Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions

SynopsisLet P(D) be a partial differential operator with constant coefficients. If P(D) has a continuous linear right inverse in the space of tempered distributions, then P is the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomial P is real and irreducible, then P(D) admits a right inverse in the tempered distributions if and only if P(×) has the property of zeros of R. Thorn.