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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 \(n>1\).

MSC:

26E10 \(C^\infty\)-functions, quasi-analytic functions
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30D55 \(H^p\)-classes (MSC2000)
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