Local to global results for spaces of BMO type
type: Item
instance of: scholarly article
title: Local to global results for spaces of BMO type
zbMATH Open document ID: 1339.43003
author: Niko Marola
author: Olli Saari
published in: Mathematische Zeitschrift
publication date: 2016-02-02T00:00:00Z
review text: Let \(Q\) be a Euclidean cube. A function \(u\in L^1\left(Q\right)\) belongs to the John-Nirenberg class \(JN_p\left(Q\right)\) if \[ K_u^p\left(Q\right):= \sup_{{\mathcal W}}\sum_{P\in{\mathcal W}}\left|P\right|\left(\int_P\!\!\!\!\!\!\!-\left|u-u_P\right|\,dx\right)^p<\infty, \] where the supremum is taken over all partitions of \(Q\) into pairwise disjoint cubes \(P\), and \[ u_E=\int_E\!\!\!\!\!\!\!-u\,d\mu=\frac{1}{\mu\left(E\right)}\int_E u\,d\mu. \] Note that \(\lim_{p\to\infty}K_u=\left\|u\right\|_{BMO}\). The article is devoted to studying local and global properties of \(JN_p\). It is motivated by corresponding relations between \(BMO\) spaces and their local counterparts \(BMO_{loc}\). Let us introduce some notations. Let \(X=\left(X,d,\mu\right)\) be a metric space with metric \(d\) and doubling measure \(\mu\). For an open subset \(\Omega\subset X\), \(\tau\geq 1\), and \(1<p<\infty\), an \(L^1\left(\Omega\right)\) function \(u\) belongs to the space \(JN_{p,\tau}\left(\Omega\right)\) if \[ K_{u,\tau}^p\left(\Omega\right):= \sup_{{\mathcal F}}\sum_{\tau B\in{\mathcal F}}\mu\left(B\right)\left(\int_B\!\!\!\!\!\!\!-\left|u-u_B\right|\,d\mu\right)^p<\infty, \] where \({\mathcal F}\) denotes any collection of pairwise disjoint balls contained in \(\Omega\). If \(\tau=1\), it can be omitted in the notation. The key tool for the authors' investigation is Boman sets. A useful estimate for them is obtained in the article. Then, the following `local to global' result is derived. Theorem. Let \(X\) be a metric measure space with a doubling measure \(\mu\) such that all balls are Boman sets with parameters \(C_1\), \(C_2\), \(C_3\), \(M\) and \(\lambda\). Then, for every open \(\Omega\subset X\) we have \(JN_{p,\tau}\left(\Omega\right)=JN_p\left(\Omega\right)\) provided \(\tau\leq C_2/C_1\). The last section of the article is devoted to a John-Nirenberg inequality. The sufficient condition obtained says that if \(\Omega\) is a Boman set with parameters \(C_1\), \(…
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 30H35
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 6538594
zbMATH Keywords: doubling measure
zbMATH Keywords: embedding
zbMATH Keywords: John-Nirenberg space
zbMATH Keywords: metric space
zbMATH Keywords: BMO space
zbMATH Keywords: local BMO space
zbMATH Keywords: John-Nirenberg inequality
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
full work available at URL: s00209-015-1549-x
OpenAlex ID: W2293372878
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DOI: 10.1007/S00209-015-1549-X
description: scientific article; zbMATH DE number 6538594