LOCALCr-RIGHT EQUIVALENCE OFCr+1FUNCTIONS

Abstract

AbstractLetf,g:(ℝn, 0) → (ℝ, 0) beCr+1functions,r∈ ℕ. We will show that if ∇f(0)=0 and there exist a neighbourhoodUof 0 ∈ ℝnand a constantC> 0 such that$$\begin{equation*} \left|\partial^m(g-f)(x)\right| ≤ C \left|\nabla f(x)\right|^{r+2-|m|} \quad \textrm{ for } x\in U, \end{equation*} $$and for anym∈ ℕ0nsuch that |m| ≤r, then there exists aCrdiffeomorphism ϕ:(ℝn, 0) → (ℝn, 0) such thatf=g° ϕ in a neighbourhood of 0 ∈ ℝn.