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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subset \\R^n$ be open, and let $K \\ge 1$.\nA map $f \\colon \\Omega \\to \\R^n$ is *$K$-quasiregular* if:\n\n1. $f$ is continuous.\n2. $f$ lies in the Sobolev space $W^{1,n}_{\\mathrm{loc}}(\\Omega; \\R^n)$, that is, its coordinates have locally $n$-th power integrable weak derivatives.\n3. $\\norm{Df(x)}^n \\le K\\, J_f(x)$ for almost every $x \\in \\Omega$.\n\nHere $Df(x)$ is the [[Jacobian matrix|0YF0VXKXGB]] of weak derivatives, $\\norm{Df(x)}$ is its operator norm, and $J_f(x) = \\det Df(x)$ is the [[Jacobian determinant|0ZCSMYHTBP]].\nA map is *quasiregular* if it is $K$-quasiregular for some $K \\ge 1$.\n\n## Remarks\n\nExample: for $n = 2$, a holomorphic function $f$ on $\\Omega \\subset \\C = \\R^2$ is $1$-quasiregular.\nIts Jacobian matrix is a rotation times $\\abs{f'(z)}$, so $\\norm{Df} = \\abs{f'}$ and $J_f = \\abs{f'}^2$.\n\nNon-example: complex conjugation $f(z) = \\bar z$ fails clause 3.\nHere $\\norm{Df} = 1$ but $J_f = -1$, so $1 \\le -K$ is false.\n\nA quasiregular homeomorphism onto its image is a [[quasiconformal map|SB8KG46NHP]].\nBy a theorem of Reshetnyak, a nonconstant quasiregular map is open and discrete.\n",
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