Local $C^r$-right equivalence of $C^{r+1}$ functions

Abstract

Let $f,g:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$ be $C^{r+1}$ functions, $r\in \mathbb{N}$. We will show that if $\nabla f(0)=0$ and there exist a neigbourhood $U$ of $0\in {\mathbb{R}}^n$ and a constant $C>0$ such that $$\left|{\partial }^m(g-f)(x)\right|\le C{\left|\nabla f(x)\right|}^{r+2-|m|},x\in U,$$ for any $m\in {\mathbb{N}}_0^n$ such that $|m|\le r$, then there exists a $C^r$ diffeomorphism $\phi :({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$ such that $f=g\circ \phi$ in a neighbourhood of $0$.

References in corpus (1)

• $C^r$-right equivalence of analytic functions