标题：Regularity and continuity of the multilinear strong maximal operators
作者：Liu F.; Xue Q.; Yabuta K.
作者机构：[Liu, F] College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China;[ Xue, Q] School 更多
通讯作者地址：[Xue, Q] School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, BeijingChin 更多
来源：Journal des Mathematiques Pures et Appliquees
关键词：Multilinear strong maximal operators; Sobolev capacity; Sobolev spaces; Triebel-Lizorkin spaces and Besov spaces
摘要：Let m≥1, in this paper, our object of investigation is the regularity and continuity properties of the following multilinear strong maximal operator MR(f→)(x)=supR∋xR∈R∏i=1m[Formula presented]∫R|fi(y)|dy, where x∈Rd and R denotes the family of all rectangles in Rd with sides parallel to the axes. When m=1, denote MR by MR. Then, MR coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that MR is bounded and continuous from the product Sobolev spaces W1,p1(Rd)×⋯×W1,pm(Rd) to W1,p(Rd), from the product Besov spaces Bs p1,q(Rd)×⋯×Bs pm,q(Rd) to Bs p,q(Rd), from the product Triebel-Lizorkin spaces Fs p1,q(Rd)×⋯×Fs pm,q(Rd) to Fs p,q(Rd). As a consequence, we further showed that MR is bounded and continuous from the product fractional Sobolev spaces to fractional Sobolev space. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of MR. In addition, we proved that MR(f→) is approximately differentiable a.e. when f→=(f1,⋯,fm) with each fj∈L1(Rd) being approximately differentiable a.e. © 2020 Elsevier Masson SAS