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This is the two-dimensional form of Ladyzhenskaya's inequality. For $u\in C_c^{\infty }({\mathbb{R}}^2)$,Apply the fundamental theorem of calculus to $|u{|}^2$ in each coordinate, multiply the resulting one-dimensional bounds, integrate, and use Cauchy--Schwarz.

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.