Quasiregularity

Quasiregularity is a concept in geometric function theory that describes a class of mappings between domains in Euclidean space that generalize holomorphic functions to higher dimensions by allowing controlled distortion. In its standard form, a mapping f: U → R^n, with U open in R^n and n ≥ 2, is called K-quasiregular if f belongs to the Sobolev space W^{1,n}_{loc}(U, R^n), is sense-preserving, and there exists a constant K ≥ 1 such that the distortion of the differential is bounded almost everywhere. Concretely, the derivative Df has operator norm |Df(x)| and the Jacobian determinant J_f(x) = det Df(x); the inequality |Df(x)|^n ≤ K J_f(x) holds for almost every x in U (equivalently, the local stretching is bounded by K times the local volume expansion). Quasiregular mappings are also known to be discrete and open, and non-constant quasiregular maps have finite multiplicity.

Quasiregularity generalizes conformality: in two dimensions, holomorphic and anti-holomorphic maps are quasiregular, and conformal maps are the K = 1 case. If a quasiregular map is additionally injective, it is called quasiconformal. Unlike quasiconformal maps, quasiregular maps may fail to be injective and can exhibit branching, making them a natural higher-dimensional analogue of branched holomorphic mappings.

Key properties include local Hölder continuity, openness and discreteness, and a rich regularity theory. Quasiregular maps play a central role in geometric function theory, analysis in all dimensions, and the study of dynamics of such maps in higher dimensions. Historical development is associated with Reshetnyak and later work by Rickman and others.