Quasiregular map
In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces Rn of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable.
The theory of holomorphic (=analytic) functions of one complex variable is one of the most beautiful and most useful parts of the whole mathematics.
One drawback of this theory is that it deals only with maps between two-dimensional spaces (Riemann surfaces). The theory of functions of several complex variables has a different character, mainly because analytic functions of several variables are not conformal. Conformal maps can be defined between Euclidean spaces of arbitrary dimension, but when the dimension is greater than 2, this class of maps is very small: it consists of Möbius transformations only. This is a theorem of Joseph Liouville; relaxing the smoothness assumptions does not help, as proved by Yurii Reshetnyak.
This suggests the search of a generalization of the property of conformality which would give a rich and interesting class of maps in higher dimension.
A differentiable map f of a region D in Rn to Rn is called K-quasiregular if the following inequality holds at all points in D:
Here K ≥ 1 is a constant, Jf is the Jacobian determinant, Df is the derivative, that is the linear map defined by the Jacobi matrix, and ||·|| is the usual (Euclidean) norm of the matrix.
The development of the theory of such maps showed that it is unreasonable to restrict oneself to differentiable maps in the classical sense, and that the "correct" class of maps consists of continuous maps in the Sobolev space W1,n
loc whose partial derivatives in the sense of distributions have locally summable n-th power, and such that the above inequality is satisfied almost everywhere. This is a formal definition of a K-quasiregular map. A map is called quasiregular if it is K-quasiregular with some K. Constant maps are excluded from the class of quasiregular maps.
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