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I wondered this, and tried to find an answer online, but the only thing I could find was a statement that the set of functions which are in all $L^p(\mathbb{R}^n)$ is well-studied. But what functions are in all $L^p(\mathbb{R}^n)$ spaces?

If the answers are very different, I’d be interested in both the $p<\infty$ and the $p \leq \infty$ case.

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As an application of interpolation (see here for a related theorem, or use Holder's inequality), this is just the set $L^1(\mathbb{R^n}) \cap L^{\infty}(\mathbb{R^n})$. So any bounded, integrable function is in every $L^p$ and vice-versa.

So the moral of the story is that the endpoints tell you everything.

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  • $\begingroup$ Does this hold regardless of whether one considered $L^\infty$ in the intersection? Or only if it is included? $\endgroup$ Oct 19, 2018 at 15:52
  • $\begingroup$ I see that I didn't really answer the second part of your question. I don't believe there's a particularly nice characterization if you don't have the actual endpoint. You may be interested in BMO, which plays a role in many endpoint estimates as a replacement of boundedness. $\endgroup$
    – user296602
    Oct 19, 2018 at 16:03

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