论提供了新的思路。尽管如此，标量理论的某些方面显然没有被成功地推广，包括M本身。事实上，怀疑已表示[20，17]是否可能存在任何有用的类似物的M。本文引入了Hardy-Littlewood极大函数的一个向量模拟，并证明了它在最简单的情形p = 2下是有界的.在这样做的时候，我们首先寻求澄清新的矢量理论和更熟悉的标量情况之间的关系，其次，开辟一个替代的方法来处理这个问题，这可能导致矢量类似的标量理论的其他功能。第二位作者[5]进一步推进了这一计划，证明了我们定理的类似物对所有1 < p < ∞都成立，并且在[3]的精神下，奇异积分算子在L（R，H，w）上的有界性可以从我们的定理推导出来。当专门研究标量情形时，我们的分析与[3]中的不同之处在于，我们研究了未加权L上的算子f7 → w1/2 M（w−1/2f），而不是w-加权L上的算子M。虽然在这个共轭框架中的直接分析比[3]稍微复杂一些，但它从标量到矢量设置的推广是透明的。2000年6月22日由编辑接收。2000年数学学科分类。第42 B25章第一作者部分由NSF资助DMS-9970660支持。他感谢中国深圳竹园酒店的工作人员在热情好客的氛围中完成了部分工作。第二作者得到了NSF研究生奖学金的支持。c ©2001年美国数学学会1995年1996年迈克尔·克里斯特和迈克尔·戈德堡1.一个极大算子设H是一个可分的Hilbert空间，可能是无穷维的.用M表示H上所有关于H的范数有界的非负埃尔米特二次型的空间。同样的符号·将表示H的一个元素的范数，或M的一个元素的范数，被认为是从H到H的线性算子。如果w：R7 → M是可测的和局部可积的，在函数x7 → <$w（x）<$$>是局部可积的意义下，定义L（R，H，w（x）dx）是可测函数f：R7 → H的所有等价类的Banach空间，其中<$Rn <$w（x）f（x），f（x）<$p/2 dx是有限的.为了简化记法，我们将系统地记为gE = 1| E|其中E ∈ R，|E|表示E的勒贝格测度，g取R、H或M中的值，积分是关于勒贝格测度进行的。对于v ∈ M或R和r ∈ R，符号v E表示（vE）。A2 = A2（R，H）定义为所有w：R 7→ M的类，使得w，w−1是局部可积的，并且存在C <∞使得对每个立方Q ≤ C。(1.1)满足这个不等式的最小常数C将被称为w的A2“范数”。Treil和Volberg [17]证明了：如果H有有限维，则Hilbert变换H映射L（R，H，w（x）dx）有界到自身，当且仅当w ∈ A2.当H具有有限维数时，（1.1）等价于
An analogue of the Hardy-Littlewood maximal function is introduced, for functions taking values in finite-dimensional Hilbert spaces. It is shown to be L2 bounded with respect to weights in the class A2 of Treil, thereby extending a theorem of Muckenhoupt from the scalar to the vector case. A basic chapter of the subject of singular integral operators is the weighted norm theory, which provides a necessary and sufficient condition on a nonnegative function w for such operators, and for the Hardy-Littlewood maximal function M , to be bounded in L with respect to the measure w(x) dx. See [3], [6], [12]. More recently, aspects of this theory have been extended first by Treil and Volberg [17, 18], then by other authors, to functions taking values in finite-dimensional Hilbert spaces, with weights taking values in the corresponding spaces of Hermitian forms. These extensions have relied on new ideas, quite different from those employed in the scalar case by earlier authors, and have shed new light on the scalar theory. Nonetheless, some aspects of the scalar theory had apparently not been successfully generalized, including M itself. Indeed, doubts have been expressed [20, 17] as to whether there could exist any useful analogue of M . In this note we introduce a vector analogue of the Hardy-Littlewood maximal function, and prove its boundedness, in the simplest case, p = 2. In doing so, we seek firstly, to clarify the relationships between the new vector theory, and the more familiar scalar case, and secondly, to open up an alternative approach to the subject, which might lead to vector analogues of other features of the scalar theory. The second author [5] has carried this program further by demonstrating that the analogue of our theorem holds for all 1 < p < ∞, and that the boundedness of singular integral operators on L(R,H, w) can be deduced from our theorem, in the spirit of [3]. When specialized to the scalar case, our analysis differs from that in [3] in that we study the operator f 7→ w1/2M(w−1/2f) on unweighted L, rather than M on w–weighted L. Although the direct analysis in this conjugated framework is slightly more intricate than [3], its generalization from the scalar to the vector setting is transparent. Received by the editors June 22, 2000. 2000 Mathematics Subject Classification. Primary 42B25. The first author was supported in part by NSF grant DMS-9970660. He thanks the staff of the Bamboo Garden hotel in Shenzhen, PRC, for the hospitable atmosphere in which a portion of this work was done. The second author was supported by an NSF graduate fellowship. c ©2001 American Mathematical Society 1995 1996 MICHAEL CHRIST AND MICHAEL GOLDBERG 1. A Maximal Operator Let H be a separable Hilbert space, possibly of infinite dimension. Denote by M the space of all nonnegative Hermitian quadratic forms on H that are bounded with respect to the norm of H. The same symbol ‖ · ‖ will denote the norm either of an element of H, or of an element of M, considered as a linear operator from H to H. If w : R 7→ M is measurable and locally integrable in the sense that the function x 7→ ‖w(x)‖ is locally integrable, define L(R,H, w(x) dx) to be the Banach space of all equivalence classes of measurable functions f : R 7→ H for which ∫ Rn〈w(x)f(x), f(x)〉 p/2 dx is finite. To simplify notation we will systematically write gE = 1 |E| ∫ E g, where E ⊂ R, |E| denotes the Lebesgue measure of E, g takes values in R,H, or M, and the integral is taken with respect to Lebesgue measure. For v ∈ M or R and r ∈ R, the notation v E means (vE) . A2 = A2(R,H) is by definition the class of all w : R 7→ M such that w,w−1 are locally integrable and there exists C <∞ such that ‖w Q (w−1) 1/2 Q ‖ ≤ C for every cube Q ⊂ R. (1.1) The smallest constant C satisfying this inequality will be called the A2 “norm” of w. Treil and Volberg [17] have proved that if H has finite dimension, then the Hilbert transform H maps L(R,H, w(x) dx) boundedly to itself, if and only if w ∈ A2. When H has finite dimension, (1.1) is equivalent to