I'm working on a problem I have been dealing with unsuccessfully for months now, so any help is greatly appreciated!
The context
I have to solve an integral of the form $$\begin{align} \int_0^{\infty}dy \; \psi\left(\frac{y}{q}\right)\phi(y) \int_{-\infty}^{\infty}\frac{dp}{\sqrt{2\pi}} p^{\gamma} e^{-ip(y-x)}\;, \end{align}$$ where $\psi$ is square-integrable and $\phi$ is undefined. When $\gamma \in \mathbb{N}$, we can write $$\begin{align} \int_0^{\infty}dy \; \psi\left(\frac{y}{q}\right)\phi(y) \int_{-\infty}^{\infty}\frac{dp}{\sqrt{2\pi}} (-i)^{-\gamma}\partial_{y}^{\gamma} \left[ e^{-ip(y-x)} \right]\;, \end{align}$$ and we obtain a Dirac delta which, upon integration by parts and discarding the surface terms thanks to $\psi$, helps us getting rid of the integral over $y$ and obtaining an analytic expression in terms of $\phi(x)$, $\psi(x/q)$ and their derivatives. I now want to do the same for $\gamma$ real. For this, I suppose that $\gamma = \nu-\rho$, with $\nu \in \mathbb{N}$ and $0<\rho<1$, because I can express the Fourier transform as $$\begin{align} \int_{-\infty}^{\infty}\frac{dp}{\sqrt{2\pi}} p^{\nu-\rho} e^{-ip(y-x)} &= \int_{-\infty}^{\infty}\frac{dp}{\sqrt{2\pi}} p^{-\rho} (-i)^{-\nu} \partial_{y}^{\nu}\left[e^{-ip(y-x)}\right] = i^{\nu}\partial_{y}^{\nu} \left[\int_{-\infty}^{\infty}\frac{dp}{\sqrt{2\pi}} p^{-\rho} e^{-ip(y-x)}\right] \\ &= i^{\nu-\rho} \partial_{y}^{\nu} \left[ \frac{\sqrt{2\pi}}{\Gamma(\rho)} \frac{H(y-x)}{(y-x)^{1-\rho}} \right] \;, \end{align}$$ where $H$ is the Heaviside function.
The problem
The trouble starts now, which is why I consider $\nu=1$ for simplicity, so that I have to evaluate an integral of the form $$\begin{align} \int_0^{\infty}dy \; \psi\left(\frac{y}{q}\right)\phi(y) \times \partial_y \left[\frac{H(y-x)}{(y-x)^{1-\rho}}\right]\;. \end{align}$$ Deriving, I obtain $$ \begin{align} \int_0^{\infty} dy \; \psi\left(\frac{y}{q}\right)\phi(y) \times \left[\frac{\delta(y-x)}{(y-x)^{1-\rho}} - (1-\rho)\frac{H(y-x)}{(y-x)^{2-\rho}}\right]\;, \end{align}$$ in which the first term is singular and the second is another Heaviside function. Amazingly, if I define $\psi$ and $\phi$, Mathematica is able to return a function of $x$, even in the presence of this singular term (I have tried with $\phi$ being a polynomial or an exponential), which convinced me that this integral should be solvable somehow (though I have asked a similar question on MathematicaSE here).
Does anybody know how I could get rid of the integral? Also, if somebody knows of an alternative way to treat this problem, that would be awesome!
Many thanks!