Hölder condition explained

In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants,, such that$$|\; f(x)\; -\; f(y)\; |\; \backslash leq\; C\backslash |\; x\; -\; y\backslash |^$$for all and in the domain of . 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 is constant (see proof below). If, then the function satisfies a Lipschitz condition. For any, 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 :

where .

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, where 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 and such that the -th partial derivatives are Hölder continuous with exponent, where . This is a locally convex topological vector space. If the Hölder coefficient$$\backslash left|\; f\; \backslash right|\_\; =\; \backslash sup\_\; \backslash frac$$

, is finite, then the function is said to be (uniformly) Hölder continuous with exponent in . In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of, then the function is said to be locally Hölder continuous with exponent in .

If the function and its derivatives up to order are bounded on the closure of Ω, then the Hölder space

C ^{k,\alpha} (\overline{\Omega}) can be assigned the norm$$\backslash left\backslash |\; f\; \backslash right\backslash |\_\; =\; \backslash left\backslash |f\backslash right\backslash |\_\; +\; \backslash max\_$$

= k

\left| D^\beta f \right|_where β ranges over multi-indices and$$\backslash |f\backslash |\_\; =\; \backslash max\_$$

\leq k

\sup_ \left |D^\beta f (x) \right |.

These seminorms and norms are often denoted simply

\left| f \right| _{0,\alpha} and \left\| f \right\| _{k,\alpha} or also \left| f \right| _{0,} and \left\| f \right\| _{k,} in order to stress the dependence on the domain of . If is open and bounded, then C ^{k,\alpha} (\overline{\Omega}) 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:$$C^(\backslash Omega)\backslash to\; C^(\backslash Omega),$$which is continuous since, by definition of the Hölder norms, we have:$$\backslash forall\; f\; \backslash in\; C^(\backslash Omega):\; \backslash qquad\; |\; f\; |\_\backslash le\; \backslash mathrm(\backslash Omega)^\; |\; f\; |\_.$$

Moreover, this inclusion is compact, meaning that bounded sets in the norm are relatively compact in the norm. This is a direct consequence of the Ascoli-Arzelà theorem. Indeed, let be a bounded sequence in . Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that uniformly, and we can also assume . Then$$\backslash left|u\_n\; -\; u\backslash right|\_\; =\; \backslash left|\; u\_n\; \backslash right|\_\; \backslash to\; 0,$$because$$\backslash frac$$

^\alpha

= \left(\frac

^\beta

\right)^ \left | u_n(x)-u_n(y) \right |^ \leq |u_n|_^ \left(2\|u_n\|_\infty\right)^ = o(1).

Examples

• If then all

C ^{0,\beta} (\overline{\Omega}) Hölder continuous functions on a bounded set Ω are also C ^{0,\alpha} (\overline{\Omega}) Hölder continuous. This also includes and therefore all Lipschitz continuous functions on a bounded set are also Hölder continuous.

• The function (with) defined on serves as a prototypical example of a function that is Hölder continuous for, but not for . Further, if we defined analogously on

[ 0 , inf ty) , it would be Hölder continuous only for .

• If a function

f is \alpha  - Hölder continuous on an interval and \alpha > 1 , then f is constant.Consider the case x < y where x,y \in R . Then \left| (f(x) - f(y))/(x - y) \right| \leq C |x - y| ^{\alpha} , 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 \{x _{i\}} ^{n} ^{i =0} where x _{k} = x + (k)/(n) (y - x) . Then |f(x) - f(y)| \leq |f(x _{0)} - f(x _{1)|} + |f(x _{1)} - f(x _{2)|} + \ldots + |f(x _{n-1} ) - f(x _{n)|} \leq \sum ^{n} _{i =1} C \left( (|x - y|)/(n) \right) ^{\alpha} = C |x - y| ^{\alpha} n ^{1-\alpha} \to 0 as n\to inf ty , due to \alpha > 1 . Thus f(x) = f(y) . Q.E.D.

• There are examples of uniformly continuous functions that are not  - Hölder continuous for any . For instance, the function defined on by and by 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: $$f(x)\; =\; \backslash sum\_^\; a^n\backslash cos\; \backslash left\; (b^n\; \backslash pi\; x\; \backslash right),$$ where

0 <a< 1 , b is an integer, b \geq 2 and ab > 1 + \tfrac{ 3 \pi}{ 2 }, is -Hölder continuous with^[1] $$\backslash alpha\; =\; -\backslash frac.$$

• The Cantor function is Hölder continuous for any exponent

\alpha \le \tfrac{ log 2 }{ log 3 }, and for no larger one. In the former case, the inequality of the definition holds with the constant .

