Systems of division problems for distributions

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.