Abstract dyadic cubes, maximal operators and Hausdorff content

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• 作者：Hiroki Saitoup>aup>up> ; up>up>j1107703@gmail.com" class="auth_mail" title="E-mail the corresponding authorup> ; Hitoshi Tanakaup>bup>up> ; up>up>htanaka@ms.u-tokyo.ac.jp" class="auth_mail" title="E-mail the corresponding authorup> ; Toshikazu Watanabeup>cup>up> ; up>up>twatana@edu.tuis.ac.jp" class="auth_mail" title="E-mail the corresponding authorup>
• 关键词：42B25
• 刊名：Bulletin des Sciences Mathematiques
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：140
• 期：6
• 页码：757-773
• 全文大小：321 K

文摘

Let μ   be a locally finite Borel measure and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si1.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=b7164e6f30fba35666d53c1c6e4447de" title="Click to view the MathML source">D$\mathcal{D}$ a family of measurable sets equipped with a certain dyadic structure. For ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si2.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=17282afcfedcd552f559c48972fb0a63" title="Click to view the MathML source">E&sub;Rup>nup> and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si3.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=dd1e5a5e56ca95fdf92a1e0bee898477" title="Click to view the MathML source">0<α≤n$0<\alpha \le n$, by α-dimensional Hausdorff content we mean where the infimum is taken over all coverings of E   by countable families of the abstract dyadic cubes ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si5.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=8c147b197bf6cde1f24d274f061d7f92" title="Click to view the MathML source">{Qub>jub>}&sub;D. In this paper we study the boundedness of the Hardy–Littlewood maximal operator urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si32.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=200b59a2885dfd7a2c5779933c7d3ae8">urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0007449716300161-si32.gif">up>MDμubsup> adapted to ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si1.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=b7164e6f30fba35666d53c1c6e4447de" title="Click to view the MathML source">D$\mathcal{D}$ and μ  , that is, we prove the strong type ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si7.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=b06e965900af6bf3d4c47fcc2ce6019e" title="Click to view the MathML source">(p,p)$(p,p)$ inequality for ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si49.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=d070c1bb580f88c889d1a85e451271de" title="Click to view the MathML source">α/n<p<∞$\alpha /n<p<\infty$, and the weak type ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0007449716300161&_mathId=si10.gif&_user=111111111&_pii=S0007449716300161&_rdoc=1&_issn=00074497&md5=248147d6eacd09b9552ac2ee858d4f36" title="Click to view the MathML source">(α/n,α/n)$(\alpha /n,\alpha /n)$ inequality where the integrals are taken in the Choquet sense.