In
mathematics, a real or complex-valued function ''f'' on ''d''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', 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 space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s. The number α is called the ''exponent'' of the Hölder condition. A function on an interval satisfying the condition with α > 1 is
constant. If α = 1, then the function satisfies a
Lipschitz condition. For any α > 0, the condition implies the function is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. The condition is named after
Otto Hölder.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line:
:
Continuously differentiable ⊂
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
⊂ α-Hölder continuous ⊂
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
⊂
continuous,
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 system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s. The Hölder space ''C''
''k'',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
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s 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 (''u
n'') be a bounded sequence in ''C''
0,β(Ω). Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that ''u
n'' → ''u'' uniformly, and we can also assume ''u'' = 0. Then
:
because
:
Examples
* If 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
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
functions on a bounded set are also ''C''
0,α Hölder continuous.
* The function ''f''(''x'') = ''x''
β (with β ≤ 1) defined on
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
serves as a prototypical example of a function that is ''C''
0,α Hölder continuous for 0 < α ≤ β, but not for α > β. Further, if we defined ''f'' analogously on
, it would be ''C''
0,α Hölder continuous only for α = β.
* For α > 1, any α–Hölder continuous function on
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
(or any interval) is a constant.
* There are examples of uniformly continuous functions that are not α–Hölder continuous for any α. For instance, the function defined on [0, 1/2] by ''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.
*The
Weierstrass function defined by:
::
:where