In mathematics, we say that a function satisfies a Hölder condition, or is -Hölder continuous or simply Hölder continuous-dimensional Euclidean space. More generally, the condition can be formulated for functions between any two metric spaces is called the exponent is constant, then the function satisfies a Lipschitz condition, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder, the function is simply bounded apart).

We have the following chain of inclusions for functions defined on a closed and bounded interval of the real line with :

where .

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 k ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up through order and such that the -th partial derivatives are Hölder continuous with exponent , where . This is a locally convex topological vector space is finite, then the function is said to be (uniformly) Hölder continuous with exponent in . In this case, the 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 .

If the function and its derivatives up to order where β ranges over multi-indices

in order to stress the dependence on the domain of . If is a Banach space with respect to the norm

Compact embedding of Hölder spaces

Let Ω be a bounded subset of some Euclidean space (or more generally, any totally bounded metric space) and let 0 < α < β ≤ 1 two Hölder exponents. Then, there is an obvious inclusion map

Moreover, this inclusion is compact, meaning that bounded sets in the norm are relatively compact in the norm. This is a direct consequence of the Ascoli-Arzelà theorem. Indeed, let be a bounded sequence in . Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that uniformly, and we can also assume

Examples

• If Hölder continuous functions on a bounded set Hölder continuous. This also includes and therefore all Lipschitz continuous functions on a bounded set are also Hölder continuous.

• The function (with ) defined on serves as a prototypical example of a function that is Hölder continuous for , but not for . Further, if we defined , it would be Hölder continuous only for .

• is constant.

is constant. Q.E.D.

Alternate idea:. Q.E.D.

• There are examples of uniformly continuous functions that are not Hölder continuous for any . For instance, the function defined on by and by otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however.

• The Weierstrass function is -Hölder continuous with

• The Cantor function is the Hausdorff dimension of the standard Cantor set.) In the former case, the inequality of the definition holds with the constant .

• Peano curves from onto the square can be constructed to be 1/2-Hölder continuous function from the unit interval to the square cannot fill the square.

• Sample paths of Brownian motion are almost surely.

• and then is Hölder continuous with exponent .

• Functions whose oscillation for some function for a fixed with and all sufficiently small values of , then is Hölder continuous.

• Functions in Sobolev space can be embedded into the appropriate Hölder space via Morrey's inequality then there exists a constant , depending only on and Thus if , then is in fact Hölder continuous of exponent , after possibly being redefined on a set of measure 0.

Properties

• A closed additive subgroup of an infinite dimensional Hilbert space , connected by Hölder continuous arcs with , is a linear subspace. There are closed additive subgroups of , not linear subspaces, connected by 1/2Hölder continuous arcs. An example is the additive subgroup of the Hilbert space .

• Any Hölder continuous function on a metric space admits a Lipschitz approximation by means of a sequence of functions such that is Conversely, any such sequence of Lipschitz functions converges to an Hölder continuous uniform limit .

• Any Hölder function on a subset of a normed space admits a uniformly continuous extension to the whole space, which is Hölder continuous with the same constant and the same exponent

• under an .

• is not separable.

• is not dense.

• satisfy on smooth arc condition on .

See also

• p-variation

Notes

References

• .