Let $(S,\mathcal{B},\mu)$ be a measure space, $Y$ be a banach space and for $1\le p <\infty$ let $L^p(\mu;Y)$ be the set of all maps $f:S\rightarrow Y$ that are measurable and for which $|f|^p$ is integrable. Let $L^{\infty}(\mu;Y)$ be the of essentially bounded maps.
I wonder if it is known for which measure spaces and for which $p,p'$ these function spaces lie in each other, i.e. for which setting there exists a (reasonably well behaved) injection $L^p(\mu;Y)\hookrightarrow L^{p'}(\mu;Y)$.
Are there known results in this generality? What if the measure space is simply a subset of $\mathbb{R}^n$ (compact, convex, or with any other property).
Or are there other restrictions one can make to get an injection?
Thank you in advance! I'd be glad for any pointers.