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I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and weak derivatives/differential operators, so that, if I wanted to be accurate, I always had to specify in words something like "where the derivative is meant in the weak sense".

As a researcher, I forgot this feeling of unease, because usually everything is clear from the context. In fact it's quite hard to devise an example in which the confusion is actually dangerous.

But then again, as a teacher, I gradually started to feel the need for a clearer notation, because when I talk to students, for purely pedagogical reasons, I'd like to make it crystal clear what I'm talking about. This is especially important precisely at the beginning, when you, introducing Sobolev spaces, have to define what a weak derivative is and how its existence should not be confused with the existence a.e. of the classical derivative.

So I wonder: is there a more or less established, effective notational tool to help distinguishing the weak derivative from the classical one?

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    $\begingroup$ I wouldn't say it is "established", but Joa Weber's lecture notes uses the $\partial$ and $D$ operators for the classical derivatives, and the subscript notation $u_{x_i}$ for weak derivatives. // I also remember seeing once some lecture notes using ${}^{(w)}\partial$ for weak differentiation, but that was a while ago and I don't remember whom by. $\endgroup$
    – Willie Wong
    Commented May 2 at 16:13
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    $\begingroup$ My guess is that you need to look for those notes which emphasize the fact that weak derivatives are only unique at the equivalence class level, and not on the function level; though Evans and Gariepy (which often distinguishes between a Sobolev function and its equivalence class) doesn't seem to use a separate notation for weak differentiation. $\endgroup$
    – Willie Wong
    Commented May 2 at 16:16
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    $\begingroup$ I'd would not recommend different notations. The situation is similar to the series notation $\sum_{k=0}^\infty f_k$: in what sense does it converge? If needed, one may say it in parentheses. $\endgroup$
    – Pietro Majer
    Commented May 2 at 19:09
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    $\begingroup$ @PietroMajer well, not so sure. By $\sum$ you still mean a sum. You may ask for different notations for the symbol indicating the kind of convergence. And indeed you have at least some: $\to$, $\rightharpoonup$, the same with * above, etc... $\endgroup$
    – Alessandro Della Corte
    Commented May 2 at 19:17
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    $\begingroup$ The decisive difference is between the classical derivative of a $C^1$-function and the distributional derivative of anything else that can be regarded as a distribtion (measure, locally integrable function, rational function $\dots$). In his elementary treatment, Sebastião e Silva used $D$ for the first, $\tilde D$ for the latter. This can be extended to the multivariate case, and higher derivatives, by using Schwartz' superscript notation. $\endgroup$
    – terceira
    Commented May 3 at 10:16

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When I introduce distributions to students for the first time I usually emphasize the difference between $f$ (locally integrable function) and $T_f$ (the corresponding distribution). For a couple of lectures I avoid to write $\partial^\alpha f$ and keep the full formalization (it's a bit tedious) $T_{\partial^\alpha f} = \partial^\alpha T_f$. Basic formula are proven with this notation and then, gradually I leave those precautions ...

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  • $\begingroup$ That's a good idea, I guess. Of course every locally $L^1$ function $f$ induces a distribution $T_f: \mathscr D(U)\to\mathbb R$, $\varphi \mapsto \int \varphi f$, so one can even introduce Sobolev spaces in this way. $\endgroup$
    – Alessandro Della Corte
    Commented May 3 at 8:26

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