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We introduce the ring of partial differential operators with constantcoefficients and commensurate time lags (we use the terminology D$\Delta$operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary$D\Delta$ operators and investigate its cohomological properties. Combiningthis ring theoretic observation with the integral representation techniquedeveloped by M. Andersson, we solve a certain type of division with bounds. Inthe last chapter, we prove the injectivity property of various function modulesover this ring as well as spectral synthesis type theorems for $D\Delta$equations.

Residue current approach to Ehrenpreis-Malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags