END-POINT INEQUALITIES FOR SPHERICAL MULTILINEAR CONVOLUTIONS

Write sigma = (sigma(1), ..., sigma(n)) for an element of the sphere S igma(n-1) and let d sigma denote Lebesgue measure on Sigma(n-1). For f unctions f(1), ..., f(n) on R. define t( f(1), ..., f(n))(x) = integra l(Sigma n-1) f(1)(x - sigma(1)) ... f(n)(x - sigma(n)) d sigma, x is a n element of R. Let R = R(n) denote the closed convex hull ill R-2 of thr 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 if n greater than or equ al to 3, then the inequality parallel to T(f<INF>1</INF>, ..., f<INF>n </INF>)parallel to q less than or equal to C parallel to f<INF>1</INF> parallel to p ... parallel to f<INF>n</INF>parallel to<INF>p </INF>hol ds if and only if ( 1/p, 1/q) is an element of R. Our results fill ill the gap in the necessary and sufficient conditions when n greater tha n or equal to 3 in Oberlin's previous work. A negative result is given along with some positive results, when n = 2. thus narrowing the gap in the necessary and sufficient conditions in this case. (C) 1998 Acad emic Press.