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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $j$ and $m$ be integers with $0 \\le j \u003c m$, and let $1 \\le q, r \\le \\infty$.\nLet $\\theta$ be a real number with $j/m \\le \\theta \\le 1$, and define $p$ by\n$$\\frac{1}{p} = \\frac{j}{n} + \\theta \\Big( \\frac{1}{r} - \\frac{m}{n} \\Big) + \\frac{1 - \\theta}{q},$$\nassuming that $1 \\le p \\le \\infty$.\nFor an integer $k \\ge 0$ write $\\abs{D^k u}$ for the Euclidean norm of the vector of all partial derivatives of $u$ of order $k$.\nAssume that it is not the case that $1 \u003c r \u003c \\infty$, $m - j - n/r$ is a nonnegative integer and $\\theta = 1$.\nThen there is a constant $C = C(n, m, j, q, r, \\theta)$ such that\n$$\\norm{D^j u}_{L^p(\\R^n)} \\le C \\, \\norm{D^m u}_{L^r(\\R^n)}^{\\theta} \\, \\norm{u}_{L^q(\\R^n)}^{1 - \\theta}$$\nfor every smooth function $u \\colon \\R^n \\to \\R$ with compact support.\n\n## Remarks\n\nThe relation defining $p$ is forced by scaling: under $u(x) \\mapsto u(\\lambda x)$ both sides must change by the same power of $\\lambda$.\nThe proof combines the [[Sobolev inequality|B03CA692K3]] 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 $\\frac{2}{p} = \\frac{1}{q} + \\frac{1}{r}$.\nThe inequality extends by density to functions in $L^q(\\R^n)$ whose [[weak derivatives|C3EC148VZD]] of order $m$ lie in $L^r(\\R^n)$, with an extra condition at infinity when $j = 0$, $rm \u003c n$ and $q = \\infty$.\n\nThe excluded case is a real exception.\nFor $n = 2$, $m = 1$, $j = 0$ 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 $\\R^2$.\n\nA 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 $n = 2$, $j = 0$, $m = 1$, $q = r = 2$, $\\theta = 1/2$.\n\n## Explanations\n\n### L1 human\n\nThe inequality bounds an intermediate quantity by two extreme ones: a lower derivative is controlled by the function and a higher derivative together.\nIt is like saying that if a curve is not too high and does not bend too much, then its slope cannot be too large.\nThe weights $\\theta$ and $1 - \\theta$ say how much each of the two extremes contributes.\n\n### L2 human\n\nIn 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.\nThe Gagliardo–Nirenberg inequality then bounds the intermediate norms that appear in the nonlinear terms.\nFor $\\theta = 1$ and $j = 0$, $m = 1$, it reduces to the [[Sobolev inequality|B03CA692K3]] on $\\R^n$.\nIt is a basic tool for the Navier–Stokes and nonlinear Schrödinger equations.\n",
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