Lp estimates for multilinear convolution operators defined with spherical measure

Let sigma = (sigma(1),sigma(2), ..., sigma(n)) is an element of Sn-1 and da denote the normalized Lebesgue measure on Sn-1, n >= 2. For functions f(1), f(2), ..., f(n) defined on R, consider the multilinear operator given by T(f(1), f(2), ..., f(n))(x) = integral(Sn-1) Pi(n)(j=1) f(j)(x-sigma(j))d sigma, x is an element of R. In this paper, we obtain necessary and sufficient conditions on exponents p(1), p(2), ..., p(n) and r for which the operator T is bounded from Pi(n)(j=1) L-pj(R) -> L-r(R), where 1 <= p(j), r <= infinity, j=1, 2, ..., n. This generalizes the results obtained in (Bak and Shim, J. Funct. Anal. 157 (1998) 534-553; Oberlin, Trans. Amer. Math. Soc. 310 (1988) 821-835).