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Let $k,\ell \geq 0$ be integers and $\alpha,\beta \in [0,1]$. Let $\Omega\subseteq \mathbb R^n$ be a closed set and set $f \in C^{k,\alpha}(\mathbb R;\mathbb R)$ and $g \in C^{\ell,\beta}(\Omega;\mathbb R)$. I am wondering whether there are any citeable references which state the largest integer $h \geq 0$ and real number $\gamma \in [0,1]$ such that $f\circ g \in C^{h, \gamma}(\Omega;\mathbb R)$. I am also interested in inequalities which upper bound $\|f \circ g\|_{C^{h,\gamma}(\Omega;\mathbb R)}$ in terms of $\|f\|_{C^{k,\alpha}(\mathbb R;\mathbb R)}$ and $\|g \|_{C^{\ell,\beta}(\Omega;\mathbb R)}$.

As mentioned in this post, when $k = \ell = 0$, one may take $\gamma = \alpha\beta$ (and, of course, $h=0$). Furthermore, when $\alpha = \beta = 0$, the chain rule would suggest that one may take $h = \min\{k,\ell\}$ and no better. Perhaps these results can be generalized to show that $f\circ g \in C^{\min\{k,\ell\},\alpha\beta}(\Omega;\mathbb R)$. For example, if $n=k=\ell=1$, then for all $x,y \in \Omega \subseteq \mathbb R$,

$$ \begin{align*} |(f \circ g)'(x) - (f \circ g)'(y)| &= |g'(x) f'(g(x)) - g'(y) f'(g(y))| \\ &\leq |g'(x) f'(g(x)) - g'(x) f'(g(y))| + |g'(x) f'(g(y)) - g'(x) f'(g(y))| \\ &\lesssim \|g\|_{C^{1,\beta}} \|f \|_{C^{1,\alpha}} |g(x) - g(y)|^\alpha + \| f\|_{C^{1,\alpha}} |g'(x) - g'(y) | \\ &\lesssim \|g\|_{C^{1,\beta}}^{1+\alpha} \|f \|_{C^{1,\alpha}} |x - y|^{\alpha\beta} + \| f\|_{C^{1,\alpha}} \| g\|_{C^{1,\beta}} |x - y |^\beta, \end{align*}$$

suggesting that $f \circ g\in C^{1,\alpha\beta}(\Omega;\mathbb R)$ and

$$\|f \circ g\|_{C^{1,\alpha\beta}} \lesssim 1+ \|g\|_{C^{1,\beta}}^{1+\alpha} \|f \|_{C^{1,\alpha}}.$$

Presumably this argument can be generalized, for instance using the Faa di Bruno formula (a similar suggestion was made in this post). But I assume this has been done somewhere already. Does anyone know of a citeable reference containing such a result?

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