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An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces. (English) Zbl 1252.42025
Let $(X,d,\mu)$ be a metric measure space with the geometric doubling property and the upper doubling condition for the measure $\mu$. In this setting, the regularized BMO space $\text{RBMO}(\mu)$ and the Hardy space $H^1(\mu)$ have been defined and studied in a number of recent papers. Here, the authors prove that any sublinear operator $T$ that is bounded from $H^1(\mu)$ to $L^{1,\infty}(\mu)$ and from $L^\infty(\mu)$ to $\text{RBMO}(\mu)$, is also bounded on $L^p(\mu)$ for all $p\in(1,\infty)$. This improves a result of {\it B. T. Anh} and {\it X. T. Duong} [“Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces”, \url{arXiv:1009.1274}, to appear in J. Geom. Anal.] who proved it for linear’ instead of sublinear’ and $L^1(\mu)$ instead of $L^{1,\infty}(\mu)$. The proof again uses the Calderón--Zygmund decomposition of Anh and Duong [op. cit.] in this setting, but also needs some new ideas.

##### MSC:
 42B35 Function spaces arising in harmonic analysis 42B25 Maximal functions, Littlewood-Paley theory 47B38 Operators on function spaces (general)
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##### References:
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