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Weak $L^{\infty}$ and BMO in Metric Spaces

Daniel Aalto — 2012

Bollettino dell'Unione Matematica Italiana

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.

John-Nirenberg lemmas for a doubling measure

Daniel Aalto; Lauri Berkovits; Outi Elina Kansanen; Hong Yue — 2011

Studia Mathematica

We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderón-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

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Weak L ∞ and BMO in Metric Spaces

John-Nirenberg lemmas for a doubling measure

Weak $L^{\infty}$ and BMO in Metric Spaces