Timeline for Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 3 at 0:39 | comment | added | rpk | *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. | |
| Apr 2 at 22:29 | comment | added | rpk | *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 $\Delta_2$. | |
| Apr 2 at 22:23 | comment | added | rpk | 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.* | |
| Apr 2 at 21:51 | comment | added | leo monsaingeon | 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! | |
| Apr 2 at 21:49 | vote | accept | leo monsaingeon | ||
| Apr 2 at 19:58 | history | edited | rpk | CC BY-SA 4.0 |
A preceding edit has introduced typos, and I am removing those in the current edit.
|
| Apr 2 at 19:45 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Added link to reviews (where existing)
|
| Apr 2 at 19:00 | history | edited | rpk | CC BY-SA 4.0 |
added 13 characters in body
|
| Apr 2 at 18:54 | history | edited | rpk | CC BY-SA 4.0 |
added 2 characters in body
|
| S Apr 2 at 18:45 | review | First answers | |||
| Apr 2 at 21:19 | |||||
| S Apr 2 at 18:45 | history | answered | rpk | CC BY-SA 4.0 |