Amemiya norm equals Orlicz norm in Musielak--Orlicz spaces (Q884887)

Let \(L^\Phi(\Omega)\) be a Musielak--Orlicz space over a \(\sigma\)-finite measure space \((\Omega, \mu)\), that is, \(L^\Phi(\Omega) = \{\mu \text{-measurable functions } u:\int_\Omega\Phi(|u(x)|/\lambda, x) \,d\mu <\infty\}\) for some \(\lambda > 0\), where \(\Phi\) is a Musielak--Orlicz function defined as follows: \(\Phi:\Omega\times [0,\infty) \rightarrow [0,\infty)\) and for almost all \(x\in\Omega\), \(\Phi(x,\cdot)\) is convex, lower semicontinuous, vanishing at zero, but not identically equal to zero or infinity, and for all \(t\in [0,\infty)\), \(\Phi(\cdot,t)\) is a \(\mu\)-measurable function. The author's main result is the demonstration that the Orlicz and Amemiya norms, \(\|\cdot\|_\Phi^0\) and \(\|\cdot\|_\Phi^A\), are equal on \(L^\Phi(\Omega)\), that is, \[ \|u\|_\Phi^0 = \sup \biggl\{|\int_\Omega u(x) v(x)\, d\mu|: I_{\Phi^*}(v) \leq 1\biggr\} =\inf_{k>0}\{(1/k)(1+ I_\Phi(ku))\} = \|u\|_\Phi^A. \] Here \(I_{\Phi^*}(v) = \int_\Omega\Phi^*(|v(x)|, x)\, d\mu\) and \(I_\Phi(u) = \int_\Omega\Phi(|u(x)|, x)\, d\mu\), where \(\Phi^*\) is the Musielak--Orlicz function conjugate to \(\Phi\). The equality between both norms was well known before for Orlicz functions, and the present result completes the problem in the case of the most general Musielak--Orlicz functions.