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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $1 \u003c p \u003c \\infty$, and let $Sf$ be the [[Littlewood–Paley square function|H4E1PW67Z1]] on $\\R^n$, built from a fixed cut-off $\\chi$.\nThen there are constants $0 \u003c c \\le C$, depending only on $n$, $p$ and $\\chi$, such that\n$$c \\norm{f}_{L^p(\\R^n)} \\le \\norm{Sf}_{L^p(\\R^n)} \\le C \\norm{f}_{L^p(\\R^n)}$$\nfor every $f$ in the [[Lp space|SASP14MA1Z]] $L^p(\\R^n)$.\n\n## Remarks\n\nFor $p = 2$ the inequality follows from Plancherel's theorem.\nFor the upper bound, view $f \\mapsto (\\Delta_j f)_{j \\in \\Z}$ as a single operator from $L^p(\\R^n)$ to $L^p(\\R^n; \\ell^2)$.\nIts kernel $(\\check\\psi_j)_j$ satisfies the size and smoothness estimates of a vector-valued [[Calderón–Zygmund operator|3SCXCW4AH3]], so the $L^2$ bound extends to $1 \u003c p \u003c \\infty$.\nThe lower bound follows by duality.\nLet $\\tilde\\Delta_j = \\Delta_{j-1} + \\Delta_j + \\Delta_{j+1}$; its multiplier equals $1$ on the support of the multiplier of $\\Delta_j$, so $\\int f \\bar g = \\sum_j \\int \\Delta_j f \\, \\overline{\\tilde\\Delta_j g}$ for nice $f$ and $g$.\nThe Cauchy–Schwarz inequality in $j$ and [[Hölder's inequality|JQ9WXC729D]] give $\\abs{\\int f \\bar g} \\le \\norm{Sf}_p \\norm{\\tilde S g}_{p'}$, and the upper bound for the square function $\\tilde S$ of the $\\tilde\\Delta_j$ in $L^{p'}$ finishes the proof.\n\nThe inequality fails for $p = 1$ and $p = \\infty$.\nAt $p = 1$ the upper bound fails already for an integrable $f$ with $\\int f \\neq 0$, and the correct substitute replaces $L^1$ by the real Hardy space $H^1$.\n\n## Explanations\n\n### L1 human\n\nIn $L^2$, a function split into frequency octaves has a norm that is exactly computable from the norms of the octaves.\nThe Littlewood–Paley inequality says that in $L^p$, for $1 \u003c p \u003c \\infty$, the norm of $f$ can still be read off, up to constants, from the square function.\nThe square function combines the sizes of the octave pieces at each point.\nSo the octave pieces of $f$ do not cancel each other too much in $L^p$.\n\n### L2 human\n\nA typical consequence: for any choice of signs $\\varepsilon_j \\in \\set{-1, 1}$, the operator $f \\mapsto \\sum_j \\varepsilon_j \\Delta_j f$ is bounded on $L^p$ with a bound independent of the signs.\nThis is the first step towards the Marcinkiewicz and Hörmander–Mikhlin multiplier theorems, where a multiplier is cut into dyadic pieces and each piece is handled separately.\nThe inequality is also the starting point for the characterization of Sobolev spaces $W^{k,p}$, Besov spaces and Triebel–Lizorkin spaces through the sizes of the $\\Delta_j f$.\nOn the torus, the analogous statement with sharp dyadic blocks of the Fourier series is the original theorem of Littlewood and Paley.\n",
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