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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nLet $\\psi_0 \\in \\mathcal{S}(\\R^n)$, the [[Schwartz space|CG9JEATW3Q]], satisfy $\\psi_0(\\xi) = 1$ for $\\abs{\\xi} \\le 1$ and $\\psi_0(\\xi) = 0$ for $\\abs{\\xi} \\ge 2$, and for $j \\ge 1$ let $\\psi_j(\\xi) = \\psi_0(2^{-j} \\xi) - \\psi_0(2^{-j+1} \\xi)$.\nFor a [[tempered distribution|AES2T5JPFP]] $f$, let $\\Delta_j f = \\mathcal{F}^{-1}(\\psi_j \\, \\mathcal{F} f)$, where $\\mathcal{F}$ is the [[Fourier transform|DPVM34ADZV]].\nLet $s \\in \\R$, $0 \u003c p \u003c \\infty$ and $0 \u003c q \\le \\infty$.\nThe *Triebel–Lizorkin space* $F^s_{p,q}(\\R^n)$ is the set of tempered distributions $f$ with\n$$\\norm{f}_{F^s_{p,q}} = \\Big\\| \\Big( \\sum_{j=0}^{\\infty} 2^{jsq} \\abs{\\Delta_j f}^q \\Big)^{1/q} \\Big\\|_{L^p(\\R^n)} \u003c \\infty,$$\nwith the inner sum replaced by $\\sup_{j \\ge 0} 2^{js} \\abs{\\Delta_j f}$ when $q = \\infty$.\n\n## Remarks\n\nA different choice of $\\psi_0$ gives the same space with an equivalent quasi-norm, and for $p, q \\ge 1$ it is a Banach space.\nCompare the [[Besov space|95QBBPRJ9E]] $B^s_{p,q}$, where the $L^p$ norm is taken before the sum over $j$.\nFor $p = q$ the two definitions coincide, so $F^s_{p,p} = B^s_{p,p}$.\n\nExample: for $1 \u003c p \u003c \\infty$, $F^0_{p,2}(\\R^n) = L^p(\\R^n)$ with equivalent norms, where $L^p$ is the [[Lp space|SASP14MA1Z]].\nThis is the Littlewood–Paley theorem: $\\norm{f}_{L^p}$ is comparable with $\\norm{(\\sum_j \\abs{\\Delta_j f}^2)^{1/2}}_{L^p}$.\nMore generally, $F^k_{p,2}(\\R^n)$ is the Sobolev space $W^{k,p}(\\R^n)$ for integers $k \\ge 0$ and $1 \u003c p \u003c \\infty$.\n\nNon-example: a nonzero constant function $c$ is not in any $F^s_{p,q}(\\R^n)$ with $p \u003c \\infty$.\nHere $\\Delta_0 c = c$, because $\\mathcal{F} c$ is a multiple of the Dirac delta and $\\psi_0(0) = 1$, and the constant $\\abs{c}$ is not in $L^p(\\R^n)$.\n",
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