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 that | f ( x ) − f ( y ) | ≤ C ‖ x − y ‖ α {\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\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 {\displaystyle \alpha } 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. If α = 0 {\displaystyle \alpha =0} , the function is simply bounded (any two values f {\displaystyle f} takes are at most C {\displaystyle C} apart).
We 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 C(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. If the Hölder coefficient | f | C 0 , α = sup x , y ∈ Ω , x ≠ y | f ( x ) − f ( y ) | ‖ x − y ‖ α , {\displaystyle \left|f\right|_{C^{0,\alpha }}=\sup _{x,y\in \Omega ,x\neq y}{\frac {|f(x)-f(y)|}{\left\|x-y\right\|^{\alpha }}},} 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 space C k , α ( Ω ¯ ) {\displaystyle C^{k,\alpha }({\overline {\Omega }})} can be assigned the norm ‖ f ‖ C k , α = ‖ f ‖ C k + max | β | = k | D β f | C 0 , α {\displaystyle \left\|f\right\|_{C^{k,\alpha }}=\left\|f\right\|_{C^{k}}+\max _{|\beta |=k}\left|D^{\beta }f\right|_{C^{0,\alpha }}} where β ranges over multi-indices and ‖ f ‖ C k = max | β | ≤ k sup x ∈ Ω | D β f ( x ) | . {\displaystyle \|f\|_{C^{k}}=\max _{|\beta |\leq k}\sup _{x\in \Omega }\left|D^{\beta }f(x)\right|.}
These seminorms and norms are often denoted simply | f | 0 , α {\displaystyle \left|f\right|_{0,\alpha }} and ‖ f ‖ k , α {\displaystyle \left\|f\right\|_{k,\alpha }} or also | f | 0 , α , Ω {\displaystyle \left|f\right|_{0,\alpha ,\Omega }\;} and ‖ f ‖ k , α , Ω {\displaystyle \left\|f\right\|_{k,\alpha ,\Omega }} in order to stress the dependence on the domain of f. If Ω is open and bounded, then C k , α ( Ω ¯ ) {\displaystyle C^{k,\alpha }({\overline {\Omega }})} is a Banach space with respect to the norm ‖ ⋅ ‖ C k , α {\displaystyle \|\cdot \|_{C^{k,\alpha }}} .
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 map of the corresponding Hölder spaces: C 0 , β ( Ω ) → C 0 , α ( Ω ) , {\displaystyle C^{0,\beta }(\Omega )\to C^{0,\alpha }(\Omega ),} which is continuous since, by definition of the Hölder norms, we have: ∀ f ∈ C 0 , β ( Ω ) : | f | 0 , α , Ω ≤ d i a m ( Ω ) β − α | f | 0 , β , Ω . {\displaystyle \forall f\in C^{0,\beta }(\Omega ):\qquad |f|_{0,\alpha ,\Omega }\leq \mathrm {diam} (\Omega )^{\beta -\alpha }|f|_{0,\beta ,\Omega }.}
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 C(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. Then | u n − u | 0 , α = | u n | 0 , α → 0 , {\displaystyle \left|u_{n}-u\right|_{0,\alpha }=\left|u_{n}\right|_{0,\alpha }\to 0,} because | u n ( x ) − u n ( y ) | | x − y | α = ( | u n ( x ) − u n ( y ) | | x − y | β ) α β | u n ( x ) − u n ( y ) | 1 − α β ≤ | u n | 0 , β α β ( 2 ‖ u n ‖ ∞ ) 1 − α β = o ( 1 ) . {\displaystyle {\frac {|u_{n}(x)-u_{n}(y)|}{|x-y|^{\alpha }}}=\left({\frac {|u_{n}(x)-u_{n}(y)|}{|x-y|^{\beta }}}\right)^{\frac {\alpha }{\beta }}\left|u_{n}(x)-u_{n}(y)\right|^{1-{\frac {\alpha }{\beta }}}\leq |u_{n}|_{0,\beta }^{\frac {\alpha }{\beta }}\left(2\|u_{n}\|_{\infty }\right)^{1-{\frac {\alpha }{\beta }}}=o(1).}