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The Gagliardo-Nirenberg interpolation inequalities bound an intermediate derivative norm by powers of a higher derivative norm and a lower norm. In two-dimensional whole space, one useful case for $f\in H^2({\mathbb{R}}^2)$ isApply Ladyzhenskaya's inequality to the components of $\nabla f$, then combine their estimates by the Cauchy-Schwarz inequality. Density of smooth functions in a Sobolev space extends the result from smooth compactly supported functions to the stated Sobolev space.

The domain and boundary assumptions matter. On a bounded domain, the affine function $f(x)=x_1$ has nonzero gradient but $D^{2}f=0$, so this homogeneous inequality cannot hold for every $H^2$ function there. A bounded-domain version needs appropriate boundary restrictions or an additional lower-order term.

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.