In mathematics, we say that a function satisfies a Hölder condition, or is α {\displaystyle \alpha } -Hölder continuous or simply Hölder continuous, if for a real or complex-valued function f {\displaystyle f} on d {\displaystyle d} -dimensional Euclidean space, i.e. f : Ω → R {\displaystyle f:\Omega \to \mathbb {R} } or C {\displaystyle \mathbb {C} } (where Ω ⊆ R d {\displaystyle \Omega \subseteq \mathbb {R} ^{d}} or C d {\displaystyle \mathbb {C} ^{d}} ), when there are real constants C ≥ 0 {\displaystyle C\geq 0} , α > 0 {\displaystyle \alpha >0} , such that
| f ( x ) − f ( y ) | ≤ C ‖ x − y ‖ α {\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}
for all x , y ∈ Ω {\displaystyle x,y\in \Omega } . More generally, the condition can be formulated for functions between any two metric spaces. The number α {\displaystyle \alpha } is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 {\displaystyle \alpha >1} is constant (see proof below). If α = 1 {\displaystyle \alpha =1} , then the function satisfies a Lipschitz condition. For any α > 0 {\displaystyle \alpha >0} , the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. If α = 0 {\displaystyle \alpha =0} , the function is simply bounded (any two values f {\displaystyle f} takes are at most C {\displaystyle C} apart). We have the following chain of inclusions for functions defined on a closed and bounded interval [a, b] of the real line with a < b:
where 0 < α ≤ 1.
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 Ck,α(Ω), where Ω 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 k and such that the k-th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient
| f | C 0 , α = sup x , y ∈ Ω , x ≠ y | f ( x ) − f ( y ) | ‖ x − y ‖ α , {\displaystyle \left|f\right|_{C^{0,\alpha }}=\sup _{x,y\in \Omega ,x\neq y}{\frac {|f(x)-f(y)|}{\left\|x-y\right\|^{\alpha }}},}
is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function f is said to be locally Hölder continuous with exponent α in Ω. If the function f and its derivatives up to order k are bounded on the closure of Ω, then the Hölder space C k , α ( Ω ¯ ) {\displaystyle C^{k,\alpha }({\overline {\Omega }})} can be assigned the norm
‖ f ‖ C k , α = ‖ f ‖ C k + max | β | = k | D β f | C 0 , α {\displaystyle \left\|f\right\|_{C^{k,\alpha }}=\left\|f\right\|_{C^{k}}+\max _{|\beta |=k}\left|D^{\beta }f\right|_{C^{0,\alpha }}}
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