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<m<M<\infty such that

m<u(x)<M \quad \forall x \in \Omega.

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.

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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.

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