Littlewood's inequality for $L^p$ spaces

Question

I have tried to prove the following inequality, but I couldn't do yet.

Prove the following interpolation estimate:

$$\| u\|_q \leq \| u\|_p^{\theta} \| u\|_r^{1- \theta}$$ where $p≤q≤r$, $θ∈[0,1]$ and $\frac{1}{q} = \frac{\theta}{p} + \frac{1-\theta}{r}$.

Note that $\| u\|_q $ denotes $L^q$ norm.

Analysis kills me. Any help will be appreciated.

Answer 1

This is known as Littlewood's inequality.

Use Holder's inequality $$\|fg\|_a\le\|f\|_b\|g\|_c$$ where $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$, on $|u|=|u|^\theta|u|^{1-\theta}$, with $a=q$, $b=p/\theta$, $c=r/(1-\theta)$, $$\|u\|_q\le\||u|^\theta\|_b\||u|^{1-\theta}\|_c=\|u\|_p^\theta\|u\|_r^{1-\theta}$$