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  "body": "\n## Statement\n\nLet $n \\ge 2$ and $1 \\le p \u003c n$.\nThen every continuously differentiable function $u \\colon \\R^n \\to \\R$ with compact support satisfies\n$$\\int_{\\R^n} \\frac{\\abs{u(x)}^p}{\\abs{x}^p} \\, dx \\le \\Big( \\frac{p}{n - p} \\Big)^p \\int_{\\R^n} \\abs{\\nabla u(x)}^p \\, dx.$$\n\n## Remarks\n\nFor $p = 2$ and $n \\ge 3$ there is a short proof.\nLet $a = (n - 2)/2$ and expand $0 \\le \\int \\abs{\\nabla u + a u \\, x / \\abs{x}^2}^2 \\, dx$.\nThe cross term is $a \\int \\nabla (u^2) \\cdot x / \\abs{x}^2 \\, dx = -a (n - 2) \\int u^2 / \\abs{x}^2 \\, dx$ by integration by parts, since $\\operatorname{div}(x / \\abs{x}^2) = (n - 2) / \\abs{x}^2$.\nCollecting terms gives $\\frac{(n-2)^2}{4} \\int u^2 / \\abs{x}^2 \\, dx \\le \\int \\abs{\\nabla u}^2 \\, dx$, which is the statement.\n\nThe constant $\\big( \\frac{p}{n - p} \\big)^p$ is the best possible, but there is no nonzero function for which equality holds.\nBy density, the inequality extends to every $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\\R^n)$.\nIt is the $n$-dimensional analogue of the one-dimensional [[Hardy's inequality|S7A8YFGG8X]].\n\nThe hypothesis $p \u003c n$ is needed.\nFor $p = n$, take $u$ smooth with compact support and $u(0) \\ne 0$: then $\\abs{u}^n / \\abs{x}^n$ is not integrable near $0$, while the right side is finite.\n\n## Explanations\n\n### L1 human\n\nThe weight $1/\\abs{x}^p$ is large near the origin, so the left side punishes functions that are large near $0$.\nThe inequality says that the gradient energy alone pays for this, as long as the dimension is larger than $p$.\nIn space with $p = 2$, it says that $\\int u^2 / \\abs{x}^2 \\le 4 \\int \\abs{\\nabla u}^2$.\n\n### L2 human\n\nHardy's inequality controls singular potentials by kinetic energy.\nIn quantum mechanics it shows that the operator $-\\Delta - c / \\abs{x}^2$ on $\\R^n$ is bounded below when $c \\le (n-2)^2 / 4$.\nIn partial differential equations it is used to treat lower-order terms with singular coefficients and to study equations with critical singular weights.\n",
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