A real-valued function f on a metric space $(X,\; d)$ satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that, $\backslash forall\; x,\; y\; \backslash in\; X$,

$|\; f(x)\; -\; f(y)\; |\; \backslash leq\; C\; d(x,y)\; ^\{\backslash alpha\}$.

This condition obviously generalises to functions between any two metric spaces. The number α is called the exponent of the Hölder condition. If $\backslash alpha\; =\; 1$, then the function satisfies a Lipschitz condition.

Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations. The Hölder space $C^\{n,\; \backslash alpha\}\; (\backslash Omega)$, where Ω is an open subset of some euclidean space, consists of those functions whose derivatives up to order n are Hölder continuous with exponent α. This is a topological vector space, where

$\backslash |\; f\; \backslash |\_\{C^\{0,\backslash alpha\}\}\; =\; \backslash sup\_\{x,y\; \backslash in\; \backslash Omega\}\; \backslash frac\{|\; f(x)\; -\; f(y)\; |\}\{|x-y|^\backslash alpha\}$,

and for $n>0$ the norm is given by

$\backslash |\; f\; \backslash |\_\{C^\{n,\; \backslash alpha\}\}\; =\; \backslash sum\_\{|\; \backslash beta\; |\; \backslash leq\; n\}\; \backslash |\; D^\backslash beta\; f\; \backslash |\_\{C^\{0,\backslash alpha\}\}$

where β ranges over multi-indices.

Examples in $C^\{0,\backslash alpha\}(\{\backslash mathbb\; R\})$

• If then all $C^\{0,\backslash beta\}$ Hölder continuous functions are also $C^\{0,\backslash alpha\}$ Hölder continuous. This also includes $\backslash beta=1$ and therefore all Lipschitz continuous functions are also $C^\{0,\backslash alpha\}$ Hölder continuous.

• The function $f(x)=\backslash sqrt\{x\}$ defined on $*$ is not Lipschitz continuous, but is $C^\{0,\backslash alpha\}$ Hölder continuous for $\backslash alpha\backslash le\backslash frac12$.

Functional_analysis