The distance from functions in BMO to BLO

Abstract

Let BMO and BLO denote the spaces of all locally integrable real-valued functions on ${\mathbb{R}}^n$ with bounded mean oscillation and bounded lower oscillation, respectively. It is well known that $$L^{\infty }({\mathbb{R}}^n)⫋\mathrm{BLO}⫋\mathrm{BMO}.$$ In 1978, Garnett and Jones gave distance formulas of $f\in \mathrm{BMO}$ to $L^{\infty }({\mathbb{R}}^n)$ and recently, Angrisani studied the distance of $f\in \mathrm{BLO}$ to $L^{\infty }({\mathbb{R}}^n)$. In this paper, we characterize the distance from any given function $f\in \mathrm{BMO}$ to BLO via the Muckenhoupt weight class $A_p$ as follows \begin{center} dist\,($f$,\ BLO)\,$\sim \inf \left\{\hat{I}¾\in (0,\infty ):e^{-\frac{f}{\hat{I}}¾}\in A_p\mathrm{for}\mathrm{some}p\in (1,\infty )\right\}$. \end{center} Two equivalent representations of this distance are also established in terms of exponential form and the infimum of the constant in a variant of John--Nirenberg inequality, respectively.