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  "body": "\n## Summary\n\nA Triebel–Lizorkin space $F^s_{p,q}(\\R^n)$ is a space of [[tempered distributions|AES2T5JPFP]] defined, like a [[Besov space|95QBBPRJ9E]], by splitting the [[Fourier transform|DPVM34ADZV]] into dyadic frequency shells.\nThe difference is the order of the norms.\nOne first takes an $\\ell^q$ sum over the shells at each point, and then the $L^p$ norm in space.\nThe scale contains the [[Lp spaces|SASP14MA1Z]] for $1 \u003c p \u003c \\infty$, the [[Sobolev spaces|M4CEZ5K9HP]] $W^{k,p}$ for $1 \u003c p \u003c \\infty$, and the local Hardy spaces.\nIt gives one framework for many function spaces of harmonic analysis and partial differential equations.\n\n## Explanations\n\n### L1 human\n\nSplit a function into pieces of frequency about $2^j$ and weight the $j$-th piece by $2^{js}$.\nA Besov norm measures each weighted piece in $L^p$ and then adds up the results.\nA Triebel–Lizorkin norm adds up the weighted pieces at each point first, through a square sum when $q = 2$, and then measures the result in $L^p$.\nFor $q = 2$ this square function is the classical tool of Littlewood–Paley theory, and it recovers the $L^p$ and Sobolev norms.\n",
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