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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
DOI: https://doi.org/10.1090/S0002-9947-1985-0808738-5
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0808738-5
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

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