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 $\| \mathcal L u \|_{G'} ≤ c(G,G’) \| u \|_G$ (with $G' \subset G$) 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.