Compact embedding of Hölder spaces

Hölder continuous. In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants , such that

for all and in the domain of . More generally, the condition can be formulated for functions between any two metric spaces. The number is called the exponent of the Hölder condition. If , then the function satisfies a Lipschitz condition. If , then the function simply is bounded.

Hölder spaces. Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space , where is an open subset of some Euclidean space and an integer, consists of those functions on having derivatives  up to order and such that the -th partial derivatives are Hölder continuous with exponent , where . This is a locally convex topological vector space.

If the Hölder coefficient

,

is finite, then the function is said to be (uniformly) Hölder continuous with exponent in . In this case, Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of , then the function is said to be locally Hölder continuous with exponent in .