• Peano curves from onto the square can be constructed to be 1/2 - Hölder continuous. It can be proved that when

\alpha > \tfrac{ 1 }{ 2 } the image of a \alpha -Hölder continuous function from the unit interval to the square cannot fill the square.

• Sample paths of Brownian motion are almost surely everywhere locally

\alpha -Hölder for every \alpha < \tfrac{ 1 }{ 2 } .

• Functions which are locally integrable and whose integrals satisfy an appropriate growth condition are also Hölder continuous. For example, if we let $$u\_\; =\; \backslash frac$$

\int_ u(y) \, dy and satisfies $$\backslash int\_\; \backslash left\; |u(y)\; -\; u\_\; \backslash right\; |^2\; dy\; \backslash leq\; C\; r^,$$ then is Hölder continuous with exponent .^[2]

• Functions whose oscillation decay at a fixed rate with respect to distance are Hölder continuous with an exponent that is determined by the rate of decay. For instance, if $$w(u,x\_0,r)\; =\; \backslash sup\_\; u\; -\; \backslash inf\_\; u$$ for some function satisfies $$w\; \backslash left\; (u,x\_0,\backslash tfrac\; \backslash right)\; \backslash leq\; \backslash lambda\; w\; \backslash left\; (u,x\_0,r\; \backslash right)$$ for a fixed with and all sufficiently small values of, then is Hölder continuous.
• Functions in Sobolev space can be embedded into the appropriate Hölder space via Morrey's inequality if the spatial dimension is less than the exponent of the Sobolev space. To be precise, if

n < p \leq inf ty then there exists a constant, depending only on and, such that: $$\backslash forall\; u\; \backslash in\; C^1\; (\backslash mathbf^n)\; \backslash cap\; L^p\; (\backslash mathbf^n):\; \backslash qquad\; \backslash |u\backslash |\_\backslash leq\; C\; \backslash |u\backslash |\_,$$ where \gamma = 1 - \tfrac{n}{p}. Thus if, then is in fact Hölder continuous of exponent, after possibly being redefined on a set of measure 0.

Properties

• A closed additive subgroup of an infinite dimensional Hilbert space, connected by  - Hölder continuous arcs with, is a linear subspace. There are closed additive subgroups of, not linear subspaces, connected by 1/2 - Hölder continuous arcs. An example is the additive subgroup of the Hilbert space .
• Any  - Hölder continuous function on a metric space admits a Lipschitz approximation by means of a sequence of functions such that is -Lipschitz and $$\backslash left\backslash |f\; -\; f\_k\backslash right\backslash |\_\; =\; O\; \backslash left\; (k^\; \backslash right).$$ Conversely, any such sequence of Lipschitz functions converges to an  - Hölder continuous uniform limit .
• Any  - Hölder function on a subset of a normed space admits a uniformly continuous extension to the whole space, which is Hölder continuous with the same constant and the same exponent . The largest such extension is: $$f^*(x)\; :=\; \backslash inf\_\backslash left\backslash .$$
• The image of any

U \subset R ^{n} under an  - Hölder function has Hausdorff dimension at most \tfrac{\dim _{H(U)}{\alpha}} , where \dim _{H(U)} is the Hausdorff dimension of U .

• The space

C ^{0,\alpha} (\Omega), 0 <\alpha\leq 1 is not separable.

• The embedding

C ^{0,\beta} (\Omega)\subset C ^{0,\alpha} (\Omega), 0 <\alpha<\beta\leq 1 is not dense.

• If

f(t) and g(t) satisfy on smooth arc the H(\mu) and H(\nu) conditions respectively, then the functions f(t) + g(t) and f(t) g(t) satisfy the H(\alpha) condition on, where \alpha = min \{ \mu, \nu\} .

See also

• p-variation

References

• Book: Lawrence C. Evans . Lawrence C. Evans . Partial Differential Equations . American Mathematical Society, Providence . 1998 . 0-8218-0772-2.
• Book: D.. Gilbarg. Neil. Trudinger. Neil Trudinger. Elliptic Partial Differential Equations of Second Order. Springer. New York. 1983. 3-540-41160-7. .
• Book: Qing. Han. Fanghua. Lin. Lin Fanghua. Elliptic Partial Differential Equations. Courant Institute of Mathematical Sciences. New York. 1997. 0-9658703-0-8. 38168365.

Notes and References

1. Hardy . G. H. . Weierstrass's Non-Differentiable Function . Transactions of the American Mathematical Society . 17 . 3 . 1916 . 301–325 . 10.2307/1989005 . 1989005 .
2. See, for example, Han and Lin, Chapter 3, Section 1. This result was originally due to Sergio Campanato.