Hölder condition

In mathematics, a real-valued function f on Rⁿ satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that, $\forall x, y \in \mathbf{R}^n$ ,

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

This condition generalizes to functions between any two metric spaces. The number α is called the exponent of the Hölder condition. If $α = 1$ , then the function satisfies a Lipschitz condition. If $α = 0$ , then the function simply is bounded.

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 $C n,α (Ω)$ , 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, with a seminorm

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

and for $n\geq 0$ the norm is given by

$\| f \|_{C^{n, \alpha}} = \|f\|_{C^n}+\max_{| \beta | = n} | D^\beta f |_{C^{0,\alpha}}$

where β ranges over multi-indices and

$\|f\|_{C^n}=\max_{| \beta | \leq n}\sup_{x\in\Omega} |D^\beta f (x)|$

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

If $0<\alpha\leq\beta\leq1$ then all $C 0,β$ Hölder continuous functions are also $C 0,α$ Hölder continuous. This also includes $β = 1$ and therefore all Lipschitz continuous functions are also $C 0,α$ Hölder continuous.

The function $f(x)=\sqrt{x}$ defined on $[0,3]$ is not Lipschitz continuous, but is $C 0,α$ Hölder continuous for $\alpha\le\frac12$ .

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Hölder condition

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

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