Sharp Gagliardo-Nirenberg inequality
= Sharp Gagliardo-Nirenberg inequality
{c}
In dimension $d$, the mass-critical sharp inequality is
$$
\|u\|_{2+4/d}^{2+4/d}
\leq\frac{d+2}{d\|Q\|_2^{4/d}}
\|u\|_2^{4/d}\|\nabla u\|_2^2,
$$
and equality holds exactly on the phase, translation, and scaling orbit of the <NLS ground state> $Q$. For $d=2$ this becomes $\|u\|_4^4\leq2\|Q\|_2^{-2}\|u\|_2^2\|\nabla u\|_2^2$.