It is known that the Poisson Integral Formula:
$f(x,y) = \dfrac{y}{\pi}\displaystyle\int\limits_{\mathbb{R}}\dfrac{U(t)}{(x-t)^{2}+y^{2}}dt$
serves as a solution to the Laplace equation on the upper half plane:
$\left\{ \begin{array}{ll} f_{xx} + f_{yy} = 0 & (x,y) \in \mathbb{R}_{\geq 0}^{2}\\ f(x,0) = U(x) & x \in \mathbb{R},\;\; U(x) \in L^{p}(\mathbb{R}),\; 1 \leq p \leq \infty \end{array} \right.$
I'm trying to justify differentiating under the integral of the solution. First I wrote the difference quotient for x and arrived at this:
$\lim\limits_{h\rightarrow 0}\dfrac{f(x+h,y) - f(x,y)}{h} = \dfrac{y}{\pi}\displaystyle\int\limits_{\mathbb{R}}U(t)\dfrac{2t-2x-h}{[(x+h-t)^{2}+y^{2}][(x-t)^{2}+y^{2}]}dt$
From this however, I can't seem to establish an upper bound for the integrand in order to use the Dominated Convergence Theorem. Would it be justifiable to do this:
$\left|U(t)\dfrac{2t-2x-h}{[(x+h-t)^{2}+y^{2}][(x-t)^{2}+y^{2}]}\right| \leq C\left|U(t)\right|$
where C is some constant independent of t and h? The same difficulty arises for the difference quotient for y.