- Hölder condition
In
mathematics , a real-valued function "f" on R"n" satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants "C", α, such that, ,:
This condition generalizes to functions between any two
metric space s. The number α is called the "exponent" of the Hölder condition. If , then the function satisfies aLipschitz condition . If , then the function simply is bounded.Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of
functional analysis relevant to solvingpartial differential equations . The Hölder space , where Ω is an open subset of some Euclidean space, consists of those functions whosederivatives up to order "n" are Hölder continuous with exponent α. This is atopological vector space , with aseminorm :
and for the norm is given by
: where β ranges over multi-indices and:
Examples in
* If
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