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Transactions of the American Mathematical Society

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

 
 

 

Systems of division problems for distributions


Author: B. Roth
Journal: Trans. Amer. Math. Soc. 155 (1971), 493-504
MSC: Primary 46F10; Secondary 58C25
DOI: https://doi.org/10.1090/S0002-9947-1971-0415310-8
MathSciNet review: 0415310
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Abstract: Suppose ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ is a $p \times p$ matrix of real-valued infinitely (respectively $m$-times continuously) differentiable functions on an open subset $\Omega$ of ${R^n}$. Then ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ maps the space of $p$-tuples of distributions on $\Omega$ (respectively distributions of order $\leqq m$ on $\Omega$) into itself. In the present paper, the $p \times p$ matrices ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ for which this mapping is onto are characterized in terms of the zeros of the determinant of ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ when the ${f_{ij}}$ are infinitely differentiable on $\Omega \subset {R^1}$ and when the ${f_{ij}}$ are $m$-times continuously differentiable on $\Omega \subset {R^n}$. Finally, partial results are obtained when the ${f_{ij}}$ are infinitely differentiable on $\Omega \subset {R^n}$ and extensions are made to $p \times q$ systems of division problems for distributions.


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Keywords: Spaces of distributions, division of distributions, zeros of finite order, the Lojasiewicz inequality
Article copyright: © Copyright 1971 American Mathematical Society