We examine the averaging operator corresponding to the manifold in
R2d of pairs of points (u,v) satisfying β£uβ£=β£vβ£=β£uβvβ£=1, so that {0,u,v} is the set of vertices of an equilateral triangle. We
establish LpΓLqβLr boundedness for T for (1/p,1/q,1/r) in the convex hull of the set of points {(0,0,0),(1,0,1),(0,1,1),(1/pdβ,1/pdβ,2/pdβ)}, where pdβ=3dβ25dβ.Comment: 15 pages, discussion on the maximal variant expande