I am having trouble with solving the following problem:
Let $\alpha \in (0, 1)$ and $T: [0, \infty) \to [0, \infty)$ be a continuous function with $\lim_{x \to \infty} x^\alpha T(x) = a_0 \in (0, \infty)$. Let $I = [0, 1]^2 \subseteq \mathbb{R}^2$ be the unit box. Decide if the the following limit exists:
$$ \lim_{\epsilon \to 0^+} \frac{1}{\epsilon^\alpha} \int_I T (\frac{|x - y|}{\epsilon}) \,dx \,dy. $$
This limit seems to be some kind of convolution notion, which is why I suspect it to be finite. I would like to use some trick to make it possible to use one of the convergence theorems, like Lebesgue Dominated Convergence, to show the limit exists, but the condition $\lim_{x \to \infty} x^\alpha T(x) = a_0$ seems a little bit too weak for us to find a integrable function dominating $T$. It does tell us that $\lim_{x \to \infty} T(x) = 0$, but I do not see how this would help with the proof. I have also tried to find a counterexample after a couple failed attempts of proving. More specifically, I tried to choose $$ T(x) := \frac{1}{x^\alpha} + 1 $$ since $x^\alpha T(x) \to a_0 > 0$ implies that $T(x)$ can not decrease too fast and $a_0 < \infty$ implies that $T$ must also decrease fast enough. However, this counterexample does not seem to work as the integral in the end seems to be finite (?).