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$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$

Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$.

The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of continuous functions $f:\RR^d \to \RR$ whose mixed partial derivatives exist up to order $j$ and that admit a finite norm $$ \|f\|_{j,\alpha} := \sum_{\substack{k \in \NN^d \\ |k|_1 \le j}} \sup_{x \in \RR^d} |D^k f(x)| + \sum_{\substack{k \in \NN^d \\ |k|_1 = j}} \sup_{\substack{x, y \in \RR^d \\ 0<|x-y|}} \frac{|D^k f(x)-D^k f(y)|}{|x-y|^\alpha}, $$ where $k = (k_1, \ldots, k_d) \in \NN^d$ is the multi-index, $|k|_1 := \sum_{i=1}^d k_i$ the $\ell^1$-norm of $k$, and $D^k f := \frac{\partial^{|k|_1} f}{\partial x_1^{k_1} \cdots \partial x_d^{k_d}}$ the mixed partial derivative. Then $(C^{j, \alpha}, \|\cdot\|_{j, \alpha})$ is a Banach space.

In the paper Propagation of chaos and fluctuations for a moderate model with smooth initial data, the authors define a variant of Hölder space as follows. I will use their notations. Let $H^{j+\alpha} := H^{j+\alpha} (\RR^d; \RR)$ be the space of continuous functions $f:\RR^d \to \RR$ whose mixed partial derivatives exist up to order $j$ and that admit a finite norm $$ \|f\|_{j+\alpha} := \sum_{\substack{k \in \NN^d \\ |k|_1 \le j}} \sup_{x \in \RR^d} |D^k f(x)| + \sum_{\substack{k \in \NN^d \\ |k|_1 = j}} \sup_{\substack{x, y \in \RR^d \\ 0<|x-y| \color{blue}{\le 1}}} \frac{|D^k f(x)-D^k f(y)|}{|x-y|^\alpha}. $$

Just as for $(C^{j, \alpha}, \|\cdot\|_{j, \alpha})$, we can prove that $(H^{j+ \alpha}, \|\cdot\|_{j+ \alpha})$ is a Banach space. Clearly, $\|\cdot\|_{j+ \alpha} \le \|\cdot\|_{j, \alpha}$ and thus $H^{j+ \alpha} \subset C^{j, \alpha}$.

My questions are

  1. Are there some reasons/advantages to restrict to $|x-y| \color{blue}{\le 1}$ in the definition of $\|\cdot\|_{j+ \alpha}$?
  2. Are there some textbooks that use the definition of $(H^{j+ \alpha}, \|\cdot\|_{j+ \alpha})$ as that of a Hölder space?
  3. Is it true that $H^{j+ \alpha} = C^{j, \alpha}$ and that $\|\cdot\|_{j+ \alpha} ,\|\cdot\|_{j, \alpha}$ are equivalent norms?

Thank you so much for your elaboration!

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    $\begingroup$ These are equivalent because when $|x - y| > 1$, you can estimate the rightmost term by one of the terms on the left and triangle inequality (if $j = 0$, essentially $|u(x) - u(y)|/|x-y|^\alpha \leq |u(x)| + |u(y)|$, etc). The corresponding seminorms are not equivalent (meaning, if you omit the left sum in the norm), as you can see using functions like $|x|^\alpha$. You would use the $C$ one if you want a scale-invariant seminorm; you might use the $H$ one if you want to avoid dealing with irrelevant behavior at infinity and are only scaling in one direction. $\endgroup$
    – user378654
    Commented Jun 29, 2023 at 2:56
  • $\begingroup$ @user378654 Thank you so much for your help! I got it. Could you post your comment as an answer so that I can accept it? $\endgroup$
    – Akira
    Commented Jun 29, 2023 at 7:04

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