Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces

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.

Answer 2

There's a famous trick attributed to Minty (sometimes also to Browder and Lions) which allows to recover at least a.e. convergence when you're able to pass to the limit through an increasing non-linearity (typically $x\mapsto x^p$). As I guess you read french (right ?) you can check " Exercice 2 " in that document.

I don't know if's applicable in your example but it is worth giving a shot (and it's friday night, so I let you check !). I would believe avoiding the regions in which $(\rho_n)_n$ gets too closer to $0$ (which are actually OK as far oscillations are concerned), you should be able to recover an increasing setting for your non-linearity.

EDIT:

If you manage to prove furthermore that the integral of $(\log(\rho_n)\rho)_n$ converges to the integral of $\log(\rho)\rho$, you should be able to recover a.e. convergence. Note that this extra assumption is somehow weaker than the one you have. If indeed you have both, you consider $f_n:=(\log(\rho_n)-\log(\rho))(\rho_n-\rho)$. Since $\log$ is non-decreasing, you have $f_n\geq 0$. Expanding the expression you get by linearity (and using both assumptions above) that $$ \int_\Omega f_n \rightarrow 0.$$ Since $f_n\geq 0$, the previous is in fact the convergence $(f_n)_n\rightarrow 0$ in $L^1(\Omega)$. Extracting a subsequence you recover a.e. convergence for $(f_n)_n$ and using that $\log$ is increasing, this a.e. convergence can be transfered to $(\rho_n)_n$ itself.