Timeline for What is the motivation of the $L^p$ differentiability?
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| May 7, 2020 at 16:38 | history | edited | Jingeon An-Lacroix | CC BY-SA 4.0 |
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| May 7, 2020 at 16:17 | vote | accept | Jingeon An-Lacroix | ||
| May 7, 2020 at 16:16 | history | edited | Willie Wong |
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| May 7, 2020 at 16:04 | comment | added | Ben McKay | @WillieWong I stand corrected. Very interesting. | |
| May 7, 2020 at 16:03 | history | edited | YCor |
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| May 7, 2020 at 16:00 | answer | added | Willie Wong | timeline score: 8 | |
| May 7, 2020 at 14:31 | comment | added | Willie Wong | For example, the function $f(x) = \sqrt{|x|}$ on $[-1,1]$ is $W^{1,p}$ for any $p \in [1,2)$. But it is not $L^p$ differentiable at $x = 0$ for any $p$: one can check that for any choice of $f'_p(0)$ necessarily the mean shown above is $O(\sqrt{h})$. // (@OP: you should probably edit this very important part of the definition into your question.) | |
| May 7, 2020 at 14:26 | comment | added | Willie Wong | @BenMcKay: I am not entirely convinced this is the Sobolev notion. The $ \| \cdot \|_p$ displayed in the question is $L^p$ mean in $h$. More precisely the definition is requiring that $$ \left( \frac{1}{h} \int_{-h}^h | f(x+s) - f(x) - f_p'(x) s|^p ds \right)^{1/p} = o(h) $$ | |
| May 7, 2020 at 8:33 | comment | added | Dave L Renfro | "The study of generalized $L^p$ differentiation began to become important in the 1950s in connection with the study of finding $L^p$ solutions for partial differential equations." However, as far as I can tell, Ash never discusses anything more, such as when the notion first appeared (but perhaps wasn't specifically singled out, and instead was buried in a theorem hypothesis or in a technical comment) and when the notion (but perhaps not using the present term) was first specifically singled out with a definition. | |
| May 7, 2020 at 8:33 | comment | added | Dave L Renfro | Several references I looked at cite Local properties of solutions of elliptic partial differential equations by Calderón/Zygmund (1961), but all the citations I've seen are for both first order $L^p$ derivatives and higher order $L^p$ derivatives, leaving open the possibility that the first order $L^p$ derivative might have appeared earlier. Some motivation can be found in Ash's 2014 paper Remarks on various generalized derivatives. The top of p. 9 (.pdf file page number) Ash writes: (continued) | |
| May 7, 2020 at 5:28 | comment | added | Ben McKay | The name Sobolev comes to mind, as it often does. en.wikipedia.org/wiki/Sobolev_space | |
| May 7, 2020 at 4:47 | history | edited | Jingeon An-Lacroix |
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| May 7, 2020 at 4:38 | history | edited | Jingeon An-Lacroix | CC BY-SA 4.0 |
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| May 7, 2020 at 4:30 | history | asked | Jingeon An-Lacroix | CC BY-SA 4.0 |