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A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. (English) Zbl 1246.30087
Summary: A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces $$(\mathbb{R}^n,\mu)$$ with $$\mu(B(x,r))\leq Cr^d$$, in which non-doubling harmonic analysis has recently been developed. It seems to be a promising framework for an abstract extension of this theory. Tolsa’s space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven.

##### MSC:
 30L99 Analysis on metric spaces 42B35 Function spaces arising in harmonic analysis 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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