Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Simple Search

Query:
Enter a query and click »Search«...
Format:
Display: entries per page entries
Zbl 0358.26010
Garnett, John B.; Jones, Peter W.
The distance in BMO to $L^\infty$.
(English)
[J] Ann. Math. (2) 108, 373-393 (1978). ISSN 0003-486X; ISSN 1939-0980/e

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 $n>1$.

Display scanned Zentralblatt-MATH page with this review.
[John B. Garnett; Peter W. Jones]
MSC 2000:
*26E10 Infinitely differentiable real functions, etc.
46J15 Banach algebras of differentiable functions
30D55 H (sup p)-classes

Highlights
Master Server