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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\R^{n+1}_+ = \\set{(y, t) : y \\in \\R^n, \\ t \u003e 0}$ be the upper half-space.\nLet $f$ be in the [[Lp space|SASP14MA1Z]] $L^p(\\R^n)$ for some $1 \\le p \\le \\infty$.\nLet $u(y, t) = (P_t * f)(y)$ be its Poisson integral, where $P_t$ is the [[Poisson kernel|TCYSWCTYYM]] of the half-space, so that $u$ is [[harmonic|0DEB0JMN0W]] on $\\R^{n+1}_+$.\nLet $\\nabla u = (\\partial_{y_1} u, \\dots, \\partial_{y_n} u, \\partial_t u)$ be its full gradient.\nFor $x \\in \\R^n$ let $\\Gamma(x) = \\set{(y, t) \\in \\R^{n+1}_+ : \\abs{y - x} \u003c t}$ be the cone with vertex $x$.\nThe *Lusin area integral* of $f$ is the function $S(f) \\colon \\R^n \\to [0, \\infty]$ given by\n$$S(f)(x) = \\left( \\iint_{\\Gamma(x)} \\abs{\\nabla u(y, t)}^2 \\, t^{1 - n} \\, dy \\, dt \\right)^{1/2}.$$\n\n## Remarks\n\nThe power $t^{1 - n}$ makes $S(f)$ invariant under dilations: $S(f(\\delta \\cdot))(x) = S(f)(\\delta x)$.\nFor $1 \u003c p \u003c \\infty$ one has $c \\norm{f}_p \\le \\norm{S(f)}_p \\le C \\norm{f}_p$ with constants depending only on $n$ and $p$.\nReplacing the cone by the vertical half-line over $x$, with the weight $t$, gives the Littlewood–Paley $g$-function.\n\nExample: the $L^2$ identity.\nIntegrating over $x$ first, $\\int_{\\R^n} S(f)(x)^2 \\, dx = \\omega_n \\iint_{\\R^{n+1}_+} t \\abs{\\nabla u(y, t)}^2 \\, dy \\, dt$, where $\\omega_n$ is the volume of the unit ball of $\\R^n$.\nWith $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i x \\cdot \\xi} dx$, one has $\\hat u(\\xi, t) = e^{-2\\pi t \\abs{\\xi}} \\hat f(\\xi)$, so $\\int \\abs{\\nabla u(y, t)}^2 dy = 8\\pi^2 \\int \\abs{\\xi}^2 e^{-4\\pi t \\abs{\\xi}} \\abs{\\hat f(\\xi)}^2 \\, d\\xi$.\nSince $\\int_0^\\infty t e^{-4\\pi t \\abs{\\xi}} \\, dt = (4\\pi\\abs{\\xi})^{-2}$, this gives $\\norm{S(f)}_2^2 = \\frac{\\omega_n}{2} \\norm{f}_2^2$.\n\nNon-example: $f = 1$ is bounded but in no $L^p$ with $p \u003c \\infty$.\nIts Poisson integral is $u = 1$, so $\\nabla u = 0$ and $S(f) = 0$ although $f \\neq 0$.\nSo the lower bound $c \\norm{f}_p \\le \\norm{S(f)}_p$ cannot hold for $p = \\infty$.\n",
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