# Ngô Quốc Anh

## June 25, 2010

### Some operations on the Hölder continuous functions

Filed under: Giải Tích 1 — Tags: — Ngô Quốc Anh @ 4:37

This entry devotes the following fundamental question: if $u$ is Hölder continuous, then how about $u^\gamma$ for some constant $\gamma$? Throughout this entry, we work on $\Omega \subset \mathbb R^n$ which is not necessarily bounded.

Firstly, we have an elementary result

Proposition. If $f$ and $g$ are $\alpha$-Hölder continuous and bounded, so is $fg$.

Proof. The proof is simple, we just observe that $\displaystyle\left| {f(x)g(x) - f(y)g(y)} \right| \leqslant \left| {f(x) - f(y)} \right||g(x)| + \left| {g(x) - g(y)} \right||f(y)|$

which yields $\displaystyle\frac{{\left| {f(x)g(x) - f(y)g(y)} \right|}}{{{{\left| {x - y} \right|}^\alpha }}} \leqslant \frac{{\left| {f(x) - f(y)} \right|}}{{{{\left| {x - y} \right|}^\alpha }}}\left( {\mathop {\sup }\limits_\Omega |g(x)|} \right) + \frac{{\left| {g(x) - g(y)} \right|}}{{{{\left| {x - y} \right|}^\alpha }}}\left( {\mathop {\sup }\limits_\Omega |f(y)|} \right)$.

Consequently,

for any positive integer number $n$ and any $\alpha$-Hölder continuous and bounded function $u$, function $u^n$ is also $\alpha$-Hölder continuous and bounded.

Let us assume , $u$ is $\alpha$-Hölder continuous and bounded, $\gamma>0$ is a constant. Let $n =\left\lfloor \gamma \right\rfloor$. Since $\gamma \in \mathbb R$, we may assume $u$ is also bounded away from zero, that means there exist two constants $0 such that $m.

We now study the $\alpha$-Hölder continuity of $u^\gamma$. Observe that function $\displaystyle f(t) = {t^{\left\lceil \gamma \right\rceil }}, \quad t>0$

is sub-additive in the sense that $\displaystyle f(t_1+t_2) \leqslant f(t_1)+f(t_2), \quad \forall t_1,t_2>0$.

## 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 $\Omega$.