A new look at the John-Nirenberg and John-Strömberg theorems for BMO
type: Item
instance of: scholarly article
title: A new look at the John-Nirenberg and John-Strömberg theorems for BMO
zbMATH Open document ID: 1254.46033
author: Michael Cwikel
author: Yoram Sagher
author: Pavel Shvartsman
published in: Journal of Functional Analysis
publication date: 2012-07-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.0766
reviewed by: Javier Soria
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 6055162
zbMATH Keywords: BMO
zbMATH Keywords: John-Nirenberg inequality
zbMATH Keywords: mean oscillation
zbMATH Keywords: John-Strömberg functional
zbMATH Keywords: dimension free inequality
MaRDI profile type: Publication
OpenAlex ID: W2029166233
arXiv ID: 1011.0766
cites work: Covering Lemmas and the Sharp Function
cites work: A Rearranged Good λ Inequality
cites work: Weak-\(L^\infty\) and BMO
cites work: Q3857021
cites work: Q4112938
cites work: \(H^p\) spaces of several variables
cites work: Monotonicity of certain functionals under rearrangement
cites work: On L(p,q) spaces
cites work: Local sharp maximal functions
cites work: Rotation and strain
cites work: On functions of bounded mean oscillation
cites work: A Mean Oscillation Inequality
cites work: ON THE CONNECTION BETWEEN MEAN OSCILLATION AND EXACT INTEGRABILITY CLASSES OF FUNCTIONS
cites work: Mean oscillations and equimeasurable rearrangements of functions
cites work: Maximal functions with respect to differential bases measuring mean oscillation
cites work: On Harnack's theorem for elliptic differential equations
cites work: Q4436159
cites work: On the John-Strömberg-Torchinsky characterization of BMO
cites work: Rearrangement-function inequalities and interpolation theory
cites work: Q4938737
cites work: Sharp results in the integral-form John–Nirenberg inequality
cites work: Q3142876
cites work: Q3866648
cites work: Sharp constants in the classical weak form of the John-Nirenberg inequality
cites work: On John and Nirenberg's theorem
DOI: 10.1016/J.JFA.2012.04.003
review text: The authors revisit the classical John-Nirenberg inequality [\textit{F. John} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)]. They start by posing an interesting open question:NEWLINENEWLINENEWLINENEWLINE Do there exist \(\tau\in(0,1/2)\) and \(s>0\), such that for every dimension \(d\in\mathbb N\) and every cube \(Q\subset\mathbb R^d\), if \(E_+,E_-\subset Q\) are disjoint sets with the property that NEWLINE\[NEWLINE \min(|E_+|,|E_-|)>\tau|Q\setminus E_+\setminus E_-|, NEWLINE\]NEWLINE then we can find another cube \(W\subset Q\) for which \(\min(|W\cap E_+|,|W\cap E_-|)\geq s|W|\)?NEWLINENEWLINENEWLINENEWLINE An affirmative answer to this geometric question would imply a \`\` dimension free\'\'\ version of the John--Nirenberg inequality: NEWLINE\[NEWLINE\Big |\Big\{x\in Q:|f(x)-m_f|\geq\alpha\Big\}\Big|\leq\max\bigg(\frac1{2\tau},\sqrt{\frac2{\tau}}\bigg)|Q|\text{exp}\bigg(-\frac{\alpha s\log\big(\frac1{2\tau}\big)}{8\| f\|_{BMO}}\bigg).\tag{1} NEWLINE\]NEWLINE Even though they can solve this problem with at least a constant depending on \(d\) (for example, \(\tau =\sqrt2 -1\) and \(s=2^{-d}(3-2\sqrt2)\)), it is interesting to observe that in the expression of the form \(C|Q|\text{exp}\big(-c\alpha/\| f\|_{BMO}\big)\) on the right-hand side of (1), they are able to reveal a quite explicit connection between the constants \(C,\;c, \;\tau\), and \(s\).NEWLINENEWLINEAn important argument used to prove their main result (Theorem 9.1), which is their most general version of (1), is the fact that they can reduce the computations to the case where the function \(f\) takes only the values 0, 1, and 2 (this was already known for \(d=1\) [\textit{V. Vasyunin} and \textit{A. Volberg}, ``Sharp constants in the classical weak form of the John-Nirenberg inequality'', \url{arXiv:1204.1782}]).NEWLINENEWLINEFurther applications are considered to the extension of the estimate NEWLINE\[NEWLINE \| f^*\|_{BMO((0,|Q|))}\leq C\| f\|_{BMO(Q)}, NE…
description: scientific article; zbMATH DE number 6055162