estimate of infinite norm by $(p,q)$ norms

Question

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$

I think this is true. I tried to prove it using integration by parts but did not succeed.

Answer 1

If the function $f$ is defined on the real line and vanishes at infinity, then combining the fundamental theorem of calculus with Hölder's inequality yields $$|f(x)|^2\leqslant 2\max\left\{\left(\int_0^{\infty}|f(t)|^p\mathrm dt\right)^{1/p}\left(\int_0^{\infty}|f'(t)|^q\mathrm dt\right)^{1/q}; \left(\int_{-\infty}^0|f(t)|^p\mathrm dt\right)^{1/p}\left( \int_{-\infty}^0|f'(t)|^q\mathrm dt\right)^{1/q}\right\}.$$ Indeed, we have $f(x)^2=2\int_{-\infty}^xf'(t)ft)\mathrm dt $ (this for $x\leqslant 0$) and a similar estimate for a positive $x$.