I would like to show that for $\phi \in C_c(\mathbb{R}^n)$, the map $$I:x \mapsto \int |\phi(x+y)-\phi(y)|^p dy$$ is continuous.
To show this, I think we can use dominated convergence theorem. Assume that the support of $\phi$ is contained in some ball $B_R(0)$. Fix some $x \in \mathbb{R}^n$. Then we have $|x| < d$ for some $d>0$. Now it suffices to show that for $x_n \to x$, we have $I(x_n) \to I(x)$. Now we have $|\phi(x_n+y)- \phi(y)|^p \le 2^p \Vert \phi \Vert_\infty^p 1_{B_{R+2d}(0)}(y)$ for all $y \in \mathbb{R}^n$ and $x_n$ for $n$ sufficiently large. Hence we can use dominated convergence to conclude that $\lim_n I(x_n) = \lim_n \int |\phi(x_n+y)-\phi(y)|^pdy = \int \lim_n |\phi(x_n+y)-\phi(y)|^pdy = \int |\phi(x+y)-\phi(y)|^pdy = I(x).$
Is this argument correct?