Endpoint Inequalities for Spherical Multilinear Convolutions

Writeσ=(σ1, …, σn) for an element of the sphereΣn−1and letdσdenote Lebesgue measure onΣn−1. For functionsf1, …, fnonR, defineT(f1, …, fn(x)=∫Σn−1 f1(x−σ1)…fn(x−σn) dσ,x∈RLetR=R(n) denote the closed convex hull inR2of the points (0, 0), (1/n, 1), ((n+1)/(n+2), 1), ((n+1)/(n+3), 2/(n+3)), ((n−1)/(n+1), 0). We show that ifn⩾3, then the inequality‖T(f1, …, fn)‖q⩽C ‖f1‖p…‖fn‖pholds if and only if (1/p, 1/q)∈R. Our results fill in the gap in the necessary and sufficient conditions whenn⩾3 in Oberlinʹs previous work. A negative result is given along with some positive results, whenn=2, thus narrowing the gap in the necessary and sufficient conditions in this case.