If $p<q$, then $L^q\subset L^p+L^\infty$

Question

In my lecture it was claimed that $L^r\subset L^p+L^q$ if $p<r<q\leq\infty$. This has been proven here for the case $q<\infty$. What about the case $q=\infty$? Is the claim still true? For example, is $L^2\subset L^1+ L^\infty$?

Answer 1

I just show $L^2 \subset L^1 + L^\infty$.

$$ f = f \, {\bf 1_{(|f| \geq 1)}} + f \, {\bf 1_{(|f| < 1)}}$$

The last term is bounded by 1 hence in $L^\infty$. Since ${\bf 1_{(|f| \geq 1)}}$ is less than $|f|$, we get the bound

$$\int |f| \, {\bf 1_{(|f| \geq 1)}} \leq \int |f|^2$$

and thus the first term is in $L^1$.