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The distance in BMO to $L^\infty$. (English) Zbl 0358.26010
Let $\varphi$ be a function in $\mathrm{BMO}(\mathbb{R}^{n} )$, and let $\mathrm{dist}(\varphi,L^{\infty}) = \mathrm{inf} \{\|\varphi - g\| : g \in L^{\infty}\}$. Let $\varepsilon (\varphi )$ be the infimum of the set of $\varepsilon >0$ such that $\sup_{Q} \frac{1}{|Q|} | \{ x \in Q : | \varphi - \varphi _{Q}|>\lambda \} | < e^{-\lambda / \varepsilon } $, for all $\lambda >\lambda_{0} (\varepsilon)$. The John-Nirenberg theorem asserts that $\varepsilon(\varphi )<\infty $, while $\varepsilon (\varphi ) = 0$ if $\varphi \in L^{\infty}$. We prove that $c_{1}\varepsilon (\varphi ) \leq \mathrm{dist} (\varphi, L^{\infty }) \leq c_{2}\varepsilon (\varphi )$ for some constants $c_{1}$ and $c_{2}$. For $n=1$ this result was known previously, but the proof would not extend to $n>1$.
Reviewer: John B. Garnett; Peter W. Jones

26E10$C^\infty$ real functions, quasi-analytic real functions
46J15Banach algebras of differentiable or analytic functions, $H^p$-spaces
30D55H (sup p)-classes (MSC2000)
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