Given $f\in L^1(\mathbb{R}^2)$, define $$T_n f(x) = \int_{\mathbb R^2} n^4 \exp\left(-n\sqrt{\left|y_1\right|+\left|y_2\right|} \right) f(x-y) dy$$ and I want to show that the limit exists for almost every $x$.
I feel like the limit should be something like $\langle \delta_x, f\rangle$ and some constant factor, since outside of any $B(0,\epsilon)$, $\lim_n T_n f(x)=0$. I tried to separate the integral
$$T_n f(x) = \frac{\pi}{\left|B\left(0,\frac{1}{n^2}\right)\right|} \int_{B\left(0,\frac{1}{n^2}\right)} \exp(-n\sqrt{|y_1|+|y_2|}) f(x-y) dy \\+ \int_{\mathbb{R}^2\setminus B\left(0,\frac{1}{n^2}\right)} n^4\exp\left(-n\sqrt{|y_1|+|y_2|}\right) f(x-y) dy $$
and I wanted to find a way to apply Lebesgue differential theorem. Could you give me some hints.