$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$

Question

For $f\in L^{p}$, $p \in [1,\infty)$ we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$

I'm not sure whether we can exchange the limit and the integral, cuz I cannot find the integral function $g(t)$ such that $|f(x-t)\frac{t}{t^2 +y^2}| \leq |g(t)|$.

How could I argue this? Appreciate with any hints!

Answer 1

If $p>1$, use Hölder's inequality. Then write $$\int_{\Bbb R}\left|\frac{t}{t^2+y^2}\right|^qdt\leqslant \int_{\{|t|\geqslant R\}}\frac 1{|t|^q}dt+(2R)^{Q+1}y^{-2q}.$$

If $p=1$, use dominated convergence theorem, where $|f(x-t)|\frac{|t|}{t^2+1}$ is a dominating function ($y>1$).