Weak \(L^\infty\) and \(BMO\) in metric spaces (Q2910081)

In this interesting article, the author characterizes the weak \(L^\infty\) space using exponential estimates for the distribution function instead of the decreasing rearrangement of a function. Moreover, a characterization of essentially bounded functions as well as of the BMO space is given by a condition similar to that of bounded mean tail oscillation. The connection between the weak \(L^\infty\) and BMO space is also given, namely, every BMO-function with a finite distribution function belongs to the weak \(L^\infty\).NEWLINENEWLINEAn important ingredient in the paper is a version of Calderón-Zygmund type covering lemma which, apparently, is different from the standard ones found in the literature.