Check whether partial derivative commutes with inverse partial derivative.

Question

Is the following true? with proof. In other words check whether $\partial^{2}_{r}$ commutes with $\partial^{-2}_{r}$. $$\partial^{-2}_{r}\partial^{2}_{r}=\partial^{2}_{r}\partial^{-2}_{r}$$ Side Note: In case of $\Box^{-1}\Box\neq \Box\Box^{-1}$.

Answer 1

Hint
Check out the most basic case: One variable and the space $\mathbb R[x]$. Is $$\frac{\mathrm d^2}{\mathrm dx^2} \int_0^x \int_0^s f(t) \mathrm dt \ \mathrm ds = \int_0^x \int_0^s \frac{\mathrm d^2}{\mathrm dt^2} f(t) \mathrm dt \mathrm ds \qquad?$$ This question is answered here (not really a duplicate, but will answer your question as well)