Lower Bound on Product of $L^p$-norms

Question

Is there an established way to get a single $L^p$-norm that bounds from below $$\int |f|^p \int |f|^q\;?$$

Answer 1

If $f$ belongs to both $L^p$ and $L^q$ then it belongs to every $L^r$ with $r$ in between $p$ and $q$. The standard estimate uses Holder's inequality: you can interpolate $r = tp + (1-t)q$ with $0 < t < 1$ and use the conjugate exponents $\dfrac 1t$, $\dfrac 1{1-t}$ to find $$\int |f|^r = \int |f|^{tp}|f|^{(1-t)q} \le \left( \int |f|^p \right)^t \left( \int |f|^q \right)^{1-t}.$$ To get the product of the integrals on the right you would require $t = \frac 12$ and so $$\left( \int |f|^{(p+q)/2} \right)^2 \le \int |f|^p \int |f|^q.$$