Timeline for Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
Current License: CC BY-SA 4.0
17 events
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| Nov 22, 2022 at 15:24 | vote | accept | Akira | ||
| Nov 20, 2022 at 17:02 | comment | added | Nik Weaver | Yeah, looks fine. Good for you. | |
| Nov 20, 2022 at 12:02 | comment | added | Akira | @NikWeaver Could you have a check on my below answer? It seems the proof depends an a special property (the Lemma) of the canonical isometric isomorphism. | |
| Nov 20, 2022 at 12:00 | answer | added | Akira | timeline score: 1 | |
| Nov 15, 2022 at 4:37 | comment | added | Nik Weaver | Well, a quick way to do it is by applying Theorem 1 to $\Omega_1 \cup \ldots \cup \Omega_n$. The isometry it gives is just the isometry for each $k$, patched together. | |
| Nov 14, 2022 at 22:56 | comment | added | Akira | @NikWeaver I'm sorry if I'm bothering you. Could you please have a look at my update and elaborate on how to finish the reverse inequality? I would like to learn it very much... | |
| Nov 14, 2022 at 11:59 | comment | added | Akira | @JochenWengenroth I have formalized your idea here. It's elegant! | |
| Nov 13, 2022 at 22:15 | comment | added | Akira | @JochenWengenroth It seems I got your idea. I will try to formalize it. On the other hand, I'm curious if my failed approach can be fixed... | |
| Nov 13, 2022 at 21:30 | comment | added | Akira | @NikWeaver I have tried but fail to prove the existence of such $f$ that $$\left [ \sum_{m=1}^n \frac{|H_m (f_m)|}{\|f_m\|_{L_{p}(\mu_m, X)}} \right ]^q \le \left [ \frac{|H (f)|}{\|f\|_{L_{p}(\mu, X)}} \right ]^q$$? Could you elaborate more? | |
| Nov 13, 2022 at 21:30 | history | edited | Akira | CC BY-SA 4.0 |
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| Nov 13, 2022 at 14:38 | comment | added | Jochen Wengenroth | Did you try the simple fact that every $\sigma$-finite measure $\mu$ is of the form $\mu= \varphi\cdot \nu$ for a finite measure $\nu$ and a density $\varphi:\Omega \to (0,\infty)$? | |
| Nov 13, 2022 at 13:57 | comment | added | Nik Weaver | Now show that $\|L\|^q \geq \sum_1^n \|L_k\|^q$ for any $n$. | |
| Nov 13, 2022 at 13:40 | comment | added | Akira | @NikWeaver I'm able to prove that $\|L\|^q \le \sum \|L_n\|^q$. However, I failed to prove the reverse inequality. Could you have a check on my update? | |
| Nov 13, 2022 at 12:27 | history | edited | Akira | CC BY-SA 4.0 |
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| Nov 12, 2022 at 17:30 | comment | added | Nik Weaver | Hint: check that $\|L\|^q = \sum \|L_n\|^q$. | |
| Nov 12, 2022 at 17:26 | history | edited | Akira | CC BY-SA 4.0 |
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| Nov 12, 2022 at 17:20 | history | asked | Akira | CC BY-SA 4.0 |