2020
DOI: 10.1016/j.matpur.2020.02.006
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Regularity and continuity of the multilinear strong maximal operators

Abstract: Let m ≥ 1, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operatorwhere x ∈ R d and R denotes the family of all rectangles in R d 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 Sobolev spacesAs a consequence, we further showed that MR is bou… Show more

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Cited by 11 publications
(11 citation statements)
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“…Subsequently, similar results was extented to multilinear fractional strong maximal operator by Cao et al [5][6][7]. For more works about M (m) n , we refer the readers to [26,33,34].…”
Section: Results For Multilinear Strong Maximal Operatorssupporting
confidence: 64%
“…Subsequently, similar results was extented to multilinear fractional strong maximal operator by Cao et al [5][6][7]. For more works about M (m) n , we refer the readers to [26,33,34].…”
Section: Results For Multilinear Strong Maximal Operatorssupporting
confidence: 64%
“…In 1997, Kinnunen proved that 𝑓 is 𝑝-quasicontinuous if 𝑓 ∈ 𝑊 1,𝑝 (ℝ 𝑑 ) for any 1 < 𝑝 < ∞. Motivated by Kinnunen's work [21], Liu et al [26] established the 𝑝-quasicontinuity of ℳ  ( ⃗ 𝑓), provided that ⃗ 𝑓 = (𝑓 1 , … , 𝑓 𝑚 ) with each 𝑓 𝑖 ∈ 𝑊 1,𝑝 𝑖 (ℝ 𝑑 ) for some 1 < 𝑝 𝑖 < ∞ and 1∕𝑝 = ∑ 𝑚 𝑖=1 1∕𝑝 𝑖 . In this paper, we shall establish the following result.…”
Section: Resultsmentioning
confidence: 99%
“…Recently, the Sobolev regularity of the multilinear strong maximal operator was investigated by Liu et al [26], who obtained the following result.…”
Section: Introductionmentioning
confidence: 99%
“…This continuity property was investigated by Luiro [25] and later extensions were given in [26,8]. More interesting works related to this topic may be found in [2,6,7,9,20,22,23], see also the nice recent survey paper given by Carneiro in [5].…”
mentioning
confidence: 90%