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.
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.
, the function is simply bounded (takes values having absolute value at most
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 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.
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
where β ranges over multi-indices and
These seminorms and norms are often denoted simply
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
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.
, so the difference quotient converges to zero as
Mean-value theorem now implies
Alternate idea: Fix