Joint continuity of division of smooth functions. I. Uniform Lojasiewicz estimates

Abstract:

In this paper we study the question of the existence of a continuous inverse to the multiplication mapping $(f,g) \to (fg,g)$ defined on pairs of ${C^\infty }$ functions on a manifold $M$. Obviously, restrictions must be imposed on the domain of such an inverse. This leads us to the study of a modified problem: Find an appropriate domain for the inverse of $(f,G) \to (f(p \circ G),G)$, where $G$ is a ${C^\infty }$ mapping of the manifold $M$ into an analytic manifold $N$ and $p$ is a fixed analytic function on $N$. We prove a theorem adequate for application to the study of inverting the mapping $(A,X) \to (A,AX)$, where $X$ is a vector valued ${C^\infty }$ function and $A$ is a square matrix valued ${C^\infty }$ function on $M$ whose determinant may vanish on a nowhere dense set.