Function spaces in harmonic analysis

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The proposed research program has as its objectives the study of function spaces of importance in the field of harmonic analysis and its applications. In general terms, many questions in analysis involve showing that certain operators are bounded, and one has to specify on which spaces. While the Lebesgue Lp spaces are the most basic, other fundamental spaces, such as Sobolev spaces, have proved essential to the study of regularity for partial differential equations. One of the major areas of modern harmonic analysis is the Calderon-Zygmund theory of singular integral operators, which also arise in the solution of PDE. In this theory, 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 interest in themselves and for their use is various applications, such as the div-curl lemma. Local versions can be defined 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 metric measure 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. The goal in transferring results from the Euclidean setting is to be able to free the proofs of their dependence on the smooth structure. For example, in order to discuss results such as the div-curl lemma, one has to be able to define notions arising from differential geometry without using derivatives. Initially it is best to consider the cases where there is extra structure, such as Lie groups or graphs, or even Euclidean space with a measure which is absolutely continuous, i.e. given by a weight. Developing the necessary tools in these settings is of importance in itself, independently of the study of function spaces. In particular, one needs to determine whether certain inequalities, such as the Poincare inequality, hold.