#### Vol. 303, No. 2, 2019

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On the boundedness of multilinear fractional strong maximal operators with multiple weights

### Mingming Cao, Qingying Xue and Kôzô Yabuta

Vol. 303 (2019), No. 2, 491–518
##### Abstract

We investigate the boundedness of multilinear fractional strong maximal operator ${\mathsc{ℳ}}_{\mathsc{ℛ},\alpha }$ associated with rectangles or related to more general basis with multiple weights ${A}_{\left(\stackrel{\to }{p},q\right),\mathsc{ℛ}}$. In the rectangular setting, we first give an end-point estimate of ${\mathsc{ℳ}}_{\mathsc{ℛ},\alpha }$, which not only extends the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extends the multilinear result of Grafakos, Liu, Pérez and Torres ($\alpha =0$) to the case $0<\alpha . Then, in the one weight case, we give several equivalent characterizations between ${\mathsc{ℳ}}_{\mathsc{ℛ},\alpha }$ and ${A}_{\left(\stackrel{\to }{p},q\right),\mathsc{ℛ}}$. Based on the Carleson embedding theorem regarding dyadic rectangles, we obtain a multilinear Fefferman–Stein type inequality, which is new even in the linear case. We present a sufficient condition for the two weighted norm inequality of ${\mathsc{ℳ}}_{\mathsc{ℛ},\alpha }$ and establish a version of the vector-valued two weighted inequality for the strong maximal operator when $m=1$. In the general basis setting, we study the properties of the multiple weight ${A}_{\left(\stackrel{\to }{p},q\right),\mathsc{ℛ}}$ conditions, including the equivalent characterizations and monotonic properties, which essentially extends previous understanding. Finally, a survey on multiple strong Muckenhoupt weights is given, which demonstrates the properties of multiple weights related to rectangles systematically.

##### Keywords
multilinear, strong maximal operator, multiple weights, two-weight inequalities, endpoint estimate
Primary: 42B25
Secondary: 47G10