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A locally compact group $G$ is said to be weakly amenable if the Fourieralgebra $A(G)$ admits completely bounded approximative units. Consider thefamily of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the$2n+1$ dimensional Heisenberg group and $SL(2,\Bbb R)$ acts via the irreduciblerepresentation of dimension $2n$ fixing the center of $H_n$. We show that thesegroups fail to be weakly amenable. Following an idea of Haagerup for the case$n=1$ one can reduce matters to the problem of obtaining nontrivial uniformbounds for a family of singular oscillatory integral operators with producttype singularities and polynomial phases.  The result on the family $G_n$ and various other previously known results areused to settle the question of weak amenability for a large class of Liegroups, including the algebraic groups; we assume that the Levi-part has finitecenter.

A family of singular oscillatory integral operators and failure of weak amenability