Weighted inequalities for multilinear fractional integral operators

Abstract

A weighted theory for multilinear fractional integral operators and maximal functions is presented. Sufficient conditions for the two weight inequalities of these operators are found, including "power and logarithmic bumps" and an A ∞ condition. For one weight inequalities a necessary and sufficient condition is then obtained as a consequence of the two weight inequalities. As an application, Poincaré and Sobolev inequalities adapted to the multilinear setting are presented.

Multilinear fractional integrals on Morrey spaces

“…This is motivated by the results due to Lacey and Thiele [6][7][8]. Recently many people are studying these operators from a various points of view [9][10][11][12][13][14].…”

Weighted endpoint estimates for commutators of multilinear fractional integral operators

“…Multilinear fractional integral operators were studied by Grafakos [7], Kenig and Stein [11], Grafakos and Kalton [8]. Recently, Chen and Xue [2], Moen [13] and Pradolini [14] studied the weighted norm inequalities for the multilinear fractional integral operators and the multilinear maximal operator.…”

Sharp weighted bounds for multilinear fractional maximal type operators with rough kernels

“…The following multiple weights class A ( P ,q) was first defined by Chen, Xue in [19], and also simultaneously by Moen in [17]. Definition 1.3.…”

“…On the other hand, the multilinear fractional maximal type operator has been studied by Chen, Xue [19], and also simultaneously by Moen [17]. We summarize some weighted norm inequalities for M α as follows.…”