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Inmathematics, a real or complex-valued functionf ond-dimensionalEuclidean space satisfies aHölder condition, or isHölder continuous, when there are real constantsC ≥ 0,α > 0, such that$${\displaystyle |f(x)-f(y)|\le C\Vert x-y{\Vert }^{\alpha }}$$for allx andy in the domain off. More generally, the condition can be formulated for functions between any twometric spaces. The number${\displaystyle \alpha }$ is called theexponent of the Hölder condition. A function on an interval satisfying the condition withα > 1 isconstant (see proof below). Ifα = 1, then the function satisfies aLipschitz condition. For anyα > 0, the condition implies the function isuniformly continuous. The condition is named afterOtto Hölder.If${\displaystyle \alpha =0}$, the function is simplybounded (any two values${\displaystyle f}$ takes are at most${\displaystyle C}$ apart).

We have the following chain of inclusions for functions defined on a closed and bounded interval[a,b] of the real line witha <b:

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

where0 <α ≤ 1.

Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas offunctional analysis relevant to solvingpartial differential equations, and indynamical systems. The Hölder spaceC^k,α(Ω), whereΩ is an open subset of some Euclidean space andk ≥ 0 an integer, consists of those functions on Ω having continuousderivatives up through orderk and such that thek-th partial derivatives are Hölder continuous with exponentα, where0 <α ≤ 1. This is alocally convex topological vector space. If the Hölder coefficient$${\displaystyle {\left|f\right|}_{C^{0,\alpha }}=\underset{x,y\in \mathrm{\Omega },x\ne y}{sup}\frac{|f(x)-f(y)|}{{\Vert x-y\Vert }^{\alpha }},}$$is finite, then the functionf is said to be(uniformly) Hölder continuous with exponentα inΩ. In this case, the Hölder coefficient serves as aseminorm. If the Hölder coefficient is merely bounded oncompact subsets ofΩ, then the functionf is said to belocally Hölder continuous with exponentα inΩ.

If the functionf and its derivatives up to orderk are bounded on the closure of Ω, then the Hölder space${\displaystyle C^{k,\alpha }(\overline{\mathrm{\Omega }})}$ can be assigned the norm$${\displaystyle {\Vert f\Vert }_{C^{k,\alpha }}={\Vert f\Vert }_{C^k}+\underset{|\beta |=k}{max}{\left|D^{\beta }f\right|}_{C^{0,\alpha }}}$$where β ranges overmulti-indices and$${\displaystyle \Vert f{\Vert }_{C^k}=\underset{|\beta |\le k}{max}\underset{x\in \mathrm{\Omega }}{sup}\left|D^{\beta }f(x)\right|.}$$

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

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:$${\displaystyle C^{0,\beta }(\mathrm{\Omega })\to C^{0,\alpha }(\mathrm{\Omega }),}$$which is continuous since, by definition of the Hölder norms, we have:$${\displaystyle \mathrm{\forall }f\in C^{0,\beta }(\mathrm{\Omega }):|f{|}_{0,\alpha ,\mathrm{\Omega }}\le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{\Omega }{)}^{\beta -\alpha }|f{|}_{0,\beta ,\mathrm{\Omega }}.}$$

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 theAscoli-Arzelà theorem. Indeed, let(u_n) be a bounded sequence inC^0,β(Ω). Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality thatu_n →u uniformly, and we can also assumeu = 0. Then$${\displaystyle {\left|u_n-u\right|}_{0,\alpha }={\left|u_n\right|}_{0,\alpha }\to 0,}$$because$${\displaystyle \frac{|u_n(x)-u_n(y)|}{|x-y{|}^{\alpha }}={\left(\frac{|u_n(x)-u_n(y)|}{|x-y{|}^{\beta }}\right)}^{\frac{\alpha }{\beta }}{\left|u_n(x)-u_n(y)\right|}^{1-\frac{\alpha }{\beta }}\le |u_n{|}_{0,\beta }^{\frac{\alpha }{\beta }}{\left(2\Vert u_n{\Vert }_{\mathrm{\infty }}\right)}^{1-\frac{\alpha }{\beta }}=o(1).}$$

• If0 <α ≤β ≤ 1 then all${\displaystyle C^{0,\beta }(\overline{\mathrm{\Omega }})}$ Hölder continuous functions on abounded set Ω are also${\displaystyle C^{0,\alpha }(\overline{\mathrm{\Omega }})}$ Hölder continuous. This also includesβ = 1 and therefore allLipschitz continuous functions on a bounded set are alsoC^0,α Hölder continuous.
• The functionf(x) =x^β (withβ ≤ 1) defined on[0, 1] serves as a prototypical example of a function that isC^0,α Hölder continuous for0 <α ≤β, but not forα >β. Further, if we definedf analogously on${\displaystyle [0,\mathrm{\infty })}$, it would beC^0,α Hölder continuous only forα =β.
• If a function${\displaystyle f}$ is${\displaystyle \alpha }$–Hölder continuous on an interval and${\displaystyle \alpha >1,}$ then${\displaystyle f}$ is constant.

