Quasiregular

Quasiregular maps are a class of mappings between domains in Euclidean space that generalize holomorphic functions to higher dimensions. A non-constant mapping f: Ω ⊂ R^n → R^n is called quasiregular if it belongs to the Sobolev space W^{1,n}_loc(Ω) and there exists a constant K ≥ 1 such that for almost every x in Ω the differential satisfies |Df(x)|^n ≤ K J_f(x), where |Df| is the operator norm and J_f(x) is the Jacobian determinant. Equivalently, f has finite distortion and is sense-preserving almost everywhere. A closely related notion is that both outer and inner distortion are bounded by K. Quasiregular maps are continuous, open, and discrete, and hence are branched coverings of their domains.

Key properties include controlled distortion and local Hölder continuity, with estimates depending only on n and K. They are open and discrete, and admit a form of the chain rule suitable for Sobolev maps. In higher dimensions, quasiregular maps satisfy various global results parallel to complex analysis, such as distortion inequalities and growth estimates. Rickman’s Picard-type theorem shows that nonconstant quasiregular maps R^n → R^n (n ≥ 3) omit only finitely many values, a higher-dimensional analogue of Picard’s theorem.

In the plane (n = 2), a fundamental result is the Stoilow factorization: every planar quasiregular map can be written as f = g ∘ φ, where φ is quasiconformal and g is holomorphic. Thus two-dimensional quasiregular maps bridge complex analysis and geometric function theory.

Typical examples include holomorphic functions (which are quasiregular with K = 1) and more general maps obtained by composing holomorphic functions with quasiconformal maps. Quasiregular mappings play a central role in geometric function theory, dynamics, and higher-dimensional analysis.