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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 14, Issue 2, 1995, pp. 249–257**

**DOI: 10.4171/ZAA/674**

Published online: 1995-06-30

On Associated and Co-Associated Complex Differential Operators

R. Heersink^{[1]}and Wolfgang Tutschke

^{[2]}(1) Technische Universität Graz, Austria

(2) Technische Universität Graz, Austria

The paper deals with initial value problems of the form
$$\frac{\partial u}{\partial t} = \mathcal L u, \ \ \ u = u_0 \ \mathrm {for} \ t = 0$$
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.

*Keywords: *Initial value problems, interior estimates, Cauchy-Kovalevskaya theorem

Heersink R., Tutschke Wolfgang: On Associated and Co-Associated Complex Differential Operators. *Z. Anal. Anwend.* 14 (1995), 249-257. doi: 10.4171/ZAA/674