Orlicz Spaces – Weak Convergence + Norm Convergence Implies Strong Convergence

Question

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?

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

Answer 1

The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $\|\cdot\|_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto\|x\|_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $\|\cdot\|_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

The answer is provided by a combination of:

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),\|\cdot\|_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

This means, in particular, that the following assumptions (included in the original post) are obsolete:

1. $\Phi$ is an N-function (i.e. $\lim_{u\rightarrow^+0}\frac{\Phi(u)}{u}=0$ and $\lim_{u\rightarrow\infty}\frac{\Phi(u)}{u}=\infty$);
2. strengthening of $\Delta_2^\infty$ to $\Delta_2$ (i.e. assuming also that $\lim_{u\rightarrow^+0}\frac{\Phi(2u)}{\Phi(u)}<\infty$).

References

[1] Anna Kamińska, 1984, "The criteria for local uniform rotundity of Orlicz spaces", Stud. Math. 79, 201–215, MR0781718, Zbl 0573.46014.

[2] Rudolf Výborný, 1956, "O slabé konvergenci v prostorech lokálně stejnoměrně konvexních", Časopis pěstov. matem. 81, 352–353, Zbl 0075.11702.

[3] Vitol'd L'vovich Shmul'yan, 1939, "On some geometrical properties of the sphere in a space of the type (B)", Dokl. Akad. nauk SSSR 24, 648–652.

[4] Marston Morse, William Transue, 1950, "Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$", Ann. Math. 51, 576–614, MR31654, Zbl 0035.07105.

[5] Hidegorō Nakano (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō, MR38565, Zbl 0041.23401.