On commutant of system of integration operators in multidimensional domains

We describe the commutant of system of integration operators in the Fréchet space $H(\Omega)$ of all functions holomorphic in a domain $\Omega$ in $\mathbb C^N,$ which is polystar with respect to the origin. In particular, among such domains, there are the products of domains $\mathbb C$ being star with respect to the origin and complete Reinhardt domains with center at the origin. As in the one–dimensional case, the operators in the commutant are the Duhamel operators. We show that $H(\Omega)$ with the Duhamel product $\ast$ is an associative and commutative topological algebra. It is topologically isomorphic to the commutant with the product, which is the composition of operators, and with the topology of bounded convergence. We obtain a similar to one–dimensional representation of the product $f\ast g$ as a sum containing one term being a multiple of $f$ and terms with the integrals at least in one variable of the function independent of the derivatives of $f$. By means of this representation we prove the criterion of $\ast$–invertibility of a function in $H(\Omega)$ and the corresponding convolution operator. We establish that the algebra $(H(\Omega), \ast)$ is local. In the case when the domain $\Omega$ is in addition convex, in the dual situation we obtain the criterion for the invertibility of operator from the commutant of system of operators of partial backward shift.