Ladyzhenskaya's inequality

Ladyzhenskaya's inequality controls the Lp norm of a function by its Lp norm and the Lp norm of its gradient: in dimension two, $\Vert u{\Vert }_4\le C\Vert u{\Vert }_2^{1/2}\Vert \nabla u{\Vert }_2^{1/2}$; in dimension three the exponents are $1/4$ and $3/4$. These estimates hold for smooth compactly supported functions and extend using density of smooth functions in a Sobolev space.