I'm trying to understand a proof on Steklov average and weak derivatives. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $T>0$, $h\neq 0$ and $u\in L^2(0,T;H^1(\Omega))$ and extend $u$ by zero. The Steklov average in time of a function $f \in L^1$ is defined as $$f_h(t) :=\frac{1}{h} \int_t^{t+h} f(x,s) ds$$
I want to prove that $\partial_{x_j} u_h =(\partial_{x_j} u)_h$ for every $h\neq 0$ and a.e. $(x,t)\in\Omega\times (0,T)$. Up to this point, the right hand side of the result is well defined. Hence my approach is to calculate the weak derivative of $u_h$. Hence let $\phi\in C_c^\infty((0,T)\times \Omega)$ and we start with (the integrals may be extended due to the domain of the functions)
\begin{aligned} \int_\Omega\int_0^T \partial_j u_h(x,t) \phi(x,t) = -\int_\Omega \int_0^T u_h(x,t) \partial_j \phi(x,t) = -\frac{1}{h}\int_\Omega \int_0^T \int_0^T \chi_{[t,t+h]}(s) u(x,s) \partial_j \phi(x,t) \end{aligned} Now using Fubini's theorem and exploiting that $\chi_{[t,t+h]}(s)=\chi_{[s-h,s]}(t)$ we can put the Steklov average on the tesfunction $\partial_j \phi(x,t)$
Thus $$...=-\frac{1}{h}\int_\Omega \int_0^T u(x,s) \int_{s-h}^{s}\partial_j \phi(x,t) =-\frac{1}{h}\int_\Omega \int_0^T u(x,s) \partial_j\int_{s-h}^{s} \phi(x,t) $$ The last step follows since $\phi$ is smooth and consequently we can apply the Leibniz rule. I hope until this point everything is correct. Now I want to integrate by parts and exploit that $u$ has a weak derivative and then put the Steklov Average again on $w:=u_j$.
However the definition of weak derivative requires that $\int w \phi=-\int u\phi_j$. The part hat I have problems with is, that $\int_{s-h}^s \phi$ may leave the space $C_c^\infty((0,T)\times \Omega)$ and consequently I need to choose $h$ prior to $\phi$ but then I'm not able to verify the definition of weak solution (since not all $\phi$ occur)
If there is something unclear feel free to ask and hopefully it's just really easy :)
I think I found an answer. The idea is essentially approximation. Take a sequence of smooth functions $u_m$ with $u_m \to u \in L^2(0,T;H^1(\Omega))$
$$\int \partial_j u_h \phi = - \int u_h \partial_j \phi \leftarrow -\int (u_m)_h \partial_j\phi = \int \partial_j (u_m)_h \phi =\int (\partial_j u_m)_h \phi \to \int (\partial_j u)_h\phi$$
The convergences should be clear. And the crucial step is to apply the Leibniz rule for smooth functions.
Any comments on this approach?