Sufficient conditions for local equivalence of mappings

Let f,g:(Rn,0)→(Rm,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$\end{document} be Cr+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{r+1}$$\end{document} mappings and let Z={x∈Rn:ν(df(x))=0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$\end{document}, 0∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in Z$$\end{document}, m≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\le n$$\end{document}. We will show that if there exist a neighbourhood U of 0∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in {\mathbb {R}}^n$$\end{document} and constants C,C′>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C,C'>0$$\end{document} and k>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>1$$\end{document} such that for x∈U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in U$$\end{document}ν(df(x))≥Cdist(x,Z)k-1,∂s(fi-gi)(x)≤C′ν(df(x))r+k-|s|,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$\end{document}for any i∈{1,⋯,m}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{1,\dots , m\}$$\end{document} and for any s∈N0n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \mathbf {\mathbb {N}}^n_0$$\end{document} such that |s|≤r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|s|\le r$$\end{document}, then there exists a Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document} diffeomorphism φ:(Rn,0)→(Rn,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$\end{document} such that f=g∘φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=g\circ \varphi $$\end{document} in a neighbourhood of 0∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in {\mathbb {R}}^n$$\end{document}. By ν(df)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu (df)$$\end{document} we denote the Rabier function.


Introduction.
Let C k (n) denote the space of C k function germs (R n , 0) → R, where k, n ∈ N and let C k (n, m) = (C k (n)) m denote the space of C k mapping germs. In the sequel instead of a function germ (a mapping germ) we will write simply a function (a mapping). Recall that C k (n) is a local ring.
Let f, g : (R n , 0) → (R m , 0) be C k mappings. We say that f and g are C r -right equivalent if there exists a C r diffeomorphism ϕ : (R n , 0) → (R n , 0) such that f = g • ϕ in a neighbourhood of 0 ∈ R n . Let N 0 = N ∪ {0}. A norm in R n we denote by | · | and by dist(x, V ) the distance of a point x ∈ R n to a set V ⊂ R n .
In this paper we will prove a sufficient condition for C r -right equivalence of mappings. The result which we obtain is a generalization of the following.
and for any m ∈ N n 0 such that |m| ≤ r, then f and g are C r -right equivalent. There exist two classical techniques to get a suitable diffeomorphism (both cases refer to a local equivalence).
The first originates from the theorem of Kuiper-Kuo [4,5] and relies on constructing a vector field and integration of this field. Kuiper and Kuo used this technique to give a sufficient condition for a topological equivalence of functions with isolated singularities (higher classes of an equivalence of isolated singularities were considered by Takens [12]). Xu generalized this theorem for the case of non-isolated singularities [14]. Theorem 1.1 is on the one hand a counterpart of the theorem of Xu in the case of equivalence of higher classes (r > 0), on the other hand a counterpart of the theorem of Takens in the case of non-isolated singularities. It is worth pointing out that the theorem of Kuiper-Kuo and the theorem of Xu are formulated in the language of jets. Obviously, there arises a natural question what happens in the case of mappings with non-isolated singularities. Rodak, Spodzieja, and the author of this paper [9] and Bekka and Koike [1] independently took that topic and obtained sufficient conditions for topological equivalence without any assumption about singularities which are straightforward generalization of the theorem of Xu.
Before we give the main theorem of this paper we have to introduce some notation. Let X, Y be Banach spaces over R. Let L(X, Y ) denote a Banach space of linear continuous mappings from X to Y . For A ∈ L(X, Y ), A * stands for the adjoint operator in L(Y , X ), where X is the dual space of X. For A ∈ L(X, Y ) we put where A is the norm of a linear mapping A (see [10]). In the case of functions we have ν(df ) = |∇f |, where df is the differential of f . The main result of this paper is the following Theorem 1.2. Let f, g : (R n , 0) → (R m , 0) be C r+1 mappings and let Z = {x ∈ R n : ν(df (x)) = 0}, 0 ∈ Z, m ≤ n. If there exist a neighbourhood U of 0 ∈ R n and constants C, C > 0 and k > 1 such that for

2)
for any i ∈ {1, . . . , m} and for any s ∈ N n 0 such that |s| ≤ r, then f and g are C r -right equivalent. In particular the mappings f and g are C r isotopic at zero.
For the definition of C r isotopy see Subsection 2.2. The above theorem is proved in the spirit of the theorem of Kuiper-Kuo. In comparison to Theorem 1.1 we had to add an extra condition (namely (1.1)) which is forced by tools used for the proof.
The second technique to get a diffeomorphism is Tougeron's implicit function theorem [13]. To formulate consequences of this theorem we need some definitions. Let f ∈ C k+1 (n, m). By M f C k (n) we will denote an ideal in C k (n) generated by all maximal minors of the Jacobi matrix of f . By mC k (n) we mean maximal ideal in C k (n).
As a corollary of Tougeron's implicit function theorem we have ). Let f, g : (R n , 0) → (R m , 0) be C r+2 mappings and let If then f and g are C r -right equivalent.
The first technique is geometrical and the second one is algebraic. Results proved in [7,Main Theorem], [8,Theorem 3] show that we can substitute the geometrical condition by the algebraic. We partially do that in Corollary 4.1.
Moreover, in view of Lemma 2.7, results which we obtain in this paper are true in the complex case.
Additionally in the last section we show the following result.
for x ∈ U , then f and g are bi-lipschitz equivalent.
In the above theorem by bi-lipschitz equivalence we mean that there exists a Lipschitz mapping ϕ : (R n , 0) → (R n , 0), the inverse of which is also a Lipschitz mapping, such that f = g • ϕ in some neighbourhood of 0 ∈ R n .

