Why is $|f|_{L^p}\le|f|^{\frac 1p}_{L^1}|f|^{\frac 1q}_{L^{\infty}}$ or in general $|f|_{L^p}\le|f|^{\frac 1p}_{L^{p'}}|f|^{\frac 1q}_{L^{q'}}$

Question

Why is $|f|_{L^p}\le|f|^{\frac 1p}_{L^1}|f|^{\frac 1q}_{L^{\infty}}$ or in general, Is it true that if $f\in L^p$ then $|f|_{L^p}\le|f|^{\frac 1p}_{L^{p'}}|f|^{\frac 1q}_{L^{q'}}$, where $\frac 1p+\frac 1q=\frac1{p'}+\frac1{q'}=1$

I have seen a special case where this holds, namely for $p'=1, q'=\infty$ maybe it holds also for some intermediate values (I also don't know why it holds for the mentioned case)

Does the generalization of Hölder's inequality (Interpolation) here imply this maybe ?

So I take $\theta_1=\frac 1p,\theta_2=\frac{p-1}{p}$ and $p_1=1,p_2=\infty$ then the formula holds ?

Answer 1

Assume that the considered measure space is finite. The wanted inequality reads $$|f|_{L^p}\leqslant\left(|f|_{L^{p'}} \right)^{1/p}|f|^{\frac{p-1}p}_{L^{q'}} $$ for any $p$ and $p',q'$ such that $1/p'+1/q'=1$. Letting $p$ going to infinity, we have $\lim_{p\to +\infty}|f|_{L^p}=|f|_{L^\infty} $ hence $|f|_{L^\infty}\leqslant |f|_{L^{q'}}$. Therefore, the wanted inequality cannot be true for any $p$ and $q$.

We can also consider the case where $f$ is the indicator function of a measurable set $A$. If $a$ is the measure of this set, then we should have $$a^{\dfrac 1p}\leqslant a^{\dfrac 1{pp'} +\dfrac 1{qq'}} ,$$ that is, $$a^{\dfrac{1}{q'}\left(1-\frac 2q \right)} \leqslant 1.$$ If $2\lt q\lt \infty$, choose $A$ of measure greater than $1$. We get also a counter-example for $q\lt 2$.