Hölder condition: Information from Answers.com

In mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that

$| f(x) - f(y) | \leq C \, |x - y|^{\alpha}$

for all x and y in the domain of ƒ. 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, then the function simply is bounded.

1 Hölder spaces
2 Compact embedding of Hölder spaces
3 Examples
4 References

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 $C k,α (Ω)$ , where Ω is an open subset of some Euclidean space and k ≥ 0 an integer, consists of those functions on Ω having 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

$| f |_{C^{0,\alpha}} = \sup_{x,y \in \Omega} \frac{| f(x) - f(y) |}{|x-y|^\alpha},$

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

If the function ƒ and its derivatives up to order k are bounded on the closure of Ω, then the Hölder space $C^{k,\alpha}(\bar{\Omega})$ can be assigned the norm

$\| f \|_{C^{k, \alpha}} = \|f\|_{C^k}+\max_{| \beta | = k} | D^\beta f |_{C^{0,\alpha}}$

where β ranges over multi-indices and

$\|f\|_{C^k} = \max_{| \beta | \leq k} \, \sup_{x\in\Omega} |D^\beta f (x)|.$

These norms and seminorms are often denoted simply $| f |_{0,\alpha}\;$ and $\| f \|_{k, \alpha}\;$ or also $| f |_{0, \alpha,\Omega}\;$ and $\| f \|_{k, \alpha,\Omega}$ in order to stress the dependence on the domain of f.

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 of the corresponding Hölder spaces:

$C^{0,\beta}(\Omega)\to C^{0,\alpha}(\Omega),$

which is continuous since, by definition of the Hölder norms, the inequality

$| f |_{0,\alpha,\Omega}\le \mathrm{diam}(\Omega)^{\beta-\alpha} | f |_{0,\beta,\Omega}$

holds for all $f\in C^{0,\beta}(\Omega).$ Moreover, this inclusion is compact, meaning that bounded sets in the $\|\cdot\|_{0,\beta}$ norm are relatively compact in the $\|\cdot\|_{0,\alpha}$ 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 $u_n\to u$ uniformly, and we can also assume $u = 0$ . Then $|u_n-u|_{0,\alpha}=|u_n|_{0,\alpha}\to0,$ because

$\frac{|u_n(x)-u_n(y)|}{|x-y|^\alpha}\le\left(\frac{|u_n(x)-u_n(y)|}{|x-y|^\beta}\right)^{\alpha/\beta}|u_n(x)-u_n(y)|^{1-\alpha/\beta} \le |u_n|_{0,\beta}^{\alpha/\beta}\,\left(2\|u_n\|_\infty\right)^{1-\alpha/\beta}=o(1).$

Examples

If 0 < α ≤ β ≤ 1 then all $C 0,β$ Hölder continuous functions on a bounded set are also $C 0,α$ Hölder continuous. This also includes β = 1 and therefore all Lipschitz continuous functions on a bounded set are also $C 0,α$ Hölder continuous. However, the function ƒ(x) = x is Lipschitz continuous on R, but does not satisfy the above definition for α < 1, for couples (x, y) with distance tending to infinity.

The function $f(x)=\sqrt{x}$ defined on [0, 3] is not Lipschitz continuous, but is $C 0,α$ Hölder continuous for α ≤ 1/2.

For α > 1, any α–Hölder continuous function on [0, 1] is a constant.

Peano curves from [0, 1] onto the square [0, 1]² can be constructed to be 1/2–Hölder continuous. It can be proved that when α > 1/2, the image of a α–Hölder continuous function from the unit interval to the square cannot fill the square.

A closed additive subgroup of an infinite dimensional Hilbert space 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 $\scriptstyle L^2(\R,\Z)$ of the Hilbert space $\scriptstyle L^2(\R,\R)$ .

Any α–Hölder continuous function $f$ on a metric space $X$ admits a Lipschitz approximation by means of a sequence of functions $(f k)$ such that $f k$ is $k$ -Lipschitz and $\|f-f_k\|_{\infty,X}=O(k^{-\frac{\alpha}{1-\alpha}}).$ Conversely, any such sequence $(f k)$ of Lipschitz functions converges to an α–Hölder continuous uniform limit $f$ .

Any α–Hölder function $f$ 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 larger such extension is:

$f^*(x):=\inf_{y\in X}\big\{f(y)+C|x-y|^\alpha\big\}.$

References

Lawrence C. Evans (1998). Partial Differential Equations. American Mathematical Society, Providence. ISBN 0-8218-0772-2.
Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7 .

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Hölder condition

Contents

Hölder spaces

Compact embedding of Hölder spaces

Examples

References