Hölder condition

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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

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.

[edit] Hölder spaces

Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations. 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 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 topological vector space. If the Hölder coefficient

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 can be assigned the norm

where β ranges over multi-indices and

[edit] 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 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.

[edit] 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 .