Weighted Lp boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by where = (y ₁,y ₂, ···, y _m ) and denotes the m-tuple (f ₁,f ₂, ···, f _m ). In this note, the one-weighted and two-weighted boundedness on L ^p (ℝ ⁿ ) space for multilinear fractional integral operator I _α^(m) and the fractional multi-sublinear maximal operator M _α^(m) are established respectively. The authors also obtain two-weighted weak type estimate for the operator M _α^(m). Supported in Part by the NNSF of China under Grant #10771110, and by NSF of Ningbo City under Grant #2006A610090.