# Questions tagged [holder-spaces]

Use this tag for concepts related to Hölder continuity (a generalisation of Lipschitz continuity) and the related Hölder spaces.

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### Hölder regularity for $f^2$

Let $f$ be Hölder-continuous, which means in that case $$\sup_{|x-y|\leq \tau}|f(x)-f(y)|\leq C\tau^\alpha,$$ for a $\alpha\in(1/2,1]$ and $\tau < 1$ as well as $x,y\in[0,1]$. I'm trying to show, ...
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### Estimate product of Hölder functions

Let $U \subseteq \mathbb{R}^n$ be the unit ball, let $k \in \mathbb{Z}_{\ge 0}$, and let $0 < \alpha < 1$. Let $f, g \in C^{k, \alpha}(U)$ be Hölder functions, i.e. $f$ and $g$ are of class $C^k$...
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### Holder regularity of Daubechies wavelets

In this pdf, table 2, the H\"older regularity of Daubechies wavelets is listed. The citation is given as Two-Scale difference equations I and Two-Scale difference equations II. However, in the ...
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### If $f \in C_{u}(U \times [0,T])$ then is it true that $f(\cdot,t) \in C_{u}([0,T])$?

Let $f:=f(x,t)$ belong to the parabolic Hölder space $C^{1,\beta}(U\times [0,T])$ where $U$ is a compact subset of $\mathbb R^2$ and $0<\beta <1$. This yields that $f$ is a Hölder continuous ...
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### Are bounded sequences in $C^{1,\alpha}(B_1)$ precompact in $C^{1,\beta}(B_1)$ for $\alpha>\beta$?

Are bounded sequences in $C^{1,\alpha}(B_1)$ precompact in $C^{1,\beta}(B_1)$ for $0<\beta<\alpha<1$? I.e., if $(u_n)_n\subset C^{1,\alpha}(B_1)$ with $||u_n||_{C^{1,\alpha}(B_1)}\leq C$ ...
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### Bounded and Continuous but not Holder [duplicate]

This is a follow-up question to this post. Does there exist an example of a bounded and continuous but not necessarily uniformly continuous function defined on all of $\mathbb{R}$, which is not H\"{o}...
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### Limit of function in $H^1(\mathbb{R}_+;\mathbb{C})$

Let $H^1(\mathbb{R}_+;\mathbb{C})$ the space of function in $L^2(\mathbb{R}_+;\mathbb{C})$ with weak derivative in the same space. Show that \begin{equation} lim_{x\rightarrow \infty} f(x) = 0 \end{...
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### Sharpness of Kolmogorov-Chentsov

The Kolmogorov-Chentsov continuity theorem is a general way to estimate the Hölder continuity of a process $X$ (up to taking a different version $\tilde{X}$ of $X$ ). However, I would like to know if ...
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### Show that if $|\hat{f}(n)|\le\frac{C}{|n|^{1+\alpha}}(n\ne0),\alpha\in(0,1)$ and $f\in C(T)$

Show that if $|\hat{f}(n)|\le\frac{C}{|n|^{1+\alpha}}(n\ne0),\alpha\in(0,1)$ and $f\in C(T)$ then $f$ is holder continuous of order $\alpha$ on $T$.
Consider the $B$-space $X = C([0, 1])$ equipped with the norm $\Vert · \Vert_{\infty}$. And let $F$ be a closed subspace of $X$ such that every function $u\in F$ is Holder continuous, namely, for each ...