Aspects of local-to-global results (Q2922868)

Based on the authors' introduction: A function space which is larger than BMO was introduced by \textit{F. John} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)]. A function in this larger space is known to belong to a weak \(L^p\) space. The authors extend this weak type inequality to the case of John domains.NEWLINENEWLINELet \(G\) be a proper open subset of \(\mathbb{R}^n\),\ \ \(n \geq 2\). Define NEWLINE\[NEWLINE \mathcal{K}_{f}^p(G) =\sup_{\mathcal{P}(G)}\sum_{Q \in \mathcal{P}(G)}|Q|\biggl(\frac{1}{|Q|}\int_{Q}|f(x)-f_{Q}|dx \biggr)^p, NEWLINE\]NEWLINE where the supremum is taken over all partitions \(\mathcal{P}(G)\) of \(G\) into cubes such that \(Q \subset G\) for each \(Q \in \mathcal{P}(G)\), and the interiors of these cubes are pairwise disjoint, and \(G=\cup_{Q \in \mathcal{P}(G)}Q\). Define further NEWLINE\[NEWLINE \mathcal{K}_{f, \mathrm{loc}}^p(G) =\sup_{\mathcal{P}_{\mathrm{loc}}(G)} \sum_{Q \in \mathcal{P}_{\mathrm{loc}}(G)}|Q| \biggl(\frac{1}{|Q|}\int_{Q}|f(x)-f_{Q}|dx \biggr)^p, NEWLINE\]NEWLINE where the supremum is taken over all partitions \(\mathcal{P}_{\mathrm{loc}}(G)\) of \(G\) into cubes such that for each \(Q \in \mathcal{P}_{\mathrm{loc}}(G)\) a dilated cube \(\lambda Q\) satisfies \(\lambda Q \subset G\) with a fixed \(\lambda >1\), and these cubes have bounded overlap, i.\,e. NEWLINE\[NEWLINE \sup_{x \in G}\sum_{Q \in \mathcal{P}_{\mathrm{loc}}(G)}\chi_{Q}(x) \leq N NEWLINE\]NEWLINE where \(N \geq 1\) is a finite constant depending on \(n\) only.NEWLINENEWLINEThe equivalence of local and global BMO norms is a well-known result, due to \textit{H. M. Reimann} and \textit{T. Rychener} [Funktionen beschränkter mittlerer Oszillation. York: Springer-Verlag (1975; Zbl 0324.46030)]. The authors prove a Reimann-Rychener type local-to-global result. More precisely, if \(f \in L^1(G)\) and \(1 < p < \infty\), then NEWLINE\[NEWLINE \mathcal{K}_{f}^p(G)\leq C\mathcal{K}_{f, \mathrm{loc}}^p(G) NEWLINE\]NEWLINE where \(C\) is a positive constant that depends on \(n,\;p\) and \(\lambda\).NEWLINENEWLINEIn the second part of this paper, the authors consider necessary and sufficient conditions for Euclidean domains to support the weak-type inequality.NEWLINENEWLINETheorem. Suppose that \(G\) is a bounded domain in \(\mathbb{R}^n, \;n \geq 2\), satisfying a separation property. Assume further that \(n/(n-1) \leq p < \infty\). Then the weak-type inequality NEWLINE\[NEWLINE \inf_{c \in \mathbb{R}}\sup_{\sigma > 0} \sigma^p |\{ x \in G: |f(x)-c| > \sigma \}| \leq C\mathcal{K}_{f, \mathrm{loc}}^p(G) NEWLINE\]NEWLINE holds for every \(f \in L^1_{\mathrm{loc}}(G)\) if and only if \(G\) is a John domain.