real analysis - Decomposition of Hölder continuous functions

I have carried out the suggestion in the last paragraph of Yemon Choi's answer. Choose $phiin C^infty(mathbb{R})$, $phige0$ and $int_{mathbb{R}}phi(x)dx=1$, and let $phi_R(x)=Rphi(Rx)$. Define

$$ f=phi_Rstareta,quad g=eta-f.$$

Then it is easy to see that

$$ |f|_{C^n}=O(R^{n-alpha}),quad |g|_infty=O(R^{-alpha}),$$

but this is not what you are asking for.

My feeling is that the constant $C$ must show some dependence on $n$.

In response to your last comment, let me prove the estimate on $|f|_{C^n}$. We have

$$f^{(n)}=(phi_R)^{(n)}stareta=R^n(phi^{(n)})_Rstareta.$$

Since $(phi^{(n)})_R$ has mean zero, for any $xinmathbb{R}$:

$$ |f^{(n)}(x)|le R^nint_{mathbb{R}}|phi^{(n)}(y)||eta(x-frac{y}{R})-eta(x)|dyle HR^{n-alpha}int_{mathbb{R}}|phi^{(n)}(y)||y|^alpha dy,$$

where $H$ is $eta$'s Hölder constant.