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An Interpolation Theorem for Sublinear Operators on Non-homogeneous Metric Measure Spaces

Let $({\mathcal X}, d, μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors establish an interpolation result that a sublinear operator which is bounded from the Hardy space $H^1(μ)$ to $L^{1,\,\infty}(μ)$ and from $L^\infty(μ)$ to the BMO-type space ${\mathop\mathrm{RBMO}}(μ)$ is also bounded on $L^p(μ)$ for all $p\in(1,\,\infty)$. This extension is not completely straightforward and improves the existing result.