An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces (Q437732)

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 \textit{B. T. Anh} and \textit{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.