Symmetrization of distributions and its application. II. Liouville type problem in convolution equations
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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.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 171 (1972), 179-194
- MSC: Primary 46F10; Secondary 35G05, 47B99
- DOI: https://doi.org/10.1090/S0002-9947-1972-0630201-6
- MathSciNet review: 0630201