Gagliardo-Nirenberg interpolation inequality

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.