On the existence of fundamental solutions of boundary problems

Abstract

In this paper we prove an existence theorem for fundamental solutions (see definition below) of a large class of boundary value problems in a half space which includes the Cauchy problem for hyperbolic and parabolic operators with constant coefficients. As a particular case, we obtain Shilov's result [4] on the existence of Green's kernels (see definition below) of Cauchy problems. The technique employed is that of Fourier transform of tempered distributions. Our theorem relies on H6rmander's result on the division of a tempered distribution by a polynomial [3]. The result of this paper is related to our previous results [1] on the existence of fundamental kernels for regular elliptic boundary problems. 1. Let P(D, Dt) where D= (D1i I * * , D.), Dj=(1/i)(/0xj) and Dt= (1/i) (a/axt) be a partial differential operator with constant coefficients. Let P(t, r) be its characteristic polynomial, assume that the highest order coefficient of r is independent of t and that all the roots of the equation in r