 Sharp Gagliardo-Nirenberg inequality

In dimension $d$, the mass-critical sharp inequality isFor nonzero $u$, equality holds exactly on the amplitude, phase, translation, and spatial-scaling orbit of the NLS ground state $Q$; $u=0$ also gives equality. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.