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Let $(M,g)$ be a smooth compact Riemannian manifold, and let $f \in W^{1,p}(M)$ for $p\ge 1$. (I don't assume $p>\dim M$).

I have seen in various sources that people refer to the weak derivative of $f$ as a linear functional $T_pM \to \mathbb{R}$, which is defined for almost every $p \in M$. (an a.e.defined section of $T^*M$).

How exactly is this object defined? I couldn't find any precise details about this.

I define $W^{1,p}(M)$ to be the completion of the space of compactly supported smooth functions $C_c^{\infty}(M)$ w.r.t the $\|\cdot\|_{1,p}$ norm.


Optional: I suggest below $2$ approaches; I would like to know if they are compatible, i.e. if they both produce the same element in $(T_pM)^*$.

(Regarding the second approach, I am not even sure if it produces a well-defined functional).

Approach 1:

Given $f \in W^{1,p}(M)$, there exist $f_n \in C_c^{\infty}(M)$, $f_n \to f$ in $W^{1,p}$.

$df_n \in \Gamma(T^*M)$ is a Cauchy sequence in $L^p(M,T^*M)$, where $L^p(M,T^*M)$ is the completion of the space of smooth sections $\Gamma(T^*M)$ w.r.t the natural $p$-norm. By completeness, $df_n$ converges to an element in $L^p(M,T^*M)$, which we can realize as a measurable section $T^*M$. We set $df=\lim_{n \to \infty} df_n$.

Approach 2 ("Local picture"):

Let $\phi:U\subseteq M \to \mathbb{R}^n$ be a surjective coordinate chart around $p \in M$, and $\phi(p)=0$. Set $f_{\phi}=f|_U \circ \phi^{-1} :\mathbb{R}^n \to \mathbb{R}$. Then $f_{\phi} \in W^{1,p}(\mathbb{R}^n)$ (we might need to shrink $U$ to ensure nothing will explode). We define $df_p$ by the equation

$$ df_p \circ d(\phi^{-1})_0(e_i):= d(f_{\phi})_0(e_i)=(\partial_i f_{\phi})(0). \tag{1}$$

Does equation $(1)$ well-defines an element in $T_p^*M$ independently of the coordinate chart? Does it coincide with $\lim_{n \to \infty} df_n$ from the previous approach?

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  • $\begingroup$ Concerning point 2: its well-definiteness relies on the validity of the chain rule for weak derivatives. This is, as far as I can tell, a bit subtle: mathoverflow.net/q/291992/13042 $\endgroup$ – Giuseppe Negro May 17 at 7:45
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You have a notion of weak derivative in this case: An almost everywhere section of $T^*M$, denoted $df$, is a weak derivative of $f$ if for every smooth, compactly supported 1-form $\phi$ you have $$ \int_M g(df,\phi) \,\text{dVol} = \int_M f\,\delta\phi \,\text{dVol}. $$ where the pairing is with respect to the metric $g$. A function is in $W^{1,p}(M)$ if it is in $L^p$, and has a weak derivative in $L^p(T^*M)$. Since this definition of weak derivative is consistent with taking limits in $L^p$, it is consistent with your approach 1. I believe that taking $\phi$ to be supported in a coordinate patch and writing the definition above in coordinates would just yield your approach 2. The question why $W^{1,p}$ as defined here is actually the same as the completion should be very similar to the same question in Euclidean spaces (the classic $H=W$ question).

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