Local structure of generalized Orlicz-Lorentz function spaces (Q2822644)

Given a quasi-Banach function space \((E,\|\cdot\|_E)\) with positive cone \(E_+\), a point \(x\in E_+\setminus \{0\}\) is called a point of lower monotonicity (LM point) if for any \(y\in E_+\) with \(y\leq x\) and \(y\neq x\) one has \(\|y\|_E<\|x\|_E\). Points of upper monotonicity (UM points) are defined analogously. \(E\) is called strictly monotone if each \(x\in E_+\setminus \{0\}\) is an LM point (equivalently, each \(x\in E_+\setminus \{0\}\) is a UM point).NEWLINENEWLINE For a Musielak-Orlicz function \(\varphi\), the author considers the corresponding Musielak-Orlicz space \(L^{\varphi}\) on \(I=[0,1)\) or \(I=[0,\infty)\), equipped with its Luxemburg norm \(\|\cdot\|_{\varphi}\), and the associated Orlicz-Lorentz space \(\Lambda^{\varphi}\), which is the symmetrisation of \(L^{\varphi}\), i.e., \(\Lambda^{\varphi}\) consists of all measurable functions \(x\) on \(I\) with \(x^*\in L^{\varphi}\) and \(\|x\|_{\Lambda^{\varphi}}:=\|x^*\|_{\varphi}\), where \(x^*\) denotes the nonincreasing rearrangement of \(x\).NEWLINENEWLINE The author proves complete characterisations of LM- and UM-points in \(\Lambda^{\varphi}\). For this purpose, he first establishes characterisations of LM\(^*\)- and UM\(^*\)-points in \(L^{\varphi}\) (these points are defined analogously to LM- and UM-points, but restricting to nonincreasing elements \(x\) and \(y\)).NEWLINENEWLINE The author also obtains a characterisation of strict monotonicity of \(\Lambda^{\varphi}\) and sufficient conditions for lower local uniform monotonicity and order continuity of a point in \(\Lambda^{\varphi}\).