Hölder condition
In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such thatfor all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces.is called the exponent of the Hölder condition.A function on an interval satisfying the condition with α > 1 is constant (see proof below).If α = 1, then the function satisfies a Lipschitz condition.For any α > 0, the condition implies the function is uniformly continuous.The condition is named after Otto Hölder., the function is simply bounded (takes values having absolute value at mostWe have the following chain of inclusions for functions defined on a closed and bounded interval [a, b] of the real line with a < b: where 0 < α ≤ 1.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 Ck,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 through order k and such that the k-th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1.This is a locally convex topological vector space.is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω.In this case, the Hölder coefficient serves as a seminorm.If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function f is said to be locally Hölder 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 Hölder spacewhere β ranges over multi-indices andThese seminorms and norms are often denoted simplyin order to stress the dependence on the domain of f. If Ω is open and bounded, thenis a Banach space with respect to the normLet 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:which is continuous since, by definition of the Hölder norms, we have: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 (un) be a bounded sequence in C0,β(Ω).Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that un → u uniformly, and we can also assume u = 0., so the difference quotient converges to zero asMean-value theorem now impliesAlternate idea: Fix