Weak $L^{\infty}$ and BMO in metric spaces

TL;DR

This paper characterizes the weak $L^{ obreak ext{ extasciitilde}}$ space and BMO in measure spaces using exponential estimates, providing new insights without relying on decreasing rearrangements.

Contribution

It offers a novel characterization of weak $L^{ obreak ext{ extasciitilde}}$ and BMO in measure spaces through exponential estimates, bypassing traditional rearrangement methods.

Findings

Exponential estimates for distribution functions are established.

New localized characterization of BMO is introduced.

Results extend understanding of weak $L^{ obreak ext{ extasciitilde}}$ and BMO in metric measure spaces.

Abstract

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 weak $L^{\infty}$ in measure spaces without using the decreasing rearrangement of a function. Instead, we obtain exponential estimates for the distribution function. In addition, we consider a localized version of the characterization that leads to a new characterization of BMO.