Characterization of the Orlicz spaces whose convergence is equivalent to convergence in measure on reflexive subspaces
Consider a (closed) subspace \(Y\) of a rearrangement invariant space of integrable functions \(X\) on \([0, 1]\). The subspace \(Y\) is said to be strongly embedded in \(X\) if the norm convergence of \(Y \) is equivalent to convergence in measure. \(F\) is an increasing convex function on \([0, \infty)\), $F(0) = 0$, and \(L_F\) is the corresponding Orlicz space on the interval \([0, 1]\). As was shown by \textit{E. Lavergne} [Colloq. Math. 113, No. 2, 333--340 (2008; Zbl 1152.46018)], the condition \[ \lim_{t \to \infty} \frac{\tilde{F}(ct)}{\tilde{F}(t)} = \infty \tag{1} \] for some \(c > 1\), where \(\tilde{F}\) is the complementary function to \(F, \) implies that every reflexive subspace of \(L_F\) is strongly embedded in \(L_F\). The goal of this article is to find necessary and sufficient conditions on \(F\) under which every reflexive subspace of \(L_F\) is strongly embedded in \(L_F\). Condition (1) on \(F\) guarantees ``proximity'' of the Orlicz space \(L_F\) to \(L_1\). Theorem~1, as the principal result of this article, demonstrates that all reflexive subspaces of an Orlicz space which are sufficiently ``close'' to \(L_\infty\) are also strongly embedded. Other interesting related results are also shown.
scientific article; zbMATH DE number 7137341
scientific article