On Associated and Co-Associated Complex Differential Operators

Abstract

The paper deals with initial value problems of the form

in $[0,T]\times G\subset {\mathbb{R}}_0^{+}\times {\mathbb{R}}^n$ where $\mathcal{L}$ is a linear first order differential operator. The desired solutions will be sought in function spaces defined as kernel of a linear differential operator $l$ being associated to $\mathcal{L}$. Mainly two assumptions are required for such initial value problems to be solvable: Firstly, the operators have to be associated, i.e. $lu=0$ implies $l(\mathcal{L}u)=0$. Secondly, an interior estimate $\Vert \mathcal{L}u{\Vert }_{G^{\prime }}\le c(G,G’)\Vert u{\Vert }_G$ (with $G^{\prime }\subset G$) must be true. Moreover, operators $\mathcal{L}$ are investigated possessing a family of associated operators $l_k$ (which then are said to be co-associated).

The present paper surveys the use of associated and co-associated differential operators for solving initial value problems of the above (Cauchy–Kovalevskaya) type. Discussing interior estimates as starting point for the construction of related scales of Banach spaces, the paper sets up a possible framework for further generalizations. E.g., that way a theorem of Cauchy–Kovalevskaya type with initial functions satisfying a differential equation of an arbitrary order k (with not necessarily analytic coefficients) is obtained.