Weighted L ^p boundedness for multilinear fractional integral on product spaces

Abstract

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

$$ I_\alpha ^{(m)} (\vec f)(x) = \int_{(\mathbb{R}^n )^m } {\frac{1} {{|\bar y|^{mn - \alpha } }}\prod\limits_{i = 1}^m {f_i (x - y_i )d\vec y} } , $$

where $\vec y$ = (y ₁,y ₂, ···, y _m ) and $\vec f$ denotes the m-tuple (f ₁,f ₂, ···, f _m ). In this note, the one-weighted and two-weighted boundedness on L ^p(ℝⁿ) space for multilinear fractional integral operator I ^(m)_α and the fractional multi-sublinear maximal operator M ^(m)_α are established respectively. The authors also obtain two-weighted weak type estimate for the operator M ^(m)_α .