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

Abstract

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 G_n = SL(2,ℝ) $\ltimes$ H_n, n ≥ 2, together with previously known results, are used to settle the question of weak amenability for all real algebraic groups. The groups G_n 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.