If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$
where $g_{(t)}(x):=g(t+x)$.
Any hints? Try to give me only hints/outlines not complete solutions
Not sure where to go from there?