Endpoint bounds for multilinear fractional integrals
Endpoint bounds for multilinear fractional integrals
复制标题
DOI：
10.4310/mrl.2012.v19.n5.a15
复制
发表时间：
2012
影响因子：
1
通讯作者：
Seungwoo Kuk;Sungyun Lee
中科院分区：
文献类型：
--
作者：
Seungwoo Kuk;Sungyun Lee
We prove that the multilinear fractional integral operator Iα(f1, . . . , fk)(x) = ∫ Rn f1(x−θ1y) . . . fk(x−θky)|y|α−ndy, where θj , j = 1, . . . , k are distinct and nonzero, (due to Grafakos [G]) has the endpoint weak-type boundedness into Lr,∞ when r = n 2n−α . Hence, we obtain by the multilinear interpolation theorem that Iα is bounded into Lr for all r > n 2n−α . Moreover, We also prove that Iα is not bounded into L r for any r < n 2n−α under some conditions on θj ’s. Similarly, we show that the multilinear Hilbert transform H(f, g, h1, . . . , hk)(x) = p.v. ∫ f(x + t)g(x − t)kj=1 hj(x − θjt) dt t , where θj = ±1 are distinct and nonzero, is not bounded into Lr for any r < 12 under some conditions on θj ’s.
