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  "body": "\n## Summary\n\nThe Lorentz spaces $L^{p,q}(\\mu)$ form a two-parameter family of function spaces that refines the scale of [[Lp spaces|SASP14MA1Z]].\nThey are defined with the [[decreasing rearrangement|BWS042NC2P]] of a function, so they only see the sizes of the level sets of the function.\nThe first index $p$ gives the main size and the second index $q$ gives a fine adjustment.\nThey contain $L^p = L^{p,p}$ and the [[weak Lp space|H4C888M5AZ]] $L^{p,\\infty}$, and they appear in interpolation theory and in sharp forms of Sobolev inequalities.\n\n## Explanations\n\n### L1 human\n\nTo measure the size of a function, rearrange its values in decreasing order, like sorting a pile of sand by height.\nThe Lorentz norm then weighs this sorted profile in a way governed by two numbers $p$ and $q$.\nWith $q = p$ one gets the usual $L^p$ norm.\nLetting $q$ grow makes the space larger, so for $p \\ge 1$ one has $L^{p,1} \\subseteq L^{p,p} \\subseteq L^{p,\\infty}$.\nFor example, $\\abs{x}^{-1}$ on $\\R$ is in $L^{1,\\infty}$ but not in $L^{1,1} = L^1$.\n",
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