The Rabier function.
where X is the dual space of X. We begin by recalling some properties of the Rabier function (cf. [11]).

Isotopy.
Let Ω ⊂ R n be a neighbourhood of 0 ∈ R n , Z ⊂ R n be a set such that 0 ∈ Z and let r ∈ N 0 . We will say that a continuous mapping H : , Ω 2 ⊂ R n are neighbourhoods of 0 ∈ R n and let Z ⊂ R n be a set such that 0 ∈ Z. We call f and g C r isotopic near Z at zero if there exists a C r isotopy near Z at zero H : Let Q = (q 1 , . . . , q m ), where q 1 , . . . , q m ∈ C k (n) and let QC k (n) denote an ideal in C k (n) generated by q 1 , . . . , q m .
Vol. 112 (2019) Sufficient conditions for local equivalence of mappings 399 Lemma 2.9. Let f, g : (R n , 0) → (R, 0) be C r+1 functions, r ∈ N. If (g − f ) ∈ (QC r (n)) r+k , then there exist a neighbourhood U of 0 ∈ R n and a constant C > 0 such that and for any s ∈ N n 0 such that |s| ≤ r. Lemma 2.10. Let U be a neighbourhood of 0 ∈ R n , f : U → R be a C r function, and Z = {x ∈ R n : f (x) = 0}, r ∈ N. There exists a C r−|α| function w α such that and for a constat C > 0 Proof. To prove the first part we will carry out induction with respect to |α|.
For simplicity of notation we assume that Let W : G → R n be the following mapping Obviously W = (W 1 , . . . , W n ), where W l = δ I W I l . Note that δ I W I is a C r mapping on G \ (R × Z) (after suitable extension). Hence W = I δ I W I is also a C r mapping on G \ (R × Z). It is easy to see, that W satisfies the equation (3.4).
We will show that W is a C r mapping. The proof of this fact we will carry out in three steps.
Step 1. The mapping W is continuous in G. Indeed, from (1.1), the definitions of A I , the choice of P , the above construction, and (3.1) there exists a constant A > 0 such that The above inequality also holds for (ξ, x) ∈ G ∩ (R × Z), therefore W is continuous in G.
In summary from Step 1, 2, and 3 we obtain that W l are C r functions in G. Therefore, W is a C r mapping in G.

Additional results.
As an immediate consequence of Theorem 1.2 and Lemma 2.9 we have Corollary 4.1. Let f, g : (R n , 0) → (R m , 0) be C r+1 mappings and let Z = {x ∈ R n : ν(df (x)) = 0}, 0 ∈ Z, m ≤ n. If there exist a neighbourhood U of 0 ∈ R n and constants C, C > 0 and k > 1 such that for for any i ∈ {1, . . . , m}, then f and g are C r -right equivalent. In particular mappings f and g are C r isotopic at zero. By (ν(df ))C r (n) we mean the ideal in C r (n) generated by the Rabier function ν(df ).
Let us pay attention that despite Theorem 1.3 and Corollary 4.1 look similar, this result possesses an essential difference. Namely Theorem 1.3 is about C r -right equivalence of C r+2 mappings while Corollary 4.1 is about C r -right equivalence of C r+1 mappings. In fact in Theorem 1.2 and Corollary 4.1 we show slightly more. Namely that two mappings are C r isotopic.
Under previous notations, in the paper [9, Corollary 1.4] it was proved that under the following conditions ν(df (x)) ≥ C dist(x, Z), |f (x) − g(x)| ≤ C 1 ν(df (x)) 2 , df (x) − dg(x) ≤ C 2 ν(df (x)), we get C 0 -right equivalence of two C 1 mappings with locally Lipschitz differentials. But in fact the first out of the above conditions is satisfied to a very narrow class of mappings. So, without changing of the proof we can improve conditions in the above corollary to get the following Proposition 4.2. Let f, g : (R n , 0) → (R m , 0) be C 1 mappings with locally Lipschitz differentials df, dg : (R n , 0) → L(R n , R m ) let Z = {x ∈ R n : ν(df (x)) = 0}, and let 0 ∈ Z, m ≤ n. If there exist a neigbourhood U ∈ R n and constants C > 0, C 1 ∈ (0, 1 2 ) and k > 1 such that df (x) − dg(x) ≤ C 1 ν(df (x)) k−1 for x ∈ U , then f and g are C 0 -right equivalent.
Moreover, in the proof of above theorem it was proved that the vector field W is a Lipschitz mapping in G\(R × Z). But under the assumption of