Smooth and Weak Synthesis of the Anti-diagonal in Fourier Algebras of Lie Groups

Let G be a Lie group of dimension n, and let A(G) be the Fourier algebra of G. We show that the anti-diagonal ∆̌G = {(g, g ) ∈ G × G | g ∈ G} is both a set of local smooth synthesis and a set of local weak synthesis of degree at most [ 2 ]+1 for A(G×G). We achieve this by using the concept of the cone property in [14]. For compact G, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in [7]. We also present similar results for sets of the form HK, where both H and K are subgroups of G×G×G×G of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.