Orlicz spaces with weakly normal structure (Q2721727)

Let \((G, \Sigma,\mu)\) be a Lebesgue measurable space in the Euclidean space with \(0 < \mu G < \infty\) and let \(M\) be an Orlicz function. Define NEWLINE\[NEWLINE\rho_M(u):=\int_G M(u(t))dtNEWLINE\]NEWLINE for every measurable function \(u :G\to \mathbb{R}\) and define NEWLINE\[NEWLINEL_M := \{u :G \to \mathbb{R}: u\text{ measurable, }\rho_M(\lambda u) < \infty\text{ for some }\lambda > 0\}.NEWLINE\]NEWLINE The authors give some conditions under which Orlicz spaces \((L_M,\|\;\|)\) and \((L_M,\|\;\|^0)\) are weakly uniformly rotund in every direction, where \(\|\;\|\) and \(\|\;\|^0\) denote the Luxemburg norm and the Orlicz norm in \(L_M\), respectively.