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

We investigate the boundedness of multilinear fractional strong maximal operator ,α associated with rectangles or related to more general basis with multiple weights A(p,q),. In the rectangular setting, we first give an end-point estimate of ,α, 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 (α = 0) to the case 0 < α < mn. Then, in the one weight case, we give several equivalent characterizations between ,α and A(p,q),. 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 ,α 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(p,q), 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.

multilinear, strong maximal operator, multiple weights, two-weight inequalities, endpoint estimate
Mathematical Subject Classification 2010
Primary: 42B25
Secondary: 47G10
Received: 5 October 2017
Accepted: 4 June 2019
Published: 4 January 2020
Mingming Cao
School of Mathematical Sciences
% Laboratory of Mathematics and Complex Systems
Beijing Normal University
Qingying Xue
School of Mathematical Sciences
Beijing Normal University
Kôzô Yabuta
Research Center for Mathematical Sciences
Kwansei Gakuin University