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.