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

DOI: 10.2140/pjm.2019.303.491

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.

Keywords

multilinear, strong maximal operator, multiple weights, two-weight inequalities, endpoint estimate

Mathematical Subject Classification 2010

Primary: 42B25

Secondary: 47G10

Milestones

Received: 5 October 2017

Accepted: 4 June 2019

Published: 4 January 2020