Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes
Identifieur interne : 001109 ( Istex/Corpus ); précédent : 001108; suivant : 001110Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes
Auteurs : A. A. Albanese ; D. Jornet ; A. OliaroSource :
- Mathematische Nachrichten [ 0025-584X ] ; 2012-03.
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Abstract
We prove the following inclusion \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ WF_* (u)\subset WF_*(Pu)\cup \Sigma , \quad u\in \mathcal {E}^\prime _\ast (\Omega ), $$ \end{document} where WF* denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.
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DOI: 10.1002/mana.201010039
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Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</mods:affiliation></affiliation></author><author><name sortKey="Jornet, D" sort="Jornet, D" uniqKey="Jornet D" first="D." last="Jornet">D. Jornet</name><affiliation><mods:affiliation>Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain</mods:affiliation></affiliation><affiliation><mods:affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</mods:affiliation></affiliation></author><author><name sortKey="Oliaro, A" sort="Oliaro, A" uniqKey="Oliaro A" first="A." last="Oliaro">A. 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Albanese</name><affiliation><mods:affiliation>Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Per Arnesano, P. O. Box 193, I‐73100 Lecce, Italy</mods:affiliation></affiliation><affiliation><mods:affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</mods:affiliation></affiliation></author><author><name sortKey="Jornet, D" sort="Jornet, D" uniqKey="Jornet D" first="D." last="Jornet">D. Jornet</name><affiliation><mods:affiliation>Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain</mods:affiliation></affiliation><affiliation><mods:affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</mods:affiliation></affiliation></author><author><name sortKey="Oliaro, A" sort="Oliaro, A" uniqKey="Oliaro A" first="A." last="Oliaro">A. Oliaro</name><affiliation><mods:affiliation>Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I‐10123 Torino, Italy</mods:affiliation></affiliation><affiliation><mods:affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Mathematische Nachrichten</title><title level="j" type="abbrev">Math. Nachr.</title><idno type="ISSN">0025-584X</idno><idno type="eISSN">1522-2616</idno><imprint><publisher>WILEY‐VCH Verlag</publisher><pubPlace>Germany</pubPlace><date type="published" when="2012-03">2012-03</date><biblScope unit="volume">285</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="411">411</biblScope><biblScope unit="page" to="425">425</biblScope></imprint><idno type="ISSN">0025-584X</idno></series><idno type="istex">31A1B00698B95DB79E29436E8E2611B7B2F46E70</idno><idno type="DOI">10.1002/mana.201010039</idno><idno type="ArticleID">MANA201010039</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0025-584X</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>35A18</term><term>35A21</term><term>MSC (2010) Primary: 46F05</term><term>Non‐quasianalytic weight function</term><term>linear partial differential operators</term><term>propagation of singularities</term><term>pseudodifferential operators</term><term>wave front set</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We prove the following inclusion \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ WF_* (u)\subset WF_*(Pu)\cup \Sigma , \quad u\in \mathcal {E}^\prime _\ast (\Omega ), $$ \end{document} where WF* denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.</div></front></TEI><istex><corpusName>wiley</corpusName><author><json:item><name>A. A. Albanese</name><affiliations><json:string>Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Per Arnesano, P. O. Box 193, I‐73100 Lecce, Italy</json:string><json:string>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</json:string></affiliations></json:item><json:item><name>D. Jornet</name><affiliations><json:string>Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain</json:string><json:string>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</json:string></affiliations></json:item><json:item><name>A. Oliaro</name><affiliations><json:string>Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I‐10123 Torino, Italy</json:string><json:string>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>Non‐quasianalytic weight function</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>pseudodifferential operators</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>linear partial differential operators</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>wave front set</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>propagation of singularities</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>MSC (2010) Primary: 46F05</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>35A18</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>35A21</value></json:item></subject><articleId><json:string>MANA201010039</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>article</json:string></originalGenre><abstract>We prove the following inclusion \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ WF_* (u)\subset WF_*(Pu)\cup \Sigma , \quad u\in \mathcal {E}^\prime _\ast (\Omega ), $$ \end{document} where WF* denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.</abstract><qualityIndicators><score>5.972</score><pdfVersion>1.3</pdfVersion><pdfPageSize>594 x 792 pts</pdfPageSize><refBibsNative>true</refBibsNative><abstractCharCount>709</abstractCharCount><pdfWordCount>8985</pdfWordCount><pdfCharCount>37115</pdfCharCount><pdfPageCount>15</pdfPageCount><abstractWordCount>81</abstractWordCount></qualityIndicators><title>Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title><refBibs><json:item><author><json:item><name>A. A. Albanese</name></json:item><json:item><name>D. Jornet</name></json:item><json:item><name>A. Oliaro</name></json:item></author><host><volume>66</volume><pages><last>181</last><first>153</first></pages><issue>2</issue><author></author><title>Integral Equations Oper. 