# Ngô Quốc Anh

## May 11, 2010

### Compact embedding of Hölder spaces

Filed under: Giải tích 8 (MA5206) — Tags: — Ngô Quốc Anh @ 2:51

Hölder continuous. 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 nonnegative real constants $C, \alpha$, such that

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

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 $\alpha$ is called the exponent of the Hölder condition. If $\alpha = 1$, then the function satisfies a Lipschitz condition. If $\alpha = 0$, 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 $C^{k,\alpha}(\Omega)$, where $\Omega$ is an open subset of some Euclidean space and $k \geqslant 0$ an integer, consists of those functions on $\Omega$ having derivatives  up to order $k$ and such that the $k$-th partial derivatives are Hölder continuous with exponent $\alpha$, where $0 <\alpha \leqslant 1$. This is a locally convex topological vector space.

If the Hölder coefficient

$\displaystyle | f |_{C^{0,\alpha}} = \sup_{x,y \in \Omega} \frac{| f(x) - f(y) |}{|x-y|^\alpha}$,

is finite, then the function $f$ is said to be (uniformly) Hölder continuous with exponent $\alpha$ in $\Omega$. In this case, Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of $\Omega$, then the function $f$ is said to be locally Hölder continuous with exponent $\alpha$ in $\Omega$.

If the function $f$ and its derivatives up to order $k$ are bounded on the closure of $\Omega$, then the Hölder space $C^{k,\alpha}(\bar{\Omega})$ can be assigned the norm

$\displaystyle \| f \|_{C^{k, \alpha}} = \|f\|_{C^k}+\max_{| \beta | = k} | D^\beta f |_{C^{0,\alpha}}$

where $\beta$ ranges over multi-indices and

$\displaystyle\|f\|_{C^k} = \max_{| \beta | \leq k} \, \sup_{x\in\Omega} |D^\beta f (x)|$.

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

Compact embeddings. Let $\Omega$ be a bounded subset of some Euclidean space (or more generally, any totally bounded metric space) and let $0 < \alpha<\beta\leqslant 1$ two Hölder exponents. Then, there is an obvious embedding 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, the inequality

$\displaystyle | f |_{0,\alpha,\Omega}\le \mathrm{diam}(\Omega)^{\beta-\alpha} | f |_{0,\beta,\Omega}$

holds for all $f\in C^{0,\beta}(\Omega)$. Moreover, this embedding is compact, meaning that bounded sets in the $\|\cdot\|_{0,\beta}$ norm are relatively compact in the $\|\cdot\|_{0,\alpha}$ norm. This is a direct consequence of the Ascoli-Arzelà theorem.

Theorem (Ascoli-Arzelà). Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$ (space of continuous functions on $X$) is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and pointwise bounded.

Indeed, let $(u_n)$ be a bounded sequence in $C^{0,\beta}(\Omega)$. Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that $u_n\to u$ uniformly, and we can also assume $u = 0$. Then

$|u_n-u|_{0,\alpha}=|u_n|_{0,\alpha}\to 0$,

because

$\displaystyle\frac{|u_n(x)-u_n(y)|}{|x-y|^\alpha}\le\left(\frac{|u_n(x)-u_n(y)|}{|x-y|^\beta}\right)^{\alpha/\beta}|u_n(x)-u_n(y)|^{1-\alpha/\beta} \le |u_n|_{0,\beta}^{\alpha/\beta}\,\left(2\|u_n\|_\infty\right)^{1-\alpha/\beta}$.

Observe that the RHS is bounded (i.e. $O(1)$).

Examples.

1. If $0 < \alpha<\beta\leqslant 1$ then all $C^{0,\beta}(\bar{\Omega})$  Hölder continuous functions on a bounded set $\Omega$ are also $C^{0,\alpha}(\bar{\Omega})$  Hölder continuous. This also includes $\beta= 1$ and therefore all Lipschitz continuous functions on a bounded set are also $C^{0,\alpha}$  Hölder continuous.
2. The function $f(x)=\sqrt{x}$ defined on $[0, 3]$ is not  Lipschitz continuous (consider $\frac{f(1/n)-f(0)}{1/n-0}$ and let $n\to \infty$), but is $C^{0,\alpha}$ Hölder continuous for $\alpha \leqslant \frac{1}{2}$ since $\frac{|\sqrt{x}-\sqrt{y}|}{\sqrt{|x-y|}} \leqslant \frac{1}{2}$.
3. In the same manner, the function $f(x) = x^\beta$ (with $\beta \leqslant 1$) defined on $[0, 3]$ serves as a prototypical example of a function that is $C^{0,\alpha}$ Hölder continuous for $0 < \alpha \leqslant \beta$, but not for $\alpha > \beta$.
4. There are examples of uniformly continuous functions that are not $\alpha$–Hölder continuous for any $\alpha$. For instance, the function defined on $[0,0.19]$ by

$f\left(x\right) = \begin{cases}0, &x=0,\\13 / \log (\log( \left| \log( x/17 ) \right| )), & x\ne 0. \end{cases}$

Such a function is uniformly continuous. It does not satisfy a Hölder condition of any order, however.

5. For $\alpha>1$, any $\alpha$ Hölder continuous function on $[0, 1]$ is a constant (try to calculate derivative using the definition).
6. Peano curves from $[0, 1]$ onto the square $[0, 1]^2$ can be constructed to be $\frac{1}{2}$ Hölder continuous. It can be proved that when $\alpha >\frac{1}{2}$, the image of a $\alpha$ Hölder continuous function from the unit interval to the square cannot fill the square.
7. A closed additive subgroup of an infinite dimensional Hilbert space $H$, connected by $\alpha$ Hölder continuous arcs with $\alpha>\frac{1}{2}$, is a linear subspace. There are closed additive subgroups of $H$, not linear subspaces, connected by $\frac{1}{2}$ Hölder continuous arcs. An example is the additive subgroup $L^2(\mathbb R,\mathbb Z)$ of the Hilbert space $L^2(\mathbb R,\mathbb R)$.
8. Any $\alpha$ Hölder continuous function f on a metric space $X$ admits a Lipschitz approximation by means of a sequence of functions $(f_k)$ such that $f_k$ is $k$-Lipschitz and

$\displaystyle \|f-f_k\|_{\infty,X}=O(k^{-\frac{\alpha}{1-\alpha}})$.

Conversely, any such sequence $(f_k)$ of Lipschitz functions converges to an $\alpha$ Hölder continuous uniform limit $f$.

9. Any $\alpha$ Hölder function $f$ on a subset $X$ of a normed space $E$ admits a uniformly continuous extension to the whole space, which is Hölder continuous with the same constant $C$ and the same exponent $\alpha$. The larger such extension is

$\displaystyle f^*(x):=\inf_{y\in X}\big\{f(y)+C|x-y|^\alpha\big\}$.

10. Functions in Sobolev space can be embedded into the appropriate Hölder space via Morrey’s inequality if the spatial dimension is less than the exponent of the Sobolev space. To be precise, if $n < p \leqslant \infty$ then there exists a constant $C$, depending only on $p$ and $n$, such that

$\displaystyle\|u\|_{C^{0,\gamma}(\mathbb R^n)}\leq C \|u\|_{W^{1,p}(\mathbb R^n)}$

for all $u \in C^1(\mathbb R^n) \cap L^p(\mathbb R^n)$, where $\gamma = 1 - \frac{n}{p}$. Thus if $u \in W^{1,p}(\mathbb R^n)$, then $u$ is in fact Hölder continuous of exponent $\gamma$, after possibly being redefined on a set of measure 0.