$\int_{\mathbb{R}^3}|f(x)|^{3/2}dx\leq (\int_{\mathbb{R}^3}|\nabla f(x)|dx)^{3/2}$

Question

This is an old prelim problem that I'm doing for practice.

So suppose that we have $f,\nabla f\in L^1(\mathbb{R}^3)$. Show that

$$\int_{\mathbb{R}^3}|f(x)|^{3/2}dx\leq (\int_{\mathbb{R}^3}|\nabla f(x)|dx)^{3/2}$$

I have absolutely no idea how to start this problem. I thought of using integration by parts or something akin to that, but didn't get anywhere.

EDIT: Note sure if it helps or not, but the inequality can be rephrased as.

$$ ||f||_{L^{3/2}}\leq ||\nabla f||_{L^1}$$

This seems like an inequality involving Sobolev spaces or the like.

Answer 1

This is the Gagliardo–Nirenberg–Sobolev inequality for $p=1$ and $n=3$.