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Title:           Non-collapsing condition and Sobolev embeddings for Hajłasz–Besov spaces
Subject:         Positivity, https://doi.org/10.1007/s11117-025-01115-1
Keywords:        Sobolev inequality; Fractional Hajłasz–Sobolev spaces; Hajłasz–Besov spaces; Metric measure spaces; Rearrangement invariant spaces; Interpolation
Author:          Joaquim Martín 
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      Title: Non-collapsing condition and Sobolev embeddings for Hajłasz–Besov spaces
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       Item: 
        Title: Abstract
        Destination: abscnt.2
       Item: 
        Title: 1 Introduction
        Destination: section.1
       Item: 
        Title: 2 Notation and preliminaries
        Destination: section.2
        Children: 
         Item: 
          Title: 2.1 Metric spaces
          Destination: subsection.2.1
         Item: 
          Title: 2.2 Rearrangements of functions
          Destination: subsection.2.2
         Item: 
          Title: 2.3 Background on rearrangement invariant spaces
          Destination: subsection.2.3
          Children: 
           Item: 
            Title: 2.3.1  α-convex q.r.i. spaces
            Destination: subsubsection.2.3.1
           Item: 
            Title: 2.3.2 Examples
            Destination: subsubsection.2.3.2
         Item: 
          Title: 2.4 Hajłasz–Sobolev and Hajłasz–Besov spaces built on r.i. spaces 
          Destination: subsection.2.4
       Item: 
        Title: 3 Interpolation theorems for Hajłasz–Besov spaces
        Destination: section.3
       Item: 
        Title: 4 The proofs of theorems 4 and 6
        Destination: section.4
       Item: 
        Title: 5 Embedding theorems for Hajłasz–Besov spaces into q.r.i. spaces 
        Destination: section.5
        Children: 
         Item: 
          Title: 5.0.1 Example: homogeneous Lorentz–Zygmund–Hajłasz–Besov spaces
          Destination: myseccnt.1
       Item: 
        Title: References
        Destination: myseccnt.2
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    Title: Non-collapsing condition and Sobolev embeddings for Hajłasz Besov spaces
    Author: Joaquim Martín 
    Subject: Positivity, https://doi.org/10.1007/s11117-025-01115-1
    Keywords: Sobolev inequality; Fractional Hajłasz Sobolev spaces; Hajłasz Besov spaces; Metric measure spaces; Rearrangement invariant spaces; Interpolation
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                           <pdfaProperty:name>CrossmarkDomainExclusive</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>CrossMarkDomains</pdfaProperty:description>
                           <pdfaProperty:name>CrossMarkDomains</pdfaProperty:name>
                           <pdfaProperty:valueType>seq Text</pdfaProperty:valueType>
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                  <pdfaSchema:namespaceURI>http://prismstandard.org/namespaces/basic/2.0/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>prism</pdfaSchema:prefix>
                  <pdfaSchema:schema>Prism Schema</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>This element provides the url for an article or unit of content. The attribute platform is optionally allowed for situations in which multiple URLs must be specified. PRISM recommends that a subset of the PCV platform values, namely “mobile” and “web”, be used in conjunction with this element. NOTE: PRISM recommends against the use of the #other value allowed in the PRISM Platform controlled vocabulary. In lieu of using #other please reach out to the PRISM group at prism-wg@yahoogroups.com to request addition of your term to the Platform Controlled Vocabulary.</pdfaProperty:description>
                           <pdfaProperty:name>url</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The Digital Object Identifier for the article.
The DOI may also be used as the dc:identifier. If used as a dc:identifier, the URI form should be captured, and the bare identifier should also be captured using prism:doi. If an alternate unique identifier is used as the required dc:identifier, then the DOI should be specified as a bare identifier within prism:doi only. If the URL associated with a DOI is to be specified, then prism:url may be used in conjunction with prism:doi in order to provide the service endpoint (i.e. the URL). </pdfaProperty:description>
                           <pdfaProperty:name>doi</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>ISSN for an electronic version of the issue in which the resource occurs. Permits publishers to include a second ISSN, identifying an electronic version of the issue in which the resource occurs (therefore e(lectronic)Issn. If used, prism:eIssn MUST contain the ISSN of the electronic version.</pdfaProperty:description>
                           <pdfaProperty:name>issn</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Volume number</pdfaProperty:description>
                           <pdfaProperty:name>volume</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Issue number</pdfaProperty:description>
                           <pdfaProperty:name>number</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Starting page</pdfaProperty:description>
                           <pdfaProperty:name>startingPage</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Ending page</pdfaProperty:description>
                           <pdfaProperty:name>endingPage</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The aggregation type specifies the unit of aggregation for a content collection. Comment PRISM recommends that the PRISM Aggregation Type Controlled Vocabulary be used to provide values for this element. Note: PRISM recommends against the use of the #other value currently allowed in this controlled vocabulary. In lieu of using #other please reach out to the PRISM group at info@prismstandard.org to request addition of your term to the Aggregation Type Controlled Vocabulary. </pdfaProperty:description>
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                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Title of the magazine, or other publication, in which a resource was/will be published. Typically this will be used to provide the name of the magazine an article appeared in as metadata for the article, along with information such as the article title, the publisher, volume, number, and cover date. Note: Publication name can be used to differentiate between a print magazine and the online version if the names are different such as “magazine” and “magazine.com.”</pdfaProperty:description>
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                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Copyright</pdfaProperty:description>
                           <pdfaProperty:name>copyright</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                  <pdfaSchema:schema>Adobe PDF Schema</pdfaSchema:schema>
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                           <pdfaProperty:name>Trapped</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                  <pdfaSchema:schema>XMP Media Management Schema</pdfaSchema:schema>
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                           <pdfaProperty:description>UUID based identifier for specific incarnation of a document</pdfaProperty:description>
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                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>The common identifier for all versions and renditions of a document.</pdfaProperty:description>
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                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
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                           <pdfaProperty:description>The common identifier for all versions and renditions of a document.</pdfaProperty:description>
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                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
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                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>A reference to the original document from which this one is derived. It is a minimal reference; missing components can be assumed to be unchanged. For example, a new version might only need to specify the instance ID and version number of the previous version, or a rendition might only need to specify the instance ID and rendition class of the original.</pdfaProperty:description>
                           <pdfaProperty:name>DerivedFrom</pdfaProperty:name>
                           <pdfaProperty:valueType>ResourceRef</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
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                           <pdfaProperty:description>Part of PDF/A standard</pdfaProperty:description>
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                           <pdfaProperty:description>Conformance level of PDF/A standard</pdfaProperty:description>
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                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>XMP08 Spec: An ordered array of plate names that are needed to print the document (including any in contained documents).</pdfaProperty:description>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
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SMUR = Submitted Manuscript Under Review
AM = Accepted Manuscript
P = Proof
VoR = Version of Record
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   Type: CRC32
  Checksum: 8a9a53957de4689620ef209a398a0909
   Type: MD5
  Checksum: 2cfa5a6f62f18630bee6aea20d90e3951027a748
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