Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

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• 作者：Paul Hagelsteinp>ap>p> ; p>p>1p>p> ; p>p>paul_hagelstein@baylor.edu" class="auth_mail" title="E-mail the corresponding authorp> ; Ioannis Parissisp>bp>p> ; p>p>2p>p> ; p>p>ioannis.parissis@gmail.com" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：primary ; 42B25 ; secondary ; 42B35
• 刊名：Advances in Mathematics
• 出版年：2015
• 出版时间：5 November 2015
• 年：2015
• 卷：285
• 期：Complete
• 页码：434-453
• 全文大小：410 K

文摘

Let pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si1.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=f24edc2160b865cde5f1128f20187d16" title="Click to view the MathML source">Bpan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> be a homothecy invariant basis consisting of convex sets in pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si2.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=15ac9110cb8fa83c369bdde1ec6c2c24" title="Click to view the MathML source">Rp>np>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>, and define the associated geometric maximal operator pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si3.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=7a6a4354dd1684f05ec678add1b20803" title="Click to view the MathML source">M_Bpan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> by and the halo function pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si5.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=d86cc131824c00d04f10a91c3e0e8a66" title="Click to view the MathML source">ϕ_B(α)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> on pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si6.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=d278edfc6c03aa9ec3271fcc666352a5" title="Click to view the MathML source">(1,∞)pan>pan class="mathContainer hidden">pan class="mathCode">$(1,\infty )$pan>pan>pan> by It is shown that if pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si5.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=d86cc131824c00d04f10a91c3e0e8a66" title="Click to view the MathML source">ϕ_B(α)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> satisfies the Solyanik estimate pan id="mmlsi8" class="mathmlsrc">pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=d8ec302c3a7da8ce0a63b098a4e0a959">pxl.gif" data-inlimgeid="1-s2.0-S0001870815002984-si8.gif">pt>p://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870815002984-si8.gif">pt>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> for pan id="mmlsi9" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si9.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=82b15d3cabcc006f57e32bfdf15fa388" title="Click to view the MathML source">α∈(1,∞)pan>pan class="mathContainer hidden">pan class="mathCode">$\alpha \in (1,\infty )$pan>pan>pan> sufficiently close to 1 then pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si10.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=924f2318364b4543dab48c32efbcb6a0" title="Click to view the MathML source">ϕ_Bpan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> lies in the Hölder class pan id="mmlsi11" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si11.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=97fd67937f214e539b04394afcb1dd13" title="Click to view the MathML source">Cp>pp>(1,∞)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>. As a consequence we obtain that the halo functions associated with the Hardy–Littlewood maximal operator and the strong maximal operator on pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si2.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=15ac9110cb8fa83c369bdde1ec6c2c24" title="Click to view the MathML source">Rp>np>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> lie in the Hölder class pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815002984&_mathId=si12.gif&_user=111111111&_pii=S0001870815002984&_rdoc=1&_issn=00018708&md5=8cd606e20072aaa7446917ea2d564eaf" title="Click to view the MathML source">Cp>1/np>(1,∞)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>.