Abstract

We investigate the boundedness of multilinear fractional strong maximal operator ${\mathcal{ℳ}}_{\mathcal{ℛ},\alpha }$ associated with rectangles or related to more general basis with multiple weights $A_{\left(\overrightarrow{p},q\right),\mathcal{ℛ}}$. In the rectangular setting, we first give an end-point estimate of ${\mathcal{ℳ}}_{\mathcal{ℛ},\alpha }$, 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 $0<\alpha <mn$. Then, in the one weight case, we give several equivalent characterizations between ${\mathcal{ℳ}}_{\mathcal{ℛ},\alpha }$ and $A_{\left(\overrightarrow{p},q\right),\mathcal{ℛ}}$. 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 ${\mathcal{ℳ}}_{\mathcal{ℛ},\alpha }$ 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 $A_{\left(\overrightarrow{p},q\right),\mathcal{ℛ}}$ 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.

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Mathematical Subject Classification 2010

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