Closed range composition operators for non-injective smooth symbols $\mathbb{R}\to\mathbb{R}^d$

In 2011 Kenessey and Wengenroth gave a full description of closed range composition operators $C_\psi:C^{\infty}(\mathbb{R}^d)\toC^{\infty}(\mathbb{R})$, $F\mapsto F\circ\psi$, corresponding to smooth injective symbols $\psi:\mathbb{R}\to\mathbb{R}^d$. In 2012 Przestacki gave a sufficient condition for the range of $C_\psi$ to be closed in case if $\psi:\mathbb{R}\to\mathbb{R}$ is a smooth not necessarily injective symbol. Using their ideas we give a sufficient condition ensuring that the range of $C_\psi$ is closed when $\psi:\mathbb{R}\to\mathbb{R}^d$ is a smooth not necessarily injective function.