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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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  • $\begingroup$ The Boltzmann entropy is only l.s.c. in the weak topology of $L^1 (\Omega)$. Do you have some situations where we have the convergence $\int_\Omega \rho_n(x)\log\rho_n(x) \mathrm d x\to \int_\Omega \rho(x)\log\rho(x) \mathrm d x$? $\endgroup$
    – Akira
    Commented Nov 24, 2023 at 16:09
  • $\begingroup$ Well, that's the whole point: for some independent reason I'm able to prove that the entropy converges. and I want to improve the convergence, i-e $\rho_n\to\rho$ should hold in a stronger sense than I initially assumed. $\endgroup$ Commented Nov 24, 2023 at 16:12
  • $\begingroup$ Let me just give a heuristic reasoning as to why this may hold true: it is well known that lack of strong convergence can only arise form oscillations. But, wild oscillations tend to encode lots of information in the sense of 1/0 bits in information theory. Such bits correspond to high entropy. But since $\limsup E(\rho_n)\leq E(\rho)$ the sequence cannot contain too much information, thus it should not oscillate too fast, and converge better than expected. OK sure, this is a very very rough picture, but I think there's some truth to it and I'm trying to make put this intuition on solid ground $\endgroup$ Commented Nov 24, 2023 at 16:18
  • $\begingroup$ Thank you so much for your elaboration! $\endgroup$
    – Akira
    Commented Nov 24, 2023 at 16:19
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    $\begingroup$ let me just mention that this property is known as Kadets-Klee or Radon-Riesz property $\endgroup$
    – erz
    Commented Nov 24, 2023 at 17:16

2 Answers 2

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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.

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  • $\begingroup$ Thank you @rpk for this clean answer. I also appreciate the historical comments, it's always good to give credit to whomever credit is due! $\endgroup$ Commented Apr 2 at 21:51
  • $\begingroup$ You are welcome! Ideally, one would like to specify characterisation of the Riesz–Radon–Shmul'yan property of $(L_\Phi(\mathcal{X},\mu),||\cdot||_\Phi)$ for nonatomic finite $(\mathcal{X},\mu)$. Unfortunately, this result seems to be missing in the literature. Yet, I am quite certain the answer is the same as the sufficient condition above.* $\endgroup$
    – rpk
    Commented Apr 2 at 22:23
  • $\begingroup$ *An evidence for this claim is provided by the characterisation of the RRS property for $\mu(\mathcal{X})=\infty$ by ($\Delta_2$ and strict convexity of $\Phi$), in Thm. (p. 341) of: Tíng Fǔ Wáng (王廷辅), Yún Ān Cuī (崔云安), Tāo Zhāng (张弢), 1998, "Kadec–Klee property in Musielak–Orlicz function spaces equipped with the Luxemburg norm", Scient. Math. 1, 339–345. It is typical in the Orlicz space theory that the change from $\mu(\mathcal{X})<\infty$ to $\mu(\mathcal{X})=\infty$ corresponds to change from $\Delta_2^\infty$ to $\Delta_2$. $\endgroup$
    – rpk
    Commented Apr 2 at 22:29
  • $\begingroup$ *More precisely, the missing thing is a characterisation for a (finite) Young function. In case if $\Phi$ is an N-function, then this characterisation (by the same conditions as in my answer) has been given independently in: Thm. 3 of: Artur M. Medzhitov, Fëdor A. Sukochev, 1992, "The property ($H$) in Orlicz spaces", Bull. Pol. Acad. Sci. Math. 40, 5–11; and Thm. 1 of: Tíng Fǔ Wáng (王廷辅), Yún Ān Cuī (崔云安), 1998, "关于Orlicz空间的H性质的注记", Acta Math. Sci. 18, 217–220. $\endgroup$
    – rpk
    Commented Apr 3 at 0:39
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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.

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  • $\begingroup$ Hi Ayman, funny to bump into you, I'm a friend and coauthor of Clément Cancès, and I'm currently trying to apply your "time compactness" trick with him and Boris. Small world. I'm well aware of Minty's trick, but unfortunately the entropy $z\log z$ is not monotone. And I don't think I have any other angle of attack for my speciic problem, the $L^p$ norms are unfortunately out of reach in my particular problem. Thanks antway, I appreciate you taking the time to answer and I hope we can meet in person some day. À plus! $\endgroup$ Commented Nov 24, 2023 at 21:31
  • $\begingroup$ Ah ! That's indeed funny ! About your problem, I was hoping for some kind of localization argument because for $\varepsilon>0$ and suitable $C_\varepsilon>0$, $z\log z + C_\varepsilon z$ is indeed increasing for $z>\varepsilon$, but it's not that clear that you can indeed localize without oscillations $\rho_n \geq \varepsilon$. I'll think about that next week and tell you if I get to anything new ! Bon week-end :) $\endgroup$ Commented Nov 24, 2023 at 21:39
  • $\begingroup$ In fact it's the increasingness of $\log$ which matters (see above the edit of my answer), however you need an extra estimate on $(\rho \log(\rho_n)_n$ to close the argument. $\endgroup$ Commented Nov 25, 2023 at 5:54
  • $\begingroup$ Merci Ayman. Unfortunately for my specific prolem I really don't see how to get started on $\int \rho\log\rho_n\to \int\rho\log\rho$. The entropy just hows up as a whoe, part of a functional that I'm minimizing, but that's it. So unless I manage to get some additional information I'm stuck. And I've tried hard, already! $\endgroup$ Commented Nov 26, 2023 at 4:46

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