H?lder condition

巴禾律 2015-11-22

展开全文

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

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

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 H?lder condition. If α = 1, then the function satisfies a Lipschitz condition. If α = 0, then the function simply is bounded. The condition is named after Otto H?lder.

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

Continuously differentiable ?Lipschitz continuous ? α-H?lder continuous ? uniformly continuous ? continuous

where 0 < α ≤1.

H?lder spacesEdit

H?lder spaces consisting of functions satisfying a H?lder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The H?lder 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 derivatives up to order k and such that the kth partial derivatives are H?lder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the H?lder coefficient

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

is finite, then the function f is said to be (uniformly) H?lder continuous with exponent α in Ω. In this case, the H?lder coefficient serves as a seminorm. If the H?lder coefficient is merely bounded on compact subsets of Ω, then the function f is said to be locally H?lder 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 H?lder space $C^{k,\alpha}(\overline{\Omega})$ can be assigned the norm

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

where β ranges over multi-indices and

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

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. If Ω is open and bounded, then $C^{k,\alpha}(\overline{\Omega})$ is a Banach space with respect to the norm $\|\cdot\|_{C^{k, \alpha}}$ .

Compact embedding of H?lder spacesEdit

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 H?lder exponents. Then, there is an obvious inclusion of the corresponding H?lder spaces:

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

which is continuous since, by definition of the H?lder norms, the inequality

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

holds for all f in C^0,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 (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

$|u_n-u|_{0,\alpha}=|u_n|_{0,\alpha}\to 0,$

because

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

ExamplesEdit

If 0 < α ≤ β ≤ 1 then all $C^{0,\beta}(\overline{\Omega})$ H?lder continuous functions on a bounded set Ω are also $C^{0,\alpha}(\overline{\Omega})$ H?lder continuous. This also includes β = 1 and therefore all Lipschitz continuous functions on a bounded set are also C^0,α H?lder continuous.

The function f(x) = x^β (with β ≤ 1) defined on [0, 1] serves as a prototypical example of a function that is C^0,α H?lder continuous for 0 < α ≤ β, but not for α > β. Further, if we defined f analogously on $[0,\infty)$ , it would be C^0,α H?lder continuous only for α = β.

For α > 1, any α–H?lder continuous function on [0, 1] (or any interval) is a constant.

There are examples of uniformly continuous functions that are not α–H?lder 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 H?lder condition of any order, however.

The Cantor function is H?lder continuous for any exponent α ≤ log(2)/log(3), and for no larger one. In the former case, the inequality of the definition holds with the constant C := 2.

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

Sample paths of Brownian motion are almost surely everywhere locally α-H?lder for every α < 1/2.

Functions which are locally integrable and whose integrals satisfy an appropriate growth condition are also H?lder continuous. For example, if we let

$u_{x,r} = \frac{1}{|B_r|} \int_{B_r(x)} u(y) dy$

and u satisfies

$\int_{B_r(x)} |u(y) - u_{x,r}|^2 dy \leq C r^{n+2\alpha},$

then u is H?lder continuous with exponent α.^[1]

Functions whose oscillation decay at a fixed rate with respect to distance are H?lder continuous with an exponent that is determined by the rate of decay. For instance, if

$w(u,x_0,r) = \sup_{B_r(x_0)} u - \inf_{B_r(x_0)} u$

for some function u(x) satisfies

$w(u,x_0,\tfrac{r}{2}) \leq \lambda w(u,x_0,r)$

for a fixed λ with 0 < λ < 1 and all sufficiently small values of r, then u is H?lder continuous.

Functions in Sobolev space can be embedded into the appropriate H?lder space via Morrey's inequality if the spatial dimension is less than the exponent of the Sobolev space. To be precise, if n < p ≤ ∞ then there exists a constant C, depending only on p and n, such that

$\|u\|_{C^{0,\gamma}(\mathbf{R}^n)}\leq C \|u\|_{W^{1,p}(\mathbf{R}^n)}$

for all u ∈ C¹(Rⁿ) ∩ L^p(Rⁿ), where γ = 1 ? (n/p). Thus if u ∈ W^{1, p}(Rⁿ), then u is in fact H?lder continuous of exponent γ, after possibly being redefined on a set of measure 0.

PropertiesEdit

A closed additive subgroup of an infinite dimensional Hilbert space H, connected by α–H?lder continuous arcs with α > 1/2, is a linear subspace. There are closed additive subgroups of H, not linear subspaces, connected by 1/2–H?lder continuous arcs. An example is the additive subgroup L²(R, Z) of the Hilbert space L²(R, R).

Any α–H?lder 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

$\|f-f_k\|_{\infty,X}=O \left (k^{-\frac{\alpha}{1-\alpha}} \right ).$

Conversely, any such sequence (f_k) of Lipschitz functions converges to an α–H?lder continuous uniform limit f.

Any α–H?lder function f on a subset X of a normed space E admits a uniformly continuous extension to the whole space, which is H?lder continuous with the same constant C and the same exponent α. The largest such extension is:

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

The image of any α–H?lder function f has Hausdorff dimension at most 1/α.

NotesEdit

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

ReferencesEdit

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. .
Han, Qing; Lin, Fanghua (1997). Elliptic Partial Differential Equations. New York: Courant Institute of Mathematical Sciences. ISBN 0-9658703-0-8. OCLC 38168365 MR 1669352