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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.

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If the measure space is finite, Hölder's inequality gives $L^{p'} \hookrightarrow L^p$ for $p \le p'$. –  martini Jul 13 '12 at 10:04
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up vote 5 down vote accepted

If $S$ has finite total measure, than Hölders inequality shows that $L^p$ embeds into $L^q$ if $p>q$ (I memorize: $L^1(S)$ is the largest one, $L^\infty(S)$ the smallest).

If $S$ has infinite total measure and free of atoms, then no $L^p$ embeds into another $L^q$. One always construct functions which are in $L^p$ but not in $L^q$ and vice versa.

If $S$ consists of atoms only (e.g. $S=\mathbb{N}$ with the counting measure), than $L^p$ embeds into $L^q$ if $p<q$ (I memorize: $\ell^\infty$ is the largest one, $\ell^1$ is the smallest).

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What do you mean by atoms? Especially "free of atoms"? –  Sh4pe Jul 13 '12 at 10:11
    
An atom is something like a "smallest set that carries some measure", see en.wikipedia.org/wiki/Atom_(measure_theory). E.g. Lebesgue measure does not have atoms but for the counting measure on an countable set, all singletons are atoms. –  Dirk Jul 13 '12 at 10:26
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