Local \(\Delta _2^E(x)\) condition as a crucial tool for local structure of Calderón-Lozanovskiĭ spaces
Given a \(\sigma\)-finite measure space \((T,\Sigma,\mu)\), a Köthe space \((E, \|\cdot\|_E)\) is a Banach space such that it is a subspace of the set of all \(\mu\)-measurable real valued functions \(L^0\) satisfying the conditions: (i) if \(x\in E\), \(y\in L^0\) and \(|y|\leq |x|\) a.e., then \(y\in E\) and \(\|y\|_E\leq \|x\|_E\); (ii) there exists a function \(x\in E\) positive on \(T\). Let \(\varphi : \mathbb{R} \rightarrow [0,\infty]\) be an Orlicz function such that it is convex, even, vanishing and continuous at zero, left continuous on \((0,\infty)\) and not identically equal to zero. Define on \(L^0\) a convex semimodular \(I_\varphi\) by \(I_\varphi(x) = \|\varphi\circ x\|_E\) if \(\varphi\circ x\in E\), and \(I_\varphi(x) = \infty\) otherwise. Then the Calderón-Lozanovskii space \(E_\varphi = \{x\in L^0: \exists{c>0} \;I_\varphi(cx) < \infty\}\) is a Banach space equipped with the norm \(\|x\|_\varphi = \inf\{\lambda>0: I_\varphi(x/\lambda) \leq 1\}\). A criterion for \(x\in E_\varphi\) is presented to be order continuous in terms of a new local growth condition \(\Delta_2(x)\). It is also proved that a Köthe sequence space \(E\) is lower locally uniformly monotone if and only if \(E\) is lower monotone and order continuous. The last result characterizes lower local uniform monotone points in \(E_\varphi\) in terms of \(\Delta_2(x)\) as well as some other useful parameters.
scientific article; zbMATH DE number 5566259
scientific article