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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $n \u003c p \\le \\infty$, and let $\\gamma = 1 - n/p$.\nThere is a constant $C = C(n, p)$ such that every continuously differentiable function $u \\colon \\R^n \\to \\R$ satisfies\n$$\\sup_{x \\in \\R^n} \\abs{u(x)} + \\sup_{x \\ne y} \\frac{\\abs{u(x) - u(y)}}{\\abs{x - y}^{\\gamma}} \\le C \\norm{u}_{W^{1,p}(\\R^n)}.$$\nConsequently, each $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\\R^n)$ agrees almost everywhere with a function in the [[Hölder space|P101AX1ZSC]] $C^{0,\\gamma}(\\R^n)$.\nThe norm of this function in $C^{0,\\gamma}(\\R^n)$ is at most $C \\norm{u}_{W^{1,p}(\\R^n)}$.\n\n## Remarks\n\nThe key step is the bound $\\frac{1}{\\abs{B}} \\int_B \\abs{u(y) - u(x)} \\, dy \\le C \\int_B \\frac{\\abs{\\nabla u(y)}}{\\abs{x - y}^{n - 1}} \\, dy$ for a ball $B$ centred at $x$.\nBy [[Hölder's inequality|JQ9WXC729D]], the kernel $\\abs{x - y}^{1 - n}$ lies in $L^{p'}(B)$ exactly when $p \u003e n$, and this gives a bound by $r^{1 - n/p} \\norm{\\nabla u}_{L^p}$ for the ball of radius $r$.\nComparing $u(x)$ and $u(y)$ with the average of $u$ on a ball containing both gives the [[Hölder continuity|012PBA0FMD]] bound.\n\nThe hypothesis $p \u003e n$ is needed.\nFor $n = 2$ and $p = 2$, the function $\\log \\log (1/\\abs{x})$, cut off away from the origin, lies in $W^{1,2}$ but is unbounded near $0$.\n\nFor a bounded open set $U$ with $C^1$ boundary, the same holds with $\\R^n$ replaced by $U$ and $C$ depending also on $U$.\nTogether with the [[Sobolev inequality|B03CA692K3]] it forms the [[Sobolev embedding theorem|GZTT6FT18G]].\n\n## Explanations\n\n### L1 human\n\nWhen the slope of a function is integrable to a high enough power, larger than the dimension, the function cannot jump or blow up.\nIt is then continuous, and even Hölder continuous with an exponent that measures how much $p$ exceeds $n$.\nIn one dimension any $p \u003e 1$ is enough, which matches the fact that $\\abs{u(x) - u(y)} \\le \\abs{x - y}^{1 - 1/p} \\norm{u'}_{L^p}$ by Hölder's inequality.\n\n### L2 human\n\nMorrey's inequality is how one shows that weak solutions are continuous functions.\nIn dimension $n$, a function in $W^{k,p}$ with $kp \u003e n$ is continuous, by combining Morrey's inequality with the Sobolev inequality for the lower derivatives.\nThis is the step that turns $L^2$ regularity results, such as those of [[elliptic regularity|2PVQY9PCT4]], into classical smoothness.\n",
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