Remarks on BMO-regularity and AK-stability
The author compares the notions of \(BMO\)-regularity and \(AK\)-stability of a couple of quasi-Banach lattices. A couple of quasi-Banach lattices \((X,Y)\) on \((\mathbb T\times \Omega, m\times \mu)\) is called \(BMO\)-regular (with constants \((C,M)\)) whenever for any \((0,0)\neq(f,g)\in X\times Y\) there exists \((u,v)\in X\times Y\) such \(|f|\leq u, |g|\leq v\) almost everywhere with \(\|u\|_X\leq M\|f\|_X, \|v\|_Y\leq M\|g\|_Y\) and \(\|\log( u(\cdot,w)/v(\cdot,w))\|_{BMO}\leq C\) for almost all \(w\in \Omega\). The couple is called weak-\(BMO\)-regular if \((XL_1,XL_1)\) is \(BMO\)-regular. One of the main results in the paper shows that for Banach lattices with the Fatou property both notions coincide, proving a conjecture posed by \textit{S. V. Kisliakov} [``Interpolation of \(H^p\)-spaces: Some recent developments'', Isr. Math. Conf. Proc. 13, 102--140 (1999; Zbl 0956.46018)]. On the other hand, a couple of quasi-Banach lattices \((X,Y)\) on \((\mathbb T\times \Omega, m\times \mu)\) is called \(AK\)-stable (with constant \(c\)) if \((X_A, Y_A)\) is \(K\)-closed in \((X,Y)\) with constant \(c\), where \(X_A=\{f\in X: f(\cdot,w)\in N^+ \) for a.e. \(w\in \Omega\}\) and \(N^+\) is the Smirnov class. We recall that \((X_1, Y_1)\) is \(K\)-closed in \((X,Y)\) if for any \(f\in X_1+Y_1\), \((g_0, h_0)\in X\times Y\) with \(f=g_0+h_0\) there exist \((g_1, h_1)\in X_1\times Y_1\) with \(f=g_1+h_1\) and \(\|g_1\|_X\leq c\|g_0\|_X\) and \(\|h_1\|_Y\leq c\|h_0\|_Y\). \((X,Y)\) is called strongly-\(AK\)-stable if the conclusion holds for \(f\in (X+Y)_A\) instead of \(X_A+Y_A\) and boundedly-\(AK\)-stable if for any \((f,g)\in X\times Y\) there exists \(U\in H^\infty(m\times \mu)\) with \(\|U\|_\infty\leq c\) such that \(\|gU\|_X\leq c \|f\|_X\) and \(\|f(1-U)\|_Y\leq c \|g\|_Y\). Connections between all these properties are presented. Among other things it is shown that for Banach lattices having some mild assumptions the couple \((X,Y)\) is \(BMO\)-regular if and only if \((X(\ell^\infty), Y(\ell^1))\) is \(AK\)-stable.
scientific article; zbMATH DE number 5924718