What is the intersection of all $L^p(\mathbb{R}^n)$ spaces?

Question

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.

Are this question and this question of any interest to you? – user505379 Oct 19 '18 at 15:43 — user505379, Oct 19 '18 at 15:43

score 8 · Accepted Answer · 2018-10-19T15:49:11.877

8

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.

edited Oct 19 '18 at 15:49

answered Oct 19 '18 at 15:42

Does this hold regardless of whether one considered $L^\infty$ in the intersection? Or only if it is included? – Sascha Baer Oct 19 '18 at 15:52
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. – Oct 19 '18 at 16:03

What is the intersection of all $L^p(\mathbb{R}^n)$ spaces?

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