A new look at the John-Nirenberg and John-Strömberg theorems for BMO
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\[NEWLINE \| f^*\|_{BMO((0,|Q|))}\leq C\| f\|_{BMO(Q)}, NEWLINE\]NEWLINE where \(f^*\) is the non-increasing rearrangement of \(f\), to the case of the so called John-Strömberg functionals (Theorems 8.1 and 8.2).
scientific article; zbMATH DE number 6055162
scientific article