ESI The Erwin Schr odinger

In the early fties L. Schwartz posed the problem to characterize those linear partial differential operators P(D) that admit a (continuous linear) right inverse on the Fr echet space E(() of all innnitely diierentiable functions on an open set in IR n respectively on the space D 0 (() of all distributions on. This problem was solved in 6], 7]. Its solution was extended to diierential complexes over convex sets by Palamodov 12] and to nonquasianalytic classes and ultradistributions in 8] and 9]. The evaluation of the general solution leads essentially to two cases that are handled by diierent methods. In the case of convex open sets , including = IR n , the existence of a right inverse for P(D) on E(() or D 0 (() is equivalent to the fact that a Phragm en-Lindell of condition depending on , holds for the plurisubharmonic functions on the variety V (P) := fz 2 C I n : P(z) = 0g. For a comprehensive study of these Phragm en-Lindell of conditions we refer to 11]. In the case of open sets with a non-empty C 1-boundary it turns out that P(D) admits a right inverse on E(() or D 0 (() only if P is hyperbolic with respect to each non-characteristic vector that is normal to @ at some point. However, the case of a characteristic half space remained open. In the present paper we give a more detailed characterization of the diierential operators P(D) of order 2 that admit a right inverse on E(IR n). For such operators, the property is equivalent to the existence of a basis fN 1 ; : : :; N n g of IR n such that P(D) admits fundamental solutions E j in D 0 (IR n) that are supported in the closed half spaces H (N j) determined by N j ; 1 j n. Moreover, it is equivalent to the existence of some bounded open convex set in IR n for which P(D) admits a right inverse on E((). An example shows that these equivalences fail for operators of order 3. The existence of suuciently many fundamental solutions supported by half spaces implies that the existing right inverse on E(IR n) can be given by a formula, involving only a nite partition of unity and convolutions with appropriate fundamental solutions that are constructed explicitely. To prove our characterization, we reduce the …