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  "body": "\n## Summary\n\nThe Lusin area integral is a square function built from the harmonic extension $u$ of a function $f$ on $\\R^n$ to the upper half-space.\nAt a point $x$ it integrates $\\abs{\\nabla u}^2$, suitably weighted, over the cone of points $(y, t)$ with $\\abs{y - x} \u003c t$.\nFor $1 \u003c p \u003c \\infty$ its $L^p$ norm is comparable to that of $f$, as for the [[Littlewood–Paley square function|H4E1PW67Z1]].\nIt was introduced by Lusin for holomorphic functions on the disc, and it plays a key role in the real-variable theory of Hardy spaces.\n\n## Explanations\n\n### L1 human\n\nExtend $f$ to a harmonic function $u$ one dimension up, using the Poisson kernel.\nOver each point $x$ stand an upside-down cone with vertex at $x$, and measure how much $u$ varies inside the cone.\nThe total variation, measured by a weighted integral of $\\abs{\\nabla u}^2$, is the area integral at $x$.\nFor a holomorphic function $F$ on the unit disc, one integrates $\\abs{F'}^2$ over a cone-shaped region.\nThe result is the area of the image of that region, counted with multiplicity, which explains the name.\n",
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