Why is this inclusion map continuous?

Question

If $1\le p<q<r\le\infty$, then I want to conclude that the inclusion map $L^p\cap L^r\to L^q$ is continuous, But I have seen this reference here, but I think that I am not understanding the definition or I am missing something, because the inclusion map always seems to me continuos.

In my case the map will act as:

$$\iota: L^p\cap L^r\to L^q$$

$$\iota (f)=f$$

where on $L^p\cap L^r$ we use the norm $\|\cdot\|_p + ||·||_r$.

So then can someone help me to prove the above result in a better way?

Thanks a lot in advance.

Answer 1

Hint: What we need to prove to prove continuity of such inclusion map is that there exists a constant $C>0$ such that, for $f \in L^p \cap L^r$, $$ \|f\|_q \leq C \|f\|, $$

where $\|f\|=\|f\|_p + \|f\|_r$ is a norm on $L^p \cap L^r$.

Answer 2

You want $i : L^p \cap L^r \to L^q$ to be continuous when you equip the domain with the $L^p$ norm and the domain with the $L^q$ norm. Since it is linear, it is equivalent for it to be bounded, i.e.

$$\| f \|_q \leq C \| f \|_p$$

for all $f \in L^p \cap L^r$. This is actually not true. Here's a proof.

Suppose $f \in L^p \setminus L^q$, without loss of generality assume $f \geq 0$. Let $f_n(x) = \min \{ f(x),n \}$. Then $f_n \in L^p \cap L^\infty$ hence $f_n \in L^q$. You want your inequality to hold for all $f_n$. Thus we should be able to take limits on both sides...but then we conclude

$$\infty \leq C \| f \|_p$$

which is not true.

On the other hand, one can equip $L^p \cap L^r$ with the norm $\| f \| = \| f \|_p + \| f \|_r$. Then the inequality is perhaps the simplest possible interpolation inequality; cf. https://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/