The L p 1 r 1 × L p 2 r 2 × … × L p k r k $L_{p_{1} r_{1}}imes L_{p_{2} r_{2}}imesdotsimes L_{p_{k}r_{k}}$ boundedness of rough multilinear fractional integral operators in the Lorentz spaces

摘要

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 Ω , α ( f ) $M_{Omega,lpha}(mathbf{f})$ and the k-linear fractional integral operator I Ω , α ( f ) $I_{Omega,lpha}(mathbf{f})$ with rough kernels from the spaces L p 1 r 1 × L p 2 r 2 × ? × L p k r k $L_{p_{1} r_{1}}imes L_{p_{2} r_{2}}imescdotsimes L_{p_{k} r_{k}}$ to L q s $L_{q s}$ , where n / ( n + α ) ≤ p 1 $p_{1},p_{2},ldots,p_{k}>1$ and r is the harmonic mean of r 1 , r 2 , … , r k > 0 $r_{1},r_{2},ldots,r_{k}>0$ .