# 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 $\Omega$.