= Gagliardo-Nirenberg interpolation inequality
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= Gagliardo-Nirenberg inequality
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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 $f\in H^2(\mathbb R^2)$ is
$$
\|\nabla f\|_4^2\leq C\|\nabla f\|_2\|D^2f\|_2.
$$
Apply <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 $D^2f=0$, 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.
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