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I was reading some papers and come up with the next definition :

A function is differentiable in the $L^p$ sense at $x$ if there exists a real number $f'_p(x)$ such that $$\bigg(\frac{1}{h}∫_{-h}^{h}|f(x+s)−f(x)−f_p'(x)s|^pds\bigg)^{1/p}=o(h)$$

And he states that many $f_p'$ are equivalent to an ordinary derivative on an almost everywhere basis. I saw this kind of differentiability is studied in some papers, but I don't know where does it started, i.e. what was it's motivation to define such operation and function space.

Ash, J. Marshall, An (L^p) differentiable non-differentiable function, Real Anal. Exch. 30(2004-2005), No. 2, 747-754 (2005). ZBL1107.26010.

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    $\begingroup$ The name Sobolev comes to mind, as it often does. en.wikipedia.org/wiki/Sobolev_space $\endgroup$
    – Ben McKay
    Commented May 7, 2020 at 5:28
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    $\begingroup$ 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) $\endgroup$
    – Dave L Renfro
    Commented May 7, 2020 at 8:33
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    $\begingroup$ "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. $\endgroup$
    – Dave L Renfro
    Commented May 7, 2020 at 8:33
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    $\begingroup$ @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 $$ \left( \frac{1}{h} \int_{-h}^h | f(x+s) - f(x) - f_p'(x) s|^p ds \right)^{1/p} = o(h) $$ $\endgroup$
    – Willie Wong
    Commented May 7, 2020 at 14:26
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    $\begingroup$ 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.) $\endgroup$
    – Willie Wong
    Commented May 7, 2020 at 14:31

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I don't know the literature, but it maybe helpful to understand exactly what this notion of differentiability gains for us.

First, unlike the Sobolev notions, this does not handle functions which belong in Holder classes. For example, based on the definition the absolute value function is not $L^p$ differentiable for any $p$, but it is certainly (locally) in the Sobolev class $W^{1,p}$ for every $p\in [1,\infty]$.

In fact, whenever $f$ is a function such that the one sided derivatives from the left and from the right independently exist, but do not agree, such a function is not $L^p$ differentiable in the sense defined in the paper.

So what does $L^p$ differentiability gain for us? It gains when your function fails to be differentiable near $x$ due to it being oscillatory in a certain way. An example:

Let the set $A = \cup_{k = 10}^\infty [ 1/k - 2^{-k}, 1/k + 2^{-k}]$, and let $f$ be the indicator function of $A$. This function is clearly not differentiable at $x = 0$. However, since

$$ \int_{-h}^h |f(s)|^p ~ds \leq \sum_{k = \lfloor 1/h \rfloor }^\infty 2^{1-k} = 2^{2-\lfloor 1/h \rfloor } $$

we see that for any $p\in [1,\infty)$, $f$ is $L^p$ differentiable at the origin with derivative $0$.

As to why measuring things with respect to $L^p$ means can be useful: there is a big hint in the paper of Calderon and Zygmund, concerning elliptic PDEs. They wrote [emphases mine]:

It seems that the notion of differentiability which is most suited to the treatment of the problems that concern us, is not the classical one. It appears that it is more convenient to estimate the remainder of the Taylor series in the mean with various exponents. This type of differentiability is much more stable ...

If you are familiar with some harmonic analysis, what this is screaming out is that a lot of analytic estimates (Sobolev regularity for elliptic PDEs, singular integrals, etc.) work generally for functions measured on the $L^p$ scale, but often fail (just by a little) at $L^\infty$ (and sometimes $L^1$).

You can of course ask why did Calderon and Zygmund not use the Sobolev class: the key is that they want to understand pointwise estimates of differentiability. As mentioned above, Sobolev classes are not so sensitive to pointwise properties.

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    $\begingroup$ Wow. Very impressive and informative answer. Thank you very much! $\endgroup$
    – Jingeon An-Lacroix
    Commented May 7, 2020 at 16:17
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    $\begingroup$ By the way, I am taking Prof.Krieger's course in EPFL, and I saw you work with him! It is an honor to get an answer from you :) $\endgroup$
    – Jingeon An-Lacroix
    Commented May 7, 2020 at 16:59
  • $\begingroup$ You are studying for an MA there? How did you come across this $L^p$ differentiability business? $\endgroup$
    – Willie Wong
    Commented May 7, 2020 at 17:31
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    $\begingroup$ Yes. I was reading some papers from who I was interested in his work, Daniel Spector. Specifically, D. Spector, On a generalization of $L^p$-differentiability, Calc. Var. Partial Differential Equations 55 (2016), no. 3, 55:62. $\endgroup$
    – Jingeon An-Lacroix
    Commented May 7, 2020 at 17:40
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    $\begingroup$ @Jingeon An: FYI, I'm a bit embarrassed that I didn't think of $L^p$ derivatives and their variations when I initially wrote, and then later expanded, this answer, given the fact that my Ph.D. supervisor's Ph.D. thesis involved $L^p$ derivatives (and I was well aware of this, even 25+ years ago). $\endgroup$
    – Dave L Renfro
    Commented May 7, 2020 at 19:55

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