# Tagged Questions

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

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### Why are $C^0(\mathbb{R})$ and $C^{0,0} (\mathbb{R})$ the same spaces?

$C^0(\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})}$. $C^{0,0} (\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})} + \sup_{x,y \in \mathbb{R}, x \neq y} |f(x) - f(y)|$. I don't ...
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### A proof of Holder continuity

I'm studying the following proof by I'm not able to understand the main step. Let $$\sigma(x)=\sqrt{\beta x(N-x)+\alpha x}$$ defined for $x \in [0, N+\frac{\alpha}{\beta}]$....
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### Hölder continuity on space variable

Is it correct to consider Hölder continuity on space variable for solution of stochastic differential equation? Let $$dx(t)=f(t,x(t))dt+g(t,x(t)dB(t)).$$ If $t\mapsto x(t)$ is Hölder continuous i.e.,...
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### Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
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### Notation for subspace of Hölder Space

As mentioned, this is largely a question on notation. I'm reading Fractional Integrals and Derivatives: Theory and Applications by Samko, Kilbas, and Marichev. I'm just starting and I'm curious about ...
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### Using Hölder condition to find upper bound on Fourier coefficients

First I want to stress that I don't want an answer, perhaps a hint. Let $f(x)$ have period $2\pi$ and let $|f(x) -f(y)| \leq c|x-y|^{\alpha}$, for some constants $c$ and $\alpha$ for all $x$ and $y$. ...
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