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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Mark Alan Mostow and Steven Shnider
Journal: Trans. Amer. Math. Soc. 292 (1985), 573-583
MSC: Primary 58C25; Secondary 26E10
MathSciNet review: 808738
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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.

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Keywords: Division of smooth functions, continuity of division, Mather Division Theorem, division of distributions, continuous dependence of solutions on parameters, matrix equations, function spaces
Article copyright: © Copyright 1985 American Mathematical Society