Let $f \in L^1(\mathbb{R}^n)$ and $T$ is a distribution. Show that $T_{D_j^hf} \rightarrow \partial x_j T_f$ as $h \rightarrow 0$ in the sense of distributions, where $$D_j^hf(x) := \dfrac{f(x + e_jh) − f(x)}{ h}, \quad h>0.$$
I try to prove that, for any $\phi \in C^{\infty}_c(\mathbb{R}^n$
$lim_{h \rightarrow 0} \int_{\mathbb{R}^n}\phi(x).\dfrac{f(x + e_ih) − f(x)}{ h}dx=-\int_{\mathbb{R^n}}\partial x_i\phi(x).f(x)dx.$
I am planning to bounded the integral
$$\int_{\mathbb{R}^n}\left\vert \phi(x).\dfrac{f(x + e_ih) − f(x)}{ h}dx + \partial x_i\phi(x).f(x)\right\vert dx < \varepsilon$$ for all $|h|<\delta.$
But I don't know how to deal with $\dfrac{f(x + e_ih) − f(x)}{ h}$ in this case. Could you please give me some ideas?
Btw, can you recommend me some good books about Distribution Theory?
As usual in the theory of distributions we want to "throw" everything onto the test function. After a change of variables (shifting the first term) we have
\begin{align*} \int_{\mathbb{R}^d} \phi(x)D_j^h(x) dx&=\int_{\mathbb{R}^d} \frac{\phi(x)}{h} f(x+he_j)dx +\int_{\mathbb{R}^d} \frac{\phi(x)}{h}f(x)dx\\ &=\int_{\mathbb{R}^d} \frac{\phi(x-he_j)-\phi(x)}{h} f(x)dx. \end{align*} Now you can us dominated convergence to conclude (as the factor with $\phi$ has compact support and is bounded by the mean value theorem).