It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\to \|x\|_X$ implies strong convergence $\|x_n-x\|_X\to 0$. The typical example arising in PDEs is that of $L^p$ spaces, which are uniformly convex Banach spaces if and only if $1<p<+\infty$. It is also known that this property fails for $p=1$, see [1, exercise 4.19], probably also for $p=\infty$ as well (although I don't have a counterexample at hand).

Question:does anyone know of a similar result in Orlicz spaces? I-e if $\Phi:\mathbb R^+\to\mathbb R^+$ is a "good" $N$-function (satisfying all the properties one might need such as $\Delta_2$ condition and so on) and $f_n\in L^\Phi$ is a sequence converging weakly $f_n\rightharpoonup f$ such that $\|f_n\|_{L^\Phi}\to \|f\|_{L^\Phi}$ then in fact $\|f_n-f\|_{L^\Phi}\to 0$?

I am mostly interested in the case of a smooth, bounded domain $\Omega\subset \mathbb R^d$ with the Orlicz space $L\log^+ L(\Omega)$. For example, my wildest dream is as follows: if $\rho_n$ is a sequence of $L^1$ probability measures converging weakly $L^1$ to some limit $\rho$ and such that the convergence in entropy $\int_\Omega \rho_n(x)\log\rho_n(x) \mathrm d x\to \int_\Omega \rho(x)\log\rho(x) \mathrm d x $ holds, then $\rho_n\to \rho$ in $L \log L$. Or at least $\rho_{n_k}(x)\to \rho(x)$ pointwise a.e. for a posisble subsequence $n_k\to \infty$. But this is probably too much to hope for. I am still wondering if anything can be said in this case?

[1] Haïm Brézis, *Functional analysis, Sobolev spaces and partial differential equations*, Universitext, New York: Springer, pp. xiii+599 (2011), MR2759829, Zbl 1220.46002.

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