Function spaces in harmonic analysis

所提出的研究方案以研究调和分析领域中的重要函数空间及其应用为目标。一般说来，分析中的许多问题都涉及到证明某些算子是有界的，并且必须指定在哪些空间上。虽然勒贝格Lp空间是最基本的，但其他基本空间，如Sobolev空间，已被证明是研究偏微分方程正则性所必需的。现代调和分析的主要领域之一是奇异积分算子的Calderon-Zygmund理论，它也出现在偏微分方程组的求解中。在这个理论中，当p为1时，Lp空间的尺度自然扩展到Hardy空间，当p为无穷大时，Lp空间的尺度自然扩展到John和Nirenberg的BMO(有界平均振荡函数)空间。这些空间本身是有趣的，它们的用途是各种应用，例如div-curl引理。我们建议将这些空间和其他相关空间的研究扩展到更一般的环境，如度量空间，并用它们来证明目前在欧几里德环境中已知的结果的类似。度量度量空间是分析和几何领域最近重要工作的主题。从欧几里得环境中转移结果的目的是能够使他们摆脱对光滑结构的依赖。例如，为了讨论像div-curl引理这样的结果，人们必须能够定义由微分几何产生的概念而不使用导数。最初，最好考虑有额外结构的情况，如李群或图，甚至欧几里德空间，其度量是绝对连续的，即由权给出。在这些环境下开发必要的工具本身就很重要，与函数空间的研究无关。特别是，人们需要确定某些不平等，如庞加莱不平等，是否成立。

The proposed research program has as its objectives the study of function spaces of importance in the field ofharmonic analysis and its applications. In general terms, many questions in analysis involve showing thatcertain operators are bounded, and one has to specify on which spaces. While the Lebesgue Lp spaces are themost basic, other fundamental spaces, such as Sobolev spaces, have proved essential to the study of regularityfor partial differential equations. One of the major areas of modern harmonic analysis is theCalderon-Zygmund theory of singular integral operators, which also arise in the solution of PDE. In thistheory, the scale of Lp spaces extends naturally to the Hardy space when p is 1, and to the space BMO(functions of bounded mean oscillation) of John and Nirenberg when p is infinite. These spaces are of interestin themselves and for their use is various applications, such as the div-curl lemma. Local versions can bedefined which are better adapted to working on domains or on manifolds.We propose to extend the study of these and other related spaces to more general settings such as metricmeasure spaces, and use them to prove analogues of results which are currently known in the Euclidean setting.Metric measure spaces are the subject of important recent work in the areas of analysis and geometry. Thegoal in transferring results from the Euclidean setting is to be able to free the proofs of their dependence on thesmooth structure. For example, in order to discuss results such as the div-curl lemma, one has to be able todefine notions arising from differential geometry without using derivatives. Initially it is best to consider thecases where there is extra structure, such as Lie groups or graphs, or even Euclidean space with a measurewhich is absolutely continuous, i.e. given by a weight. Developing the necessary tools in these settings is ofimportance in itself, independently of the study of function spaces. In particular, one needs to determinewhether certain inequalities, such as the Poincare inequality, hold.