The Lp1r1 x Lp2r2 x center dot center dot center dot x L-pkrk boundedness of rough multilinear fractional integral operators in the Lorentz spaces

Abstract

In this paper, we prove the O'Neil inequality for the k-linear convolution operator in the Lorentz spaces. As an application, we obtain the necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator M-Omega,M-alpha(f) and the k-linear fractional integral operator /(Omega,alpha)(f) with rough kernels from the spaces L-p1r1 x L-p2r2 x center dot center dot center dot x L-pkrk to L-qs, where n/(n + alpha) <= p < q < infinity, 0 < r <= s < infinity, p is the harmonic mean of p(1), p(2),..,p(k) > 1 and r is the harmonic mean of r(1),r(2),...,r(k) >0.