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 ) → ( f g , g ) (f,g) \to (fg,g) defined on pairs of C ∞ {C^\infty } functions on a manifold M 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 ) → ( f ( p ∘ G ) , G ) (f,G) \to (f(p \circ G),G) , where G G is a C ∞ {C^\infty } mapping of the manifold M M into an analytic manifold N N and p p is a fixed analytic function on N N . We prove a theorem adequate for application to the study of inverting the mapping ( A , X ) → ( A , A X ) (A,X) \to (A,AX) , where X X is a vector valued C ∞ {C^\infty } function and A A is a square matrix valued C ∞ {C^\infty } function on M M whose determinant may vanish on a nowhere dense set.