Geodesic
Weak $L^\infty$ and BMO in Metric Spaces
Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 369-385.

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Bennett, DeVore and Sharpley introduced the space weak $L^{\infty}$ in 1981 and studied its relationship with functions of bounded mean oscillation. Here we characterize the weak $L^{\infty}$ in measure spaces without using the decreasing rearrangement of a function. Instead, we use exponential estimates for the distribution function. In addition, we consider a localized version of the characterization that leads to a new characterization of BMO. 
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Aalto, Daniel. Weak $L^\infty$ and BMO in Metric Spaces. Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 369-385. http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_2_a10/

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