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Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K acompact group of automorphisms of N, and the algebra D(N)^K of left-invariantand K-invariant differential operators on N is commutative. In thesehypotheses, N is necessarily of step at most two. We say that (N,K) satisfiesVinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N).  Fixing a system D of d formally self-adjoint generators of D(N)^K, theGelfand spectrum of the commutative convolution algebra L^1(N)^K can becanonically identified with a closed subset S_D of R^d. We prove that, on anilpotent Gelfand pair satisfying Vinberg's condition, the spherical transformestablishes an isomorphism from the space of $K$-invariant Schwartz functionson N and the space of restrictions to S_D of Schwartz functions in R^d.

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition