Transactions of the American Mathematical Society

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



Symmetrization of distributions and its application. II. Liouville type problem in convolution equations

Author: Kuang-ho Chen
Journal: Trans. Amer. Math. Soc. 171 (1972), 179-194
MSC: Primary 46F10; Secondary 35G05, 47B99
MathSciNet review: 0630201
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Abstract: The symmetrization of distributions corresponding to a bounded $ n - 1$ dimensional $ {C^\infty }$-submanifold of a $ {C^\infty }$-manifold is constructed. This device reduces the consideration of distributions in $ {R^n}$ to the one of distributions in $ {R^1}$, i.e. the symmetrized distributions. Using the relation between the inverse Fourier transform of a symmetrized distribution and the one of the original (nonsymmetrized) distribution, we determine the rate of decay at infinity of solutions to a general convolution equation necessary to assure uniqueness. Using a result in the division problem for distributions, we achieve the following result: If $ u \in C({R^n})$ is a solution of the convolution equation $ S \ast u = f,f \in \mathcal{D}({R^n})$, with some suitable $ S \in \mathcal{E}'({R^n})$, then $ u \in \mathcal{D}({R^n})$, provided $ u$ decays sufficiently fast at infinity.

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Keywords: Symmetrization of distributions corresponding to a manifold in distribution (or function) sense, convolution equation, Liouville type problem, $ {C^\infty }$-diffeomorphism, division problem
Article copyright: © Copyright 1972 American Mathematical Society