# Composition of $C^{k,\alpha}$ Holder Functions

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?

• Could you please explain the reasoning for the last line before "suggesting that..."? When $\Omega$ is bounded, and $g \in C^{1,\beta}$, we know that $g$ is Lipschitz. Thus, we shall have $|g(x)-g(y)|\leq |x-y|$, right? Commented Apr 3 at 18:29