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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $1 \\le p \u003c \\infty$.\nLet $W^{1,p}(\\R^n)$ be the Sobolev space of functions $u \\in L^p(\\R^n)$ whose first weak derivatives lie in $L^p(\\R^n)$.\n\n1. If $p \u003c n$, let $p^* = np/(n - p)$. There is a constant $C = C(n, p)$ such that $\\norm{u}_{L^{p^*}} \\le C \\norm{\\nabla u}_{L^p}$ for all $u \\in W^{1,p}(\\R^n)$. So $W^{1,p}(\\R^n)$ embeds continuously in the [[Lp space|SASP14MA1Z]] $L^{p^*}(\\R^n)$.\n2. If $p \u003e n$, let $\\gamma = 1 - n/p$. Each $u \\in W^{1,p}(\\R^n)$ agrees almost everywhere with a bounded function that is [[Hölder continuous|012PBA0FMD]] with exponent $\\gamma$. Its supremum norm and Hölder constant are at most $C(n, p) \\norm{u}_{W^{1,p}}$.\n\n## Remarks\n\nPart 1 is the Gagliardo–Nirenberg–Sobolev inequality, and part 2 is Morrey's inequality.\nFor $p \u003c n$, the space $W^{1,p}(\\R^n)$ also embeds in $L^q(\\R^n)$ for every $q$ between $p$ and $p^*$, by interpolation.\n\nThe exponent $p^*$ is forced by scaling.\nFor $u_\\lambda(x) = u(\\lambda x)$, one has $\\norm{u_\\lambda}_{L^q} = \\lambda^{-n/q} \\norm{u}_{L^q}$ and $\\norm{\\nabla u_\\lambda}_{L^p} = \\lambda^{1 - n/p} \\norm{\\nabla u}_{L^p}$.\nThe inequality of part 1 can hold for all $\\lambda \u003e 0$ only if $-n/q = 1 - n/p$, that is $q = p^*$.\n\nFor $p = n \\ge 2$, functions in $W^{1,n}(\\R^n)$ need not be bounded.\nOn the ball $\\abs{x} \u003c 1/2$ in $\\R^2$, the function $\\log \\log (1/\\abs{x})$ is in $W^{1,2}$ but unbounded near $0$.\n\n## Explanations\n\n### L1 human\n\nIf a function and its derivative are both small in an average sense, then the function cannot be too large anywhere on a big set.\nThe theorem says how much better the function is than its derivative.\nWith enough integrability of the derivative compared with the dimension, the function is even continuous.\n\n### L2 human\n\nSobolev embeddings are the main tool to get regularity of weak solutions of partial differential equations.\nOne first finds a solution in a Sobolev space by energy methods, then uses embeddings to show that it is continuous or smooth.\nOn bounded domains with regular boundary the same embeddings hold, and the Rellich–Kondrachov theorem makes them compact for $q \u003c p^*$.\n",
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