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Transactions of the American Mathematical Society

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

 

 

Symmetrization of distributions and its application


Author: Kuang-ho Chen
Journal: Trans. Amer. Math. Soc. 162 (1971), 455-471
MSC: Primary 46F10
DOI: https://doi.org/10.1090/S0002-9947-1971-0415308-X
MathSciNet review: 0415308
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Abstract: Let P be a polynomial such that k of the $ n - 1$ principal curvatures are different from zero at each point of $ N(P) = \{ s \in {R^n}:P(s) = 0\} ;N(P)$ is assumed to be nonempty, bounded, and $ n - 1$ dimensional. If $ {\text{Supp}}\;\varphi \subset {U^\delta } = \{ s \in {R^n}:\vert P(s)\vert < \delta \} $ with $ \delta $ small and $ \varphi \in C_c^\infty ({R^n})$, let $ {\varphi ^\rho }$ be the integral of $ \varphi $ over $ N(P - q)$ if $ q \in [ - \delta ,\delta ]$ and $ {\varphi ^\sigma }(s) = {\varphi ^\rho }(P(s))$ on $ {U^\delta }$ and $ = 0$ outside $ {U^\delta }$. Then $ {\varphi ^\sigma } \in C_c^\infty ({R^n})$. We define the symmetrization $ {v^\sigma }$ of a distribution v, with $ {\text{Supp}}\;v \subset {U^\delta }$, in a natural way. Setting $ u = {\mathcal{F}^{ - 1}}\{ v\} $ and $ {u_0} = {\mathcal{F}^{ - 1}}\{ {v^\sigma }\} $, we prove that $ {u_0}$ is the integral of the product of u with some function $ w(,)$ which depends only on P. This result is used to prove a Liouville type theorem for entire solutions of $ P( - i{D_x})u(x) = f(x)$, with $ f \in C_c^\infty ({R^n})$.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0415308-X
Keywords: Distribution, support of a distribution, fast-decreasing function, tempered distribution, symmetrization of a distribution (or function) with respect to a manifold, diffeomorphism, Dirac-measure on a manifold, quadratic symmetrization of a polynomial, Liouville type theorem
Article copyright: © Copyright 1971 American Mathematical Society