"Interpolating" between $L^1$ and $L^\infty$.

Question

I know how to do "interpolation" for this: If $f$ belong to both $L^{p_1}$ and $L^{p_2}$, with $1\leq p_1<p_2<\infty$, then $f\in L^p$ for all $p_1\leq p\leq p_2$. Basically, one trick is to write $p=\alpha p_1+(1-\alpha)p_2$, and use Holder's inequality.

However, how do I show that if $f\in L^1\cap L^\infty$, then $f\in L^q$ for any $1\leq q\leq \infty$? I realise I can't use the same trick.

Thanks a lot.

Answer 1

Hint: If $f \in L^{\infty}$ then $|f|$ is bounded a.e., say $|f| \leq M$. Now $|f|^q \leq M^{q-1} |f|$, so ...