Math 2.0

Functions of a complex variable Definition

Quasiregular map

Builds on Jacobian matrixJacobian determinant

Definition

Let n≥2n \ge 2, let Ω⊂ℝn\Omega \subset \R^n be open, and let K≥1K \ge 1. A map f ⁣:Ω→ℝnf \colon \Omega \to \R^n is KK-quasiregular if:

  1. ff is continuous.
  2. ff lies in the Sobolev space Wloc1,n(Ω;ℝn)W^{1,n}_{\mathrm{loc}}(\Omega; \R^n), that is, its coordinates have locally nn-th power integrable weak derivatives.
  3. ∥Df(x)∥n≤K Jf(x)\norm{Df(x)}^n \le K\, J_f(x) for almost every x∈Ωx \in \Omega.

Here Df(x)Df(x) is the Jacobian matrix of weak derivatives, ∥Df(x)∥\norm{Df(x)} is its operator norm, and Jf(x)=det⁡Df(x)J_f(x) = \det Df(x) is the Jacobian determinant. A map is quasiregular if it is KK-quasiregular for some K≥1K \ge 1.

Remarks

Example: for n=2n = 2, a holomorphic function ff on Ω⊂ℂ=ℝ2\Omega \subset \C = \R^2 is 11-quasiregular. Its Jacobian matrix is a rotation times ∣f′(z)∣\abs{f'(z)}, so ∥Df∥=∣f′∣\norm{Df} = \abs{f'} and Jf=∣f′∣2J_f = \abs{f'}^2.

Non-example: complex conjugation f(z)=zˉf(z) = \bar z fails clause 3. Here ∥Df∥=1\norm{Df} = 1 but Jf=−1J_f = -1, so 1≤−K1 \le -K is false.

A quasiregular homeomorphism onto its image is a quasiconformal map. By a theorem of Reshetnyak, a nonconstant quasiregular map is open and discrete.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionJacobian matrix, concept, no evidence levelJacobian matrixconceptJacobian matrix, definition, no evidence levelJacobian matrixdefinitionJacobian determinant, concept, no evidence levelJacobian determinantconceptQuasiregular map, definition, no evidence levelQuasiregular mapdefinition
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  1. Euclidean space object
  2. Function concept
  3. Total derivative concept
  4. Function of several variables concept
  5. Limit of a function of several variables concept
  6. Partial derivative concept
  7. Differentiable function of several variables definition
  8. Jacobian matrix concept
  9. Jacobian matrix definition
  10. Jacobian determinant concept
  11. Quasiregular map definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quasiregular_map
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DefinesQuasiregular map
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Changed2 November 2026

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