Endpoint inequalities for spherical multilinear convolutions

Abstract

Write sigma = (sigma(1), ..., sigma(n)) for an element of the sphere Sigma(n-1) and let d sigma denote Lebesgue measure on Sigma(n-1). For functions f(1), ..., f(n) on R, define T( f(1), ..., f(n))(x) = integral(Sigma n-1) f(1)(x - sigma(1)) ... f(n)(x - sigma(n)) d sigma, x is an element of R. Let R = R(n) denote the closed convex hull in R-2 of 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 if n >= 3, then the inequality parallel to T(f(1), ..., f(n))parallel to(q) <= C parallel to f(1) parallel to (p) ... parallel to f(n) parallel to(p) holds 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 >= 3 in Oberlin&apos;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 Academic Press.