If $f(y-x)$ is in $L^p(\mathbb R^d\times\mathbb R^d)$, then I seem to conclude that $f=0$ a.e. (which seems wrong). My reasoning is that by Fubini and the integral's shift invariance (assume $p=1$ for convenience) $$\infty>\iint|f(y-x)|dydx = \int\left( \int |f(y-x)|dy\right) dx = \int\left( \int |f(y)|dy\right) dx = \int \text{const}\ dx $$ The const is non-negative but can't be positive else the right hand side will diverge, thus const $=0$. But this implies that $f=0$.
Where am I going wrong?