I had a conversation with my supervisor once who was talking about square-integrable functions and he said that all functions of the form $\frac{1}{x^a}$ are square integrable (over the whole real line) for $a>1/2$, I.e. $$\int_{\mathbb{R}} |\frac{1}{x^a}|^2 \ dx < \infty\quad a>\frac{1}{2}.$$
Due to the presence of a pole at $x=0$, this doesn’t seem immediately obvious to me and I’m not sure if he was meaning it in some kind of distributional sense. In my own work currently, I want to check that the following type of integral converges:
$$\int_{\mathbb{R}} \frac{f(x)}{(cx+d)^2} e^{iu \frac{ax+d}{cx+d}} \ dx $$
where $a,b,c,d \in \mathbb{R}$ are constants and $f$ is a Schwartz function. The method I considered is by estimating the integral and applying Hölders inequality as follows:
$$\left| \int_{\mathbb{R}} \frac{f(x)}{(cx+d)^2} e^{iu \frac{ax+d}{cx+d}} \ dx \right| \leq \| f\|_2 \left\| \frac{1}{(cx+d)^2} \right\|_2 < \infty $$
Where $\| \cdot \|_2$ is the $L^2(\mathbb{R})$ norm. If my previous query is correct, then the last statement is also correct. If none of this is correct, then could someone point me in the right direction as to proving the convergence of this integral, please?