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

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.

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