Hölder condition 1 Hölder condition In mathematics, a real or complex-valued functionƒon 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 ofƒ. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponentof the Hölder condition. Ifα = 1, then the function satisfies a Lipschitz condition. Ifα = 0, then the function simply is bounded. 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 , 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 kand 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 is finite, then the functionƒis 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 functionƒis said to be locally Hölder continuous with exponent α in Ω. If the functionƒand its derivatives up to order kare bounded on the closure of Ω, then the Hölder space can be assigned the norm where β ranges over multi-indices and These norms and seminorms are often denoted simply and or also and in order to stress the dependence on the domain off. If is open and bounded, then is a Banach space with respect to the norm . 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: which is continuous since, by definition of the Hölder norms, the inequality holds for all Moreover, this inclusion is compact, meaning that bounded sets in the norm are relatively compact in the norm. This is a direct consequence of the Ascoli-Arzelà theorem. Indeed, let be a bounded sequence in . Thanks to the Ascoli-Arzelà theorem we can assume without loss ofgenerality that uniformly, and we can also assume . Then because
