Division by holomorphic functions and convolution equations in infinite dimension

Abstract:

Let $E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $\Omega$ a connected open set in $E$ and $\mathcal {E}(\Omega )$ the space of the ${C^\infty }$ functions on $\Omega$ (in the real sense). Then we show that any element of $\mathcal {E}’(\Omega )$ may be divided by any nonzero holomorphic function on $\Omega$ with the quotient as an element of $\mathcal {E}’(\Omega )$. This result has for standard consequence a new proof of the surjectivity of any nonzero convolution operator on the space $\operatorname {Exp} (E’)$ of entire functions of exponential type on the dual $E’$ of $E$. As an application of the above division result and of a result of ${C^\infty }$ solvability of the $\overline \partial$ equation in strong duals of nuclear Fréchet spaces we study the solutions of the homogeneous convolution equations in $\operatorname {Exp} (E’)$ in terms of the zero set of their characteristic functions.