I guess the chances are slim but still curious about the integral in the title.

Let $f : [0, \infty) \to \mathbb{R}$ be a locally "square-integrable" function on $[0,\infty)$.

Then, for any $\epsilon \in (0,1)$, is it possible to estimate the following integral?: \begin{equation} \int_{\epsilon}^1 \Bigl\lvert \int_0^x \frac{f(y)}{\lvert x-y\rvert^{1/2}} dy\Bigr\rvert^2 dx \end{equation}

In particular, is this integral finite in general? Naive application of Jensen's inequality of course leads to divergent estimate, but I wonder if there is anything more precise..