Overdetermined systems of linear partial differential operators with normal principal symbols (Q1304652)

Spencer introduced for an involutive, symbol surjective linear partial differential operator a complex of first order linear differential operators. He conjectured that this complex was exact for elliptic operators. The author shows that known methods yield a proof of this conjecture for a special class of elliptic operators satisfying certain assumptions. These include in particular, that the principal symbols of the components of the operator must be normal. The author provides two proofs. One is based on a pointwise estimate allowing for the use of results of \textit{W. J. Sweeney} [Ann. Math., II. Ser. 90, 353-360 (1969; Zbl 0182.42603)]. The other one uses Hörmander's method of weighted \(L^2\)-estimates.