Differential Operators of Constant Strength

Summary

In this chapter we shall study differential operators which in the spaces B _p.k can be considered as bounded perturbations of differential operators with constant coefficients. This requires that the constant coefficient operators obtained by “freezing” the argument in the coefficients at a point ϰ ₀ have a strength independent of ϰ ₀. By means of a simple perturbation argument most of the results which we have proved for differential operators with constant coefficients can be extended locally to differential operators having constant strength in this sense.

After a discussion of local existence theorems in Sections 13.2 and 13.3 and of hypoellipticity in Section 13.4 we turn in Section 13.5 to global existence questions. It turns out that solution exist globally if and only if the adjoint operator and its localizations at infinity are injective. When the coefficients are real analytic this is always true by Holmgren’s uniqueness theorem (see Section 8.6). However, no such uniqueness theorem is valid when the coefficients are just in C ^∞. In fact, we construct in Section 13.6 a number of examples of non-uniqueness for the Cauchy problem including an elliptic equation which has a non-trivial solution of compact support.