Derivations and Alberti representations

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• 作者：Andrea Schioppa¹ ; ^{andrea.schioppa@math.ethz.ch" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C23 ; 46J15 ; 58C20
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：30 April 2016
• 年：2016
• 卷：293
• 期：Complete
• 页码：436-528
• 全文大小：1129 K

文摘

We relate generalized Lebesgue decompositions of measures in terms of curve fragments (“Alberti representations”) and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000669&_mathId=si1.gif&_user=111111111&_pii=S0001870816000669&_rdoc=1&_issn=00018708&md5=fb984a4cc161b83515904e79f77fb854" title="Click to view the MathML source">(X,μ)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mo stretchy="false">(mo><mi>Xmi><mo>,mo><mi>μmi><mo stretchy="false">)mo>math>: the local norm of a form m>dfm> “sees” how fast m>fm> grows on curve fragments “seen” by m>μm>. This implies a new characterization of differentiability spaces in terms of the m>μm>-a.e. equality of the local norm of m>dfm> and the local Lipschitz constant of m>fm>. As a consequence, the “Lip–lip” inequality of Keith must be an equality. We also provide dimensional bounds for the module of derivations in terms of the Assouad dimension of m>Xm>.