On Associated and Co-Associated Complex Differential Operators

  • R. Heersink

    Technische Universität Graz, Austria
  • Wolfgang Tutschke

    Technische Universität Graz, Austria


The paper deals with initial value problems of the form

ut=Lu,   u=u0 for t=0\frac{\partial u}{\partial t} = \mathcal L u, \ \ \ u = u_0 \ \mathrm {for} \ t = 0

in [0,T]×GR0+×Rn[0,T] \times G \subset \mathbb R^+_0 \times \mathbb R^n where L\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 ll being associated to L\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=0lu = 0 implies l(Lu)=0l(\mathcal L u) = 0. Secondly, an interior estimate LuGc(G,G)uG\| \mathcal L u \|_{G'} ≤ c(G,G’) \| u \|_G (with GGG' \subset G) must be true. Moreover, operators L\mathcal L are investigated possessing a family of associated operators lkl_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.

Cite this article

R. Heersink, Wolfgang Tutschke, On Associated and Co-Associated Complex Differential Operators. Z. Anal. Anwend. 14 (1995), no. 2, pp. 249–257

DOI 10.4171/ZAA/674