Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$.
More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$.
I am wondering whether the composition $g \circ f:\mathbb{R}^n \to \mathbb{R}$ is Holder of exponent $\min \{\alpha,\beta \}$, i.e, if $g \circ f \in C^{\min\{\alpha,\beta \}}(\mathbb{R}^n;\mathbb{R})$.
This is claimed on a paper I am reading but I don't know how to prove it. The naive approach would be:
$$
|g(f(x+h))-g(f(x))| \le [g]_{\beta}|f(x+h)-f(x)|^{\beta} \le [g]_{\beta} |h|^{\beta \alpha} [f]^{\beta}_{\alpha}
$$
so we have $g \circ f \in C^{\alpha \beta }(\mathbb{R}^n)$. But $\alpha \beta < \min \{\alpha,\beta\}$, so this is not very helpful.
Any help would be welcome.
Thak you.
Best Answer
In fact, $\alpha\beta$ is optimal. To prove it, let $f,g:[0,1]\to\mathbb{R}$ with $f(x)=x^\alpha$ and $g(x)=x^\beta$.
Note that $f\circ g\in C^{\alpha\beta}$, however, for all $\gamma>\alpha\beta$, $f\circ g$ does not belong to $C^\gamma$.