If $M(x,y)$ is differentiable on a bounded subset $\Omega\times \Omega\subset \mathbb{R}^2\times \mathbb{R}^2$, can we move the derivative under the integral sign in \begin{equation}\frac{\partial}{\partial x_j}\int_\Omega \frac{M(x,y)}{|x-y|^\alpha}d^2y=\int_\Omega \frac{\partial}{\partial x_j}\frac{M(x,y)}{|x-y|^\alpha}d^2y\end{equation}where $0<\alpha<1/2$, $j\in \{1,2\}$? I tried to use Leibniz's integral rule, but it requires an integrable dominating function $g(y)$ such that\begin{align}\left|\frac{\partial}{\partial x_j}\frac{M(x,y)}{|x-y|^\alpha}\right|\leq g(y)\end{align}for $x,y\in \Omega^2$, which doesn't exist due to the singularity in $x$. Is there a version of Leibniz's integral rule for this type of weakly singular integral? Am I missing something?
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