Endpoint Inequalities for Spherical Multilinear Convolutions

AbstractWriteσ=(σ1,…,σn) for an element of the sphereΣn−1and letdσdenote Lebesgue measure onΣn−1. For functionsf1,…,fnonR, defineT(f1,…,fn(x)=∫Σn−1f1(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 Oberlin's 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