Gagliardo–Nirenberg interpolation inequality

Remarks

The relation defining $p$ is forced by scaling: under $u(x) \mapsto u(\lambda x)$ both sides must change by the same power of $\lambda$. The proof combines the Sobolev inequality with one-dimensional bounds obtained by integration by parts, such as $\norm{u'}_{L^{p}(\R)}^2 \le C \norm{u}_{L^{q}(\R)} \norm{u''}_{L^{r}(\R)}$ for $\frac{2}{p} = \frac{1}{q} + \frac{1}{r}$. The inequality extends by density to functions in $L^q(\R^n)$ whose weak derivatives of order $m$ lie in $L^r(\R^n)$, with an extra condition at infinity when $j = 0$, $rm < n$ and $q = \infty$.

The excluded case is a real exception. For $n = 2$, $m = 1$, $j = 0$ and $r = 2$, the choice $\theta = 1$ gives $p = \infty$, and the inequality $\norm{u}_{L^\infty} \le C \norm{\nabla u}_{L^2}$ fails on $\R^2$.

A common special case is Ladyzhenskaya's inequality in the plane: $\norm{u}_{L^4(\R^2)}^2 \le C \norm{u}_{L^2(\R^2)} \norm{\nabla u}_{L^2(\R^2)}$, which is the case $n = 2$, $j = 0$, $m = 1$, $q = r = 2$, $\theta = 1/2$.

Going deeper

For a student

In nonlinear evolution equations one usually controls the $L^q$ norm of a solution by a conservation law and some higher derivative by an energy estimate. The Gagliardo–Nirenberg inequality then bounds the intermediate norms that appear in the nonlinear terms. For $\theta = 1$ and $j = 0$, $m = 1$, it reduces to the Sobolev inequality on $\R^n$. It is a basic tool for the Navier–Stokes and nonlinear Schrödinger equations.