Hölder condition

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 nonnegative real constants C, α, 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 α is called the exponent of the Hölder condition. If α = 1, then the function satisfies a Lipschitz condition. If α > 0, the condition implies the function is continuous. If α = 0, the function need not be continuous, but it is bounded. The condition is named after Otto Hölder.

We have the following chain of inclusions for functions over a compact subset of the real line

Continuously differentiableLipschitz continuousα-Hölder continuousuniformly continuouscontinuous

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(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 to order k and such that the kth 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 norms and seminorms 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, the inequality

holds for all f in C0,β(Ω). 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 unu uniformly, and we can also assume u = 0. Then



and u satisfies
then u is Hölder continuous with exponent α.[1]
for some function u(x) satisfies
for a fixed λ with 0 < λ < 1 and all sufficiently small values of r, then u is Hölder continuous.
for all uC1(Rn) ∩ Lp(Rn), where γ = 1 − (n/p). Thus if uW1, p(Rn), then u is in fact Hölder continuous of exponent γ, after possibly being redefined on a set of measure 0.


Conversely, any such sequence (fk) of Lipschitz functions converges to an αHölder continuous uniform limit f.


  1. See, for example, Han and Lin, Chapter 3, Section 1. This result was originally due to Sergio Campanato.


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