4
votes
1answer
32 views

Decay of Fourier Coefficients implies Holder Continuity?

This is an exercise problem. I got stuck here and would like to get a hint. The problem is Suppose $f$ is continuous and $2\pi$-periodic, and $|\hat{f}(n)|\leq |n|^{-3/2}$ for all non-zero ...
3
votes
2answers
79 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
1
vote
1answer
23 views

Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?

We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
1
vote
1answer
20 views

Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
2
votes
1answer
44 views

Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
3
votes
5answers
507 views

$\sqrt{x}$ isn't Lipschitz function

A function f such that $$ |f(x)-f(y)| \leq C|x-y| $$ for all $x$ and $y$, where $C$ is a constant independent of $x$ and $y$, is called a Lipschitz function show that $f(x)=\sqrt{x}\hspace{3mm} ...
3
votes
0answers
35 views

Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
7
votes
1answer
299 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...
1
vote
1answer
99 views

Show: $C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)$.

Show that $$ C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)~~~~~~~\forall0<\lambda\leq 1. $$ Hello, I have some problems to show these ...
2
votes
0answers
64 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
1
vote
1answer
220 views

Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$

The sobolev-space $H^s([-\pi,\pi])$ can be embedded into $C^{(\alpha)}([-\pi,\pi])$ (space of $\alpha$-Hölder-continuous functions) and vice-versa. My question is for which exponents $s, \alpha$ can ...
1
vote
1answer
266 views

Hölder Continuity between metric spaces

Hölder continuity has never appeared in my formal education and the wikipedia article seems insufficiently general. I want to make sure that this definition of Hölder continuity is correct and ...
1
vote
1answer
93 views

Regarding Hölder continuity

Let $\alpha \geq 0$. We say that $f \colon D \to \mathbb{R}^m$ is $\alpha$-Hölder continuous if there is a constant $c$ such that for each $x,x_0\in D$, $|f(x) - f(x_0)| \leqslant c\cdot |x - ...
3
votes
1answer
701 views

Space of all Lipschitz-continuous functions and the compact and separable sets

Let $C^{0,1}([a,b])$ be the space of all Lipschitz-continuous functions $x\colon [a,b] \to \mathbb{R}$ with the metric $$ d_{0,1}(x,y) := \sup_{a \le t \le b} |x(t) - y(t)| + \sup_{a \le s,t \le b, ...
1
vote
1answer
85 views

Function extending a Lipschitz function

Let $X$ be a metric space with a metric $d$, let $E\subset X$. We have a function $f:E \rightarrow \mathbb R$ satisfying for some $M>0$: $$ |f(x)-f(y)|\leq M d(x,y) \quad \text{for } x,y \in E. $$ ...
0
votes
1answer
60 views

Are there nonconstant Hoelder type function?

It is known that nonconstant Hoelder functions $f:R \rightarrow R$ with power $>1$ non exist. Are there functions $f: R\rightarrow R$ satisfying $$ |f(x)-f(y)| \leq C|x-y|^p +D$$ for all $x,y ...
3
votes
0answers
214 views

Convergence of sequence in $C^{k, \alpha}$ composed with $C^\infty$ function

Is it true that if a sequence $u_n \to u$ in $C^{k, \alpha}$ norm, and if you have a function $f \in C^\infty$, then $f(u_n) \to f(u)$ in $C^{k, \alpha}$? I think so, since this is true for ordinary ...