Let $1
I know that from Hölder's inequality, we can get that $||f||_{L^1([0,\infty))} \leq ||f||_{L^p([0,\infty))} x^{1-\frac{1}{p}}, \; \forall x >0$, and I also think that this inequality must be strict, since equality would imply that $f$ is constant, which would contradict the fact that $f \in L^p$ on a domain with infinite measure.
That's only a necessary condition though, and I'm not really sure how to proceed any further.