Hölder condition

Type of continuity of a complex-valued function

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 $${\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}$$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 ${\displaystyle \alpha }$ 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.

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:

Continuously differentiable ⊂ Lipschitz continuous ⊂ ${\displaystyle \alpha }$-Hölder continuous ⊂ uniformly continuous = continuous,

where 0 < α ≤ 1.

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 C^k,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. If the Hölder coefficient $${\displaystyle \left|f\right|_{C^{0,\alpha }}=\sup _{x,y\in \Omega ,x\neq y}{\frac {|f(x)-f(y)|}{\left\|x-y\right\|^{\alpha }}},}$$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 ${\displaystyle C^{k,\alpha }({\overline {\Omega }})}$ can be assigned the norm $${\displaystyle \left\|f\right\|_{C^{k,\alpha }}=\left\|f\right\|_{C^{k}}+\max _{|\beta |=k}\left|D^{\beta }f\right|_{C^{0,\alpha }}}$$where β ranges over multi-indices and $${\displaystyle \|f\|_{C^{k}}=\max _{|\beta |\leq k}\sup _{x\in \Omega }\left|D^{\beta }f(x)\right|.}$$

These seminorms and norms are often denoted simply ${\displaystyle \left|f\right|_{0,\alpha }}$ and ${\displaystyle \left\|f\right\|_{k,\alpha }}$ or also ${\displaystyle \left|f\right|_{0,\alpha ,\Omega }\;}$ and ${\displaystyle \left\|f\right\|_{k,\alpha ,\Omega }}$ in order to stress the dependence on the domain of f. If Ω is open and bounded, then ${\displaystyle C^{k,\alpha }({\overline {\Omega }})}$ is a Banach space with respect to the norm ${\displaystyle \|\cdot \|_{C^{k,\alpha }}}$.

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 map of the corresponding Hölder spaces: $${\displaystyle C^{0,\beta }(\Omega )\to C^{0,\alpha }(\Omega ),}$$which is continuous since, by definition of the Hölder norms, we have: $${\displaystyle \forall f\in C^{0,\beta }(\Omega ):\qquad |f|_{0,\alpha ,\Omega }\leq \mathrm {diam} (\Omega )^{\beta -\alpha }|f|_{0,\beta ,\Omega }.}$$

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 (u_n) be a bounded sequence in C^0,β(Ω). Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that u_n → u uniformly, and we can also assume u = 0. Then $${\displaystyle \left|u_{n}-u\right|_{0,\alpha }=\left|u_{n}\right|_{0,\alpha }\to 0,}$$because $${\displaystyle {\frac {|u_{n}(x)-u_{n}(y)|}{|x-y|^{\alpha }}}=\left({\frac {|u_{n}(x)-u_{n}(y)|}{|x-y|^{\beta }}}\right)^{\frac {\alpha }{\beta }}\left|u_{n}(x)-u_{n}(y)\right|^{1-{\frac {\alpha }{\beta }}}\leq |u_{n}|_{0,\beta }^{\frac {\alpha }{\beta }}\left(2\|u_{n}\|_{\infty }\right)^{1-{\frac {\alpha }{\beta }}}=o(1).}$$