Taking the Divergence of integral equality

Let's have the equality (look to the the original question) $$\int \limits_{-\infty}^{\infty} \left[ [\nabla_{\mathbf r'} \times \mathbf A (\mathbf r' )] \times \frac{\mathbf r' - \mathbf r}{|\mathbf r' - \mathbf r |^{3}}\right]d^{3}\mathbf r' = 4 \pi\mathbf A (\mathbf r). \qquad (.1)$$ I need to find the condition on the field, to which it must satisfy in order to satisfy the $(.1)$. By taking divergence $\nabla_{\mathbf r}$ of left and right sides of the equation, and changing the sequence of integration and divergence operations, I can get zero in the left side and $(\nabla \cdot \mathbf A)$ in the right side. But are integration and divergence operations commuting operations in case of $(.1)$?

This also solves my linked question.

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Since $d^3\mathbf{r}'$ seems to be a volume element, what exactly is the domain of integration? Depending on wether the domain is 3-dimensional, the LHS should be scalar while the RHS is a vector field. – gofvonx Sep 16 '13 at 17:29
@gofvonx . The left side refer to vector field, as right. $d^{3}\mathbf r' = dx'dy'dz'$. – John Taylor Sep 16 '13 at 17:34