# Revista Matemática Iberoamericana

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**Volume 33, Issue 2, 2017, pp. 555–572**

**DOI: 10.4171/RMI/949**

On multilinear fractional strong maximal operator associated with rectangles and multiple weights

Mingming Cao^{[1]}, Qingying Xue

^{[2]}and Kôzô Yabuta

^{[3]}(1) School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, 100875, BEIJING, CHINA

(2) School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, 100875, BEIJING, CHINA

(3) Research Center for Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1, 669-1337, SANDA, JAPAN

In this paper, we introduce the multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ and the corresponding multiple weights $A_{({\vec{\hskip.5pt p}},q),\mathcal{R}}$ associated with rectangles. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we present a new proof for the weighted estimates of multilinear fractional integral operator and fractional maximal operator associated with cubes, which is quite different from and simpler than the proof that has been presented previously.

*Keywords: *Multilinear fractional strong maximal operators, $A_{({\vec{\hskip.5pt p}},q),\mathcal{R}}$ weights, dyadic reverse doubling condition, two-weight inequalities

Cao Mingming, Xue Qingying, Yabuta Kôzô: On multilinear fractional strong maximal operator associated with rectangles and multiple weights. *Rev. Mat. Iberoamericana* 33 (2017), 555-572. doi: 10.4171/RMI/949