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  "body": "\n## Summary\n\nThe Littlewood–Paley square function of a function $f$ on $\\R^n$ is $Sf = \\left( \\sum_{j \\in \\Z} \\abs{\\Delta_j f}^2 \\right)^{1/2}$, where the $\\Delta_j f$ are the pieces of the [[Littlewood–Paley decomposition|D1PAT0K446]].\nBy Plancherel's theorem its $L^2$ norm is comparable to that of $f$, and by the [[Littlewood–Paley inequality|6GYQNB2862]] the same holds in $L^p$ for $1 \u003c p \u003c \\infty$.\nIt turns the frequency information of $f$ into a pointwise quantity that can be estimated with real-variable tools.\nIt is a model for many other square functions, such as the [[Lusin area integral|KHY35P4AKF]].\n\n## Explanations\n\n### L1 human\n\nSplit $f$ into octaves of frequency, look at the size of each octave at a point $x$, and combine these sizes by a square root of a sum of squares.\nThe result is a new nonnegative function $Sf$, which ignores the phases of the pieces and only remembers their sizes.\nChanging the signs of the pieces does not change $Sf$, which explains why such sign changes do not change the $L^p$ norm much.\nThe square root of a sum of squares is used because it matches the $L^2$ norm exactly when the pieces are orthogonal.\n",
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