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An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces. (English) Zbl 1252.42025
Let (X,d,μ) be a metric measure space with the geometric doubling property and the upper doubling condition for the measure μ. In this setting, the regularized BMO space RBMO(μ) and the Hardy space H 1 (μ) have been defined and studied in a number of recent papers. Here, the authors prove that any sublinear operator T that is bounded from H 1 (μ) to L 1, (μ) and from L (μ) to RBMO(μ), is also bounded on L p (μ) for all p(1,). This improves a result of B. T. Anh and X. T. Duong [“Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces”, arXiv:1009.1274, to appear in J. Geom. Anal.] who proved it for ‘linear’ instead of ‘sublinear’ and L 1 (μ) instead of L 1, (μ). The proof again uses the Calderón–Zygmund decomposition of Anh and Duong [op. cit.] in this setting, but also needs some new ideas.
MSC:
42B35Function spaces arising in harmonic analysis
42B25Maximal functions, Littlewood-Paley theory
47B38Operators on function spaces (general)