Stability of Hölder continuity under nonlinear maps

Question

Let $f, g \in C^{\alpha}(\mathbb{R}^d, \mathbb{R})$ be Hölder continuous of degree $\alpha$ and let $F \in C^2_b$ be a bounded, twice continuously differentiable function with bounded derivatives. I would like to show that then also $F(f) - F(g) \in C^{\alpha}$ with an estimate in the direction of $$\|F(f) - F(g)\|_{C^{\alpha}} \leq C(1 + \|f\|_{C^{\alpha}})(1 + \|g\|_{C^{\alpha}}) \|f-g\|_{\alpha}.$$ Here the fact that $F$ is really 'twice' continuously differentiable should be very important. Maybe it is also necessary that, moreover, $F''$ is Hölder continous of degree $\epsilon$. Unfortunately, I couldn't make out at which point I would be able to exploit this. Help would be very much appreciated.

Answer 1

I will use the notation $$ \|f\|_\infty=\sup_{x\in\Bbb R^n}|f(x)|,\quad\|f\|_\alpha=\|f\|_\infty+\sup_{x\ne y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}. $$ Using the mean value theorem is easy to see that $$ \|F(f)\|_\alpha\le\|F\|_\infty+\|F'\|_\infty\|f\|_\alpha $$ and $$ \|F(f)-F(g)\|_\infty\le\|F'\|_\infty\|f-g\|_\infty. $$ Now, given $x,y\in\Bbb R^n$ and $f,g\in C_b^\alpha$ \begin{align} F(f(x))-F(g(x))&=F'(\xi)(f(x)-g(x)),\quad\xi\text{ between }f(x)\text{ and }g(x),\\ F(f(y))-F(g(y))&=F'(\eta)(f(y)-g(y)),\quad\eta\text{ between }f(y)\text{ and }g(y). \end{align} Then \begin{align} &|(F(f(x))-F(g(x)))-(F(f(y))-F(g(y)))|\\ &\qquad=|F'(\xi)(f(x)-g(x))-F'(\eta)(f(y)-g(y))|\\ &\qquad\le|F'(\xi)|((f(x)-g(x))-(f(y)-g(y)))|+|F'(\xi)-F'(\eta)|\,|f(y)-g(y)|\\ &\qquad\le\|F'\|_\infty\|f-g\|_\alpha|x-y|^\alpha+\|F''\|_\infty\|f-g\|_\infty|\xi-\eta|\\ &\qquad\le\|F'\|_\infty\|f-g\|_\alpha|x-y|^\alpha+\|F''\|_\infty\|f-g\|_\infty|(\|f\|_\alpha+\|g\|_\alpha)|x-y|^\alpha, \end{align} where in the last line I have used that $|\xi-\eta|\le|f(x)-f(y)|+|g(x)-g(y)|$.