Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups

Pseudo H-type Lie groups G(r,s) of signature (r, s) are defined via a module action of the Clifford algebra Cl-r,Cl-s e on a vector space V congruent to R-2(n). They form a subclass of all 2-step nilpotent Lie groups. Based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let N-r,N-s denote the Lie algebra corresponding to G(r,s). In the case s > 0 a choice of left-invariant vector fields [X-1, ..., X-2n] which generate a complement of the center of N(r,s )gives rise to a second order differential operator Delta(r,s) := (X-1(2)+ ... + X-n(2)) - (X-n+1(2) + ... +X-2n(2)), which we call ultra-hyperbolic. We prove that Delta(r,s) is locally solvable if and only if r = 0. In particular, it follows that Delta(r,s) does not admit a fundamental solution in the space D' (G(r,s)) of Schwartz distributions whenever r > 0. In terms of classical special functions we present families of fundamental solutions of A(0,s) in the class of tempered distributions S'(G(0,s)) and study their properties. (C) 2020 Elsevier Inc. All rights reserved.