Lipschitzään

Lipschitzään is a mathematical concept that generalizes Lipschitz continuity by allowing the local bound on how a function can vary to depend on the base point. It is used to describe functions between metric spaces that are not globally Lipschitz but still exhibit controlled, location-dependent variation.

A function f: X → Y between metric spaces (X, d_X) and (Y, d_Y) is said to be Lipschitzään with a Lipschitz gauge L: X → [0, ∞) if for all x, y in X, d_Y(f(x), f(y)) ≤ max{L(x), L(y)} · d_X(x, y). The gauge L is typically assumed to be locally bounded (finite on every compact set) and may be taken to be upper semicontinuous in practical settings. If the gauge is bounded, f is globally Lipschitz with constant equal to the essential bound of L.

Basic properties include that Lipschitzään implies local Lipschitz, since on any compact subset the gauge is bounded, yielding a local constant. The notion is compatible with standard Lipschitz conditions: if L is bounded, the usual Lipschitz condition holds with a constant sup L.

Examples: The map f(x) = x^2 on the real line is Lipschitzään with gauge L(x) = 2|x|, because |x^2 − y^2| = |x − y| |x + y| ≤ 2 max(|x|, |y|) |x − y|. A constant function is Lipschitzään with gauge identically zero.

Origins and use: Lipschitzään appears in theoretical contexts such as geometric analysis and the study of variable-order regularity, where smoothness constraints can vary across a domain. It relates to local Lipschitz and Hölder-type notions.

See also: Lipschitz continuity, local Lipschitz, Hölder continuity, variable exponent spaces.