In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0 , α > 0 , such that for all x and y in the domain of f . More generally, the condition can be formulated for functions between any two metric spaces. The number \alpha is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant (see proof below). If α = 1 , then the function satisfies a Lipschitz condition. For any α > 0 , the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. We have the following chain of inclusions for functions defined on a closed and bounded interval 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 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 can be assigned the norm where β ranges over multi-indices and These seminorms and norms are often denoted simply and or also and in order to stress the dependence on the domain of f . If Ω is open and bounded, then 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 of the corresponding Hölder spaces: which is continuous since, by definition of the Hölder norms, we have: Moreover, this inclusion is compact, meaning that bounded sets in the ‖ · ‖0,β norm are relatively compact in the ‖ · ‖0,α norm. This is a direct consequence of the Ascoli-Arzelà theorem. Indeed, let (un) be a bounded sequence in C0,β(Ω) . Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that un → u uniformly, and we can also assume u = 0 . Then because

Examples

0 < α ≤ β ≤ 1 then all Hölder continuous functions on a bounded set Ω are also Hölder continuous. This also includes β = 1 and therefore all Lipschitz continuous functions on a bounded set are also C0,α Hölder continuous. f(x) = xβ (with β ≤ 1 ) defined on serves as a prototypical example of a function that is C0,α Hölder continuous for 0 < α ≤ β , but not for α > β . Further, if we defined f analogously on [0,\infty), it would be C0,α Hölder continuous only for α = β . Consider the case x < y where. Then, so the difference quotient converges to zero as |x-y| \to 0. Hence f' exists and is zero everywhere. Mean-value theorem now implies f is constant. Q.E.D. Alternate idea: Fix x < y and partition [x,y] into where. Then as n\to \infty, due to \alpha > 1. Thus f(x) = f(y). Q.E.D. f(0) = 0 and by f(x) = 1/log(x) otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however. C := 2 . [0, 1]2 can be constructed to be 1/2–Hölder continuous. It can be proved that when the image of a \alpha-Hölder continuous function from the unit interval to the square cannot fill the square. u satisfies then u is Hölder continuous with exponent α. u(x) satisfies for a fixed λ with 0 < λ < 1 and all sufficiently small values of r, then u is Hölder continuous. C , depending only on p and n, such that: where Thus if u ∈ W1, p(Rn) , then u is in fact Hölder continuous of exponent γ, after possibly being redefined on a set of measure 0.

Properties

H , connected by α–Hölder continuous arcs with α > 1/2 , is a linear subspace. There are closed additive subgroups of H , not linear subspaces, connected by 1/2–Hölder continuous arcs. An example is the additive subgroup L2(R, Z) of the Hilbert space L2(R, R) . f on a metric space X admits a Lipschitz approximation by means of a sequence of functions (fk) such that fk is k-Lipschitz and Conversely, any such sequence (fk) of Lipschitz functions converges to an α–Hölder continuous uniform limit f . f on a subset X of a normed space E admits a uniformly continuous extension to the whole space, which is Hölder continuous with the same constant C and the same exponent α. The largest such extension is: L , where.