Local to global results for spaces of BMO type
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 \(C_1\), \(C_2\), \(C_3\), \(M\) and \(\lambda\), then for any \(u\in JN_p\left(\Omega\right)\), \[ \lambda^p\mu\left(\left\{x\in\Omega:\,\left|u\left(x\right)-u_\Omega\right|>\lambda\right\}\right)\leq C K_u^p\left(\Omega\right), \] where \(C\) depends only on \(p\), the doubling constant for the measure \(\mu\), and the Boman parameters of \(\Omega\). Finally, a necessary condition for a weak type inequality is obtained: Theorem. Let \(1<p<\infty\) and \(\tau_0\geq 1\) be fixed and let \(X\) be a metric measure space with a doubling measure \(\mu\) and which supports a \(\left(p,p\right)\)-Poincaré inequality with dilatation constant \(\tau\leq\tau_0\). Suppose that for every \(u\in L^1\left(\Omega\right)\) \[ \inf_{c\in{\mathbb R}}\sup_{\lambda>0}\lambda^p\mu\left(\left\{x\in\Omega:\,\left|u\left(x\right)-c\right|>\lambda\right\}\right) \leq C K_{u,\tau_0}^p\left(\Omega\right). \] If \(g\) is an upper gradient of \(u\), then \[ \inf_{c\in{\mathbb R}}\left(\int_\Omega\left|u-c\right|^p\,d\mu\right)^{1/p}\leq C\left(\int_\Omega g^p\,d\mu\right)^{1/p}. \] Let us mention that the authors also provide very good historical references for the results considered, and proper motivation. The article should be interesting for specialists in harmonic, complex, and functional analysis.
scientific article; zbMATH DE number 6538594
scientific article