Proving $\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha} \|g\|_{L^q}^{1-\alpha}$

Question

I am trying to prove the following inequality: $$\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha}\|g\|_{L^q}^{1-\alpha}$$ where $$\quad 1= \frac1{p} +\frac1{q}, \quad 1 \leq p, q, \leq \infty.$$ In particular, we may choose $\alpha = 1/p$.

This seems very closely related to Hölder's inequality and some kind of $L^p$ interpolation, but I have been unsuccessful so far, especially with getting the correct exponents $\alpha$ and $1-\alpha$. How can I prove this? Moreover, is there a name for this inequality? I am thinking that this may only be true in bounded domains.

Answer 1

Let $f$ and $g$ be any positive functions for which all the integrals here are finite. Replacing $f$ by $nf$ we see that we must have $1 \leq \alpha$. Replacing $g$ by $ng$ we see that $1 \leq 1-\alpha$. We have obtained a contradiction.