Timeline for If $f:U_1\to\mathcal L^p(\mu;E_2)$ is Fréchet differentiable, can we say anything about the Fréchet differentiability of $u\mapsto f(u)(\omega)$?
Current License: CC BY-SA 4.0
11 events
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| S Aug 3, 2020 at 12:04 | history | bounty ended | CommunityBot | ||
| S Aug 3, 2020 at 12:04 | history | notice removed | CommunityBot | ||
| Jul 26, 2020 at 13:57 | comment | added | 0xbadf00d | @BartMichels Convergence in $L^1$ at least implies convergence in measure and hence convergence almost everywhere along a subsequence. | |
| Jul 26, 2020 at 11:53 | comment | added | Bart Michels | This is very similar to "convergence in $L^1$ $\implies$ convergence almost everywhere?", and it seems you can use this to produce counterexamples. FWIW, one situation where I think the implication holds, is when $\Omega$ has a nice topology, $E_1 = \mathbb C$, $f$ is jointly continuous and $L^p$-holomorphic. When you then write $f(u, \omega) - f(0, \omega) = u f'(0, \omega) + u R(u, \omega)$; you can use Cauchy's integral formula to prove that $f'$ and $R$ are jointly continuous, and then $R \to 0$ in $L^p$ implies $R \to 0$ everywhere; i.e. $f$ is pointwise holomorphic (=for fixed $\omega$). | |
| S Jul 26, 2020 at 10:21 | history | bounty started | 0xbadf00d | ||
| S Jul 26, 2020 at 10:21 | history | notice added | 0xbadf00d | Canonical answer required | |
| Jul 24, 2020 at 18:35 | history | edited | 0xbadf00d | CC BY-SA 4.0 |
edited title
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| Jul 24, 2020 at 12:25 | comment | added | 0xbadf00d | @DanieleTampieri Thanks for noting. Didn't saw that for some reason. | |
| Jul 24, 2020 at 12:24 | history | edited | 0xbadf00d | CC BY-SA 4.0 |
added 2 characters in body
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| Jul 24, 2020 at 11:03 | comment | added | Daniele Tampieri | There are two dollar sign missing in the text below formula (1). | |
| Jul 24, 2020 at 9:46 | history | asked | 0xbadf00d | CC BY-SA 4.0 |