CONTINUOUS LINEAR RIGHT INVERSES FOR PARTIAL-DIFFERENTIAL OPERATORS OF ORDER-2 AND FUNDAMENTAL-SOLUTIONS IN HALF-SPACES

Let P be a complex polynomial in n variables of degree 2 and P(D) the corresponding partial differential operator with constant coefficients . It is shown that P(D) : C-infinity(IR(n)) --> C-infinity(IR(n)) admi ts a continuous linear right inverse if and only if after a separation of variables and up to a complex factor for some c is an element of C the polynomial P has the form P(x(1),lll,x(n)) = Q(x(1),...,x(r)) + L (x(r)+1,...,x(n)) + c where either r = 1 and L = 0 or r > 1, Q and L a re real and Q is indefinite. The proof of this characterization is bas ed on the general solution of the right inverse problem for such opera tors and the fact that for each operator P(D) of the given form and ea ch characteristic vector N there exists a fundamental solution for P(D ) supported by {x is an element of IR(n) : (x, N) greater than or equa l to 0}, which can be constructed explicitely using partial Fourier tr ansform. The existence of sufficiently many fundamental solutions with support in closed half spaces implies that some right inverse can be given by a concrete formula. An example shows that the present charact erization is restricted to operators of order 2.