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I know from Triebl, Theory of Function Spaces II, that for $\alpha \notin \mathbb{N}$ Hölder-Zygmund Spaces on $\mathbb{R}$ are equal to the classical Hölder Spaces. However, I have two questions regarding this matter:

  1. Does this hold on compact subsets of $\mathbb{R}$ aswell? For example is the Hölder-Zygmund Space for functions on $[0,1]$ equal to the Hölder space $C^{\alpha}[0,1]$?
  2. I am looking for connections between the classical Hölder Space and the Hölder-Zygmund Space for integer values and functions on $[0,1]$. In particular I would like to estimate $\sup\limits_{|x-y| \leq h} |f(x)-f(y)|$ for a function from a Hölder-Zygmund space with smoothness parameter $\alpha \in \mathbb{N}$ by something like $C \cdot h^{\alpha}$. Is there any possibility to do this?

Thanks a lot!

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A functions $f:[0,1]\to\mathbb R$ admits a natural extension to $\mathbb R$: $$ F(x) = \begin{cases} f(0),\quad & x<0 \\ f(x)\quad & 0\le x\le 1 \\ f(1),\quad & x\ge 1 \end{cases} $$ Observe that the Hölder and Zygmund seminorms of $F$ are comparable to those of $f$. This allows us to translate the results known for $\mathbb R$ to the spaces on $[0,1]$.

Concerning your second question: yes, you can do it. The Zygmund seminorm $$\sup_{x,h} \frac{|f(x-h)-2f(x)+f(x+h)|}{h} \tag{1}$$ dominates the $C^\alpha$ seminorm for every $\alpha\in (0,1)$. That is, $\Lambda^1$ is continuously embedded in $C^\alpha$ for $\alpha\in (0,1)$. One way to see this is to replace the denominator in (1) by $h^\alpha$ and apply the equality of spaces for fractional exponents.

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