What kind of functional analytic results can be said to hold for all 1<p<2 if it holds for *some* 1<p<2? Same with $2<q<\infty$

Question

One obvious fact is that $L_p(\mu)$ spaces behave very, very similarly for all $p\in(1,2)$. The same is true for $L_q(\mu)$ spaces for $q\in(2,\infty)$. Has this observation ever been formalized, and if so, to what extent?

I guess what I'm looking for is a "meta-theorem" along the following lines: Suppose there is $p_0\in(1,2)$ such that $Ap_0$, where $A$ is a predicate satisfying some particular condition(s); then $Ap$ for all $p\in(1,2)$.

Answer 1

I think in most cases, this is happening because we have something like

For all $p \in (1,\infty)$, the following are equivalent:

• The space $L^p(\mu)$ has a property "A".
• The measure $\mu$ has a property "B".

behind the scenes.

For example:

• The separability of $L^p(\mu)$ is equivalent to the separability of the measure $\mu$.
• $1 \in L^p(\mu)$ iff $\mu$ is finite.
• $L^p(\mu)$ is finite dimensional iff $\mu$ consists of at most finitely many atoms.