I'm reading through Terry Tao's 'Why are solitons stable?' and I don't understand one of the bounds he's constructed on the $H^{1}$ norm of the solution $u(x,t)$ to the gKdV, $u_{t} + u_{xxx} + (u^{k})_{x} = 0$.
At the start of section 4 he bounds $||u(t)||^{2}_{H^{1}_{x}(\mathbb{R})}$ using this integral inequality: $ \int_{\mathbb{R}}v^{k+1} \leq C(p)\left(\int_{\mathbb{R}}v^{2}\right)^{\frac{k+3}{4}}\left(\int_{\mathbb{R}}v_{x}^{2}\right)^{\frac{k-1}{4}}.$ He calls this the Gagliardo-Nirenberg inequality. Given this inequality the $H^{1}$ norm of the solution is bounded by its mass and energy - great.
However, I thought that the Gagliardo-Nirenberg inequality was $ ||u||_{L^{p*}(\mathbb{R}^{n})} \leq C(n,p) ||Du||_{L^{p}(\mathbb{R}^{n})} $ with $1\leq p < n$ and $p* = \frac{pn}{n-p} > p$, which I don't think can be used here since the gKdV has one spatial dimension, i.e. $n = 1$ precluding any use of this result. How do we use Galiardo-Nirenberg here?