The Triangle Operator

We examine the averaging operator $T$ corresponding to the manifold in ${\mathbb{R}}^{2d}$ of pairs of points $(u,v)$ satisfying $|u| = |v| = |u - v| = 1$,sothat${0,u,v}$ is the set of vertices of an equilateral triangle. We establish $L^p\times L^q\to L^r$ boundedness for $T$ for $(1/p, 1/q, 1/r)$intheconvexhullofthesetofpoints$\lbrace (0, 0, 0) ,, (1, 0 , 1) ,, (0, 1, 1) , , ({1}/{p_d}, {1}/{p_d}, {2}/{p_d}) \rbrace$,where$p_d = \frac{19d-4}{11d - 12}$and$d\geq 7$.