• There are examples of uniformly continuous functions that are notα–Hölder continuous for anyα. For instance, the function defined on[0, 1/2] byf(0) = 0 and byf(x) = 1/log(x) otherwise is continuous, and therefore uniformly continuous by theHeine-Cantor theorem. It does not satisfy a Hölder condition of any order, however.
• TheWeierstrass function defined by:$${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}^n\cos \left(b^n\pi x\right),}$$ where is an integer,${\displaystyle b\ge 2}$ and${\displaystyle ab>1+\frac{3\pi }{2},}$ isα-Hölder continuous with^[1]$${\displaystyle \alpha =-\frac{\log (a)}{\log (b)}.}$$
• TheCantor function is Hölder continuous for any exponent${\displaystyle \alpha \le \frac{\log 2}{\log 3},}$ and for no larger one. (The number${\displaystyle \frac{\log 2}{\log 3}}$ is the Hausdorff dimension of the standard Cantor set.) In the former case, the inequality of the definition holds with the constantC := 2.
• Peano curves from[0, 1] onto the square[0, 1]² can be constructed to be 1/2–Hölder continuous. It can be proved that when${\displaystyle \alpha >\frac{1}{2}}$ the image of a${\displaystyle \alpha }$-Hölder continuous function from the unit interval to the square cannot fill the square.
• Sample paths ofBrownian motion are almost surely everywhere locally${\displaystyle \alpha }$-Hölder for every.
• Functions which are locally integrable and whose integrals satisfy an appropriate growth condition are also Hölder continuous. For example, if we let$${\displaystyle u_{x,r}=\frac{1}{|B_r|}{\int }_{B_r(x)}u(y)dy}$$ andu satisfies$${\displaystyle {\int }_{B_r(x)}{\left|u(y)-u_{x,r}\right|}^{2}dy\le C{r}^{n+2\alpha },}$$ thenu is Hölder continuous with exponentα.^[2]
• Functions whoseoscillation decay at a fixed rate with respect to distance are Hölder continuous with an exponent that is determined by the rate of decay. For instance, if$${\displaystyle w(u,x_0,r)=\underset{B_r(x_0)}{sup}u-\underset{B_r(x_0)}{inf}u}$$ for some functionu(x) satisfies$${\displaystyle w\left(u,x_0,\frac{r}{2}\right)\le \lambda w\left(u,x_0,r\right)}$$ for a fixedλ with0 <λ < 1 and all sufficiently small values ofr, thenu is Hölder continuous.
• Functions inSobolev space can be embedded into the appropriate Hölder space viaMorrey's inequality if the spatial dimension is less than the exponent of the Sobolev space. To be precise, if then there exists a constantC, depending only onp andn, such that:$${\displaystyle \mathrm{\forall }u\in C^1({\mathbf{R}}^n)\cap L^p({\mathbf{R}}^n):\Vert u{\Vert }_{C^{0,\gamma }({\mathbf{R}}^n)}\le C\Vert u{\Vert }_{W^{1,p}({\mathbf{R}}^n)},}$$ where${\displaystyle \gamma =1-\frac{n}{p}.}$ Thus ifu ∈W^1,p(Rⁿ), thenu is in fact Hölder continuous of exponentγ, after possibly being redefined on a set of measure 0.

• A closed additive subgroup of an infinite dimensional Hilbert spaceH, connected byα–Hölder continuous arcs withα > 1/2, is a linear subspace. There are closed additive subgroups ofH, not linear subspaces, connected by 1/2–Hölder continuous arcs. An example is the additive subgroupL²(R,Z) of the Hilbert spaceL²(R,R).
• Anyα–Hölder continuous functionf on a metric spaceX admits aLipschitz approximation by means of a sequence of functions(f_k) such thatf_k isk-Lipschitz and$${\displaystyle {\Vert f-f_k\Vert }_{\mathrm{\infty },X}=O\left(k^{-\frac{\alpha }{1-\alpha }}\right).}$$ Conversely, any such sequence(f_k) of Lipschitz functions converges to anα–Hölder continuous uniform limitf.
• Anyα–Hölder functionf on a subsetX of a normed spaceE admits auniformly continuous extension to the whole space, which is Hölder continuous with the same constantC and the same exponentα. The largest such extension is:$${\displaystyle f^{\ast }(x):=\underset{y\in X}{inf}\left\{f(y)+C|x-y{|}^{\alpha }\right\}.}$$
• The image of any${\displaystyle U\subset {\mathbb{R}}^n}$ under anα–Hölder function has Hausdorff dimension at most${\displaystyle \frac{{\dim }_H(U)}{\alpha }}$, where${\displaystyle {\dim }_H(U)}$ is the Hausdorff dimension of${\displaystyle U}$.
• The space is not separable.
• The embedding is not dense.
• If${\displaystyle f(t)}$ and${\displaystyle g(t)}$ satisfy on smooth arcL the${\displaystyle H(\mu )}$ and${\displaystyle H(\nu )}$ conditions respectively, then the functions${\displaystyle f(t)+g(t)}$ and${\displaystyle f(t)g(t)}$ satisfy the${\displaystyle H(\alpha )}$ condition onL, where${\displaystyle \alpha =min\{\mu ,\nu \}}$.

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