A locally compact group G is said to be weakly amenable if the Fourier algebra A(G) admits completely bounded approximative units. New results concerning the family of semidirect products Gn = SL(2,ℝ) $\ltimes$ Hn, n ≥ 2, together with previously known results, are used to settle the question of weak amenability for all real algebraic groups. The groups Gn fail to be weakly amenable. To show this, one follows an idea of Haagerup for the case n = 1, and one is led to the estimation of certain singular Radon transforms with product-type singularities. By representation theory, matters are reduced to a problem of obtaining rather nontrivial L2-bounds for a family of singular oscillatory integral operators in the plane, with product-type singularities and polynomial phases.
Michael Cowling. Brian Dorofaeff. Andreas Seeger. James Wright. "A family of singular oscillatory integral operators and failure of weak amenability." Duke Math. J. 127 (3) 429 - 486, 15 April 2005. https://doi.org/10.1215/S0012-7094-04-12732-0