Joint Continuity of Division of Smooth Functions. I: Uniform Lojasiewicz Estimates

In this paper we study the question of the existence of a continuous inverse to the multiplication mapping (/, g) -» (fg, g) defined on pairs of C°° 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 (/, G) -» (/(/> ° G), G), where G is a C°° 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) -» (A, AX), where X is a vector valued C°° function and A is a square matrix valued C"5 function on M whose determinant may vanish on a nowhere dense set.