Lp estimates for multilinear convolution operators defined with spherical measure.

Let σ=(σ1,σ2,⋯,σn)∈Sn−1 and dσ denote the normalized Lebesgue measure on Sn−1,n⩾2. For functions f1,f2,⋯,fn defined on R, consider the multilinear operator given by T(f1,f2,⋯,fn)(x)=∫Sn−1∏j=1nfj(x−σj)dσ,x∈R.In this paper, we obtain necessary and sufficient conditions on exponents p1,p2,⋯,pn and r for which the operator T is bounded from ∏j=1nLpj(R)→Lr(R), where 1⩽pj,r⩽∞,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). [ABSTRACT FROM AUTHOR]