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

TL;DR

This paper develops a residue current approach to analyze linear differential equations with constant coefficients and commensurate time lags, extending cohomological and spectral methods to this class of operators.

Contribution

It introduces the ring of partial differential operators with commensurate time lags and proves new cohomological and spectral synthesis results for these operators.

Findings

Proved injectivity of function modules over the ring.

Established spectral synthesis theorems for DΔ equations.

Solved division problems with bounds using integral representations.

Abstract

We introduce the ring of partial differential operators with constant coefficients 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. Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.