# 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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### Extension of a Hölder function on the product space

Let $k\geq0, \alpha>0$ and let $\Omega\subset \mathbb{R}^d$ be a $C^{k,\alpha}$ convex domain. Suppose $f:[0,1]\times \overline{\Omega} \to \mathbb{R}$ satisfies the following: For each $t\in[0,1]$...
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### If $f$ is Holder of order $\alpha$, then $f'$ is holder of order $\alpha-1$

Let $D\subset\mathbb{R}$. We say that $f:D\to \mathbb{R}$ is Holder of order $\alpha>1$ if for the largest integer $l<\alpha$ (i.e. $\alpha-l\in (0,1)$: $f$ is $l$ times differentiable, and ...
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### Regularity of functions approximated by polynomials

I want to ask how to prove the following result: Suppose $u\in C(\overline{B_{1}})$, if there exists a universal constant M>0, such that for any $x_{0}\in \overline{B_{1}}$, we have a quadratic ...
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1 vote
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### Reverse Holder functions

For a function defined on $[0,1]\subset \mathbb{R}$, we call it reverse-Holder with exponent $\alpha >0$ if there exists some $c > 0$ such that $$|f(x) - f(y)| \geq c|x-y|^\alpha.$$ The ...
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### Does differential operator $D^{\alpha} : W^{m+|\alpha|, p} \rightarrow W^{m, p}$ have a closed image in $W^{m, p}$

I actually have have $\Omega \subseteq \mathbb{R}^{d_1+d_2}$ and set multiindex set $$\mathcal{A}:= \{\alpha = (\alpha_1, \alpha_2)\in \mathbb{N}^{d_1+d_2} : 2|\alpha_1| + |\alpha_2|\leq2 \}.$$ And ...
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### Showing that spaces $C_b^{(1,2)}$ and $W^{(1,2),p}$ are Banach spaces.

So in essence there are a few propositions in the beginning. For arbitrary $T$ the space $B(T;\mathbb{C})$ is a space of all (real or) complex functions $f: T\rightarrow \mathbb{C}$ with sup norm, ...
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### Does $C^{1,\alpha}$ imply Holder's continuity with exponent $1+\alpha$ in some cases? [duplicate]

My analysis professor said that if $V \colon \mathbb R^n \to \mathbb R$ is $\mathcal C^{1,\alpha}$ for $\alpha >0$, i.e. it is differentiable with Holder's continuous gradient of order $\alpha$ ...
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