Garnett, John B.; Jones, Peter W. The distance in BMO to \(L^\infty\). (English) Zbl 0358.26010 Ann. Math. (2) 108, 373-393 (1978). 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 6 ReviewsCited in 17 Documents MSC: 26E10 \(C^\infty\)-functions, quasi-analytic functions 46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces 30D55 \(H^p\)-classes (MSC2000) PDFBibTeX XMLCite \textit{J. B. Garnett} and \textit{P. W. Jones}, Ann. Math. (2) 108, 373--393 (1978; Zbl 0358.26010) Full Text: DOI