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Padova</title></host><title>Iterati di operatori e regolarità Gevrey microlocale anisotropa</title></json:item></refBibs><genre><json:string>article</json:string></genre><host><volume>285</volume><publisherId><json:string>MANA</json:string></publisherId><pages><total>15</total><last>425</last><first>411</first></pages><issn><json:string>0025-584X</json:string></issn><issue>4</issue><subject><json:item><value>Research Article</value></json:item></subject><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1522-2616</json:string></eissn><title>Mathematische Nachrichten</title><doi><json:string>10.1002/(ISSN)1522-2616</json:string></doi></host><categories><wos><json:string>science</json:string><json:string>mathematics</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>mathematics & statistics</json:string><json:string>general mathematics</json:string></scienceMetrix></categories><publicationDate>2012</publicationDate><copyrightDate>2012</copyrightDate><doi><json:string>10.1002/mana.201010039</json:string></doi><id>31A1B00698B95DB79E29436E8E2611B7B2F46E70</id><score>0.022082126</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/31A1B00698B95DB79E29436E8E2611B7B2F46E70/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/31A1B00698B95DB79E29436E8E2611B7B2F46E70/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/31A1B00698B95DB79E29436E8E2611B7B2F46E70/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>WILEY‐VCH Verlag</publisher><pubPlace>Germany</pubPlace><availability><p>Copyright © 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim</p></availability><date>2012</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title><author xml:id="author-1"><persName><forename type="first">A. A.</forename><surname>Albanese</surname></persName><affiliation>Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Per Arnesano, P. O. Box 193, I‐73100 Lecce, Italy</affiliation><affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</affiliation></author><author xml:id="author-2"><persName><forename type="first">D.</forename><surname>Jornet</surname></persName><affiliation>Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain</affiliation><affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</affiliation></author><author xml:id="author-3"><persName><forename type="first">A.</forename><surname>Oliaro</surname></persName><affiliation>Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I‐10123 Torino, Italy</affiliation><affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</affiliation></author></analytic><monogr><title level="j">Mathematische Nachrichten</title><title level="j" type="abbrev">Math. Nachr.</title><idno type="pISSN">0025-584X</idno><idno type="eISSN">1522-2616</idno><idno type="DOI">10.1002/(ISSN)1522-2616</idno><imprint><publisher>WILEY‐VCH Verlag</publisher><pubPlace>Germany</pubPlace><date type="published" when="2012-03"></date><biblScope unit="volume">285</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="411">411</biblScope><biblScope unit="page" to="425">425</biblScope></imprint></monogr><idno type="istex">31A1B00698B95DB79E29436E8E2611B7B2F46E70</idno><idno type="DOI">10.1002/mana.201010039</idno><idno type="ArticleID">MANA201010039</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2012</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>We prove the following inclusion \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ WF_* (u)\subset WF_*(Pu)\cup \Sigma , \quad u\in \mathcal {E}^\prime _\ast (\Omega ), $$ \end{document} where WF* denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.</p></abstract><textClass xml:lang="en"><keywords scheme="keyword"><list><head>keywords</head><item><term>Non‐quasianalytic weight function</term></item><item><term>pseudodifferential operators</term></item><item><term>linear partial differential operators</term></item><item><term>wave front set</term></item><item><term>propagation of singularities</term></item><item><term>MSC (2010) Primary: 46F05</term></item><item><term>35A18</term></item><item><term>35A21</term></item></list></keywords></textClass><textClass><keywords scheme="Journal Subject"><list><head>article-category</head><item><term>Research Article</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2010-02-11">Received</change><change when="2010-05-04">Registration</change><change when="2012-03">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/31A1B00698B95DB79E29436E8E2611B7B2F46E70/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Wiley, elements deleted: body"><istex:xmlDeclaration>version="1.0" encoding="UTF-8" standalone="yes"</istex:xmlDeclaration><istex:document><component version="2.0" type="serialArticle" xml:lang="en"><header><publicationMeta level="product"><publisherInfo><publisherName>WILEY‐VCH Verlag</publisherName><publisherLoc>Germany</publisherLoc></publisherInfo><doi registered="yes">10.1002/(ISSN)1522-2616</doi><issn type="print">0025-584X</issn><issn type="electronic">1522-2616</issn><idGroup><id type="product" value="MANA"></id></idGroup><titleGroup><title type="main" xml:lang="de" sort="MATHEMATISCHE NACHRICHTEN">Mathematische Nachrichten</title><title type="short">Math. Nachr.</title></titleGroup></publicationMeta><publicationMeta level="part" position="40"><doi origin="wiley" registered="yes">10.1002/mana.v285.4</doi><numberingGroup><numbering type="journalVolume" number="285">285</numbering><numbering type="journalIssue">4</numbering></numberingGroup><coverDate startDate="2012-03">March 2012</coverDate></publicationMeta><publicationMeta level="unit" type="article" position="5" status="forIssue"><doi origin="wiley" registered="yes">10.1002/mana.201010039</doi><idGroup><id type="unit" value="MANA201010039"></id></idGroup><countGroup><count type="pageTotal" number="15"></count></countGroup><titleGroup><title type="articleCategory">Research Article</title><title type="tocHeading1">Research Articles</title></titleGroup><copyright ownership="publisher">Copyright © 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim</copyright><eventGroup><event type="manuscriptReceived" date="2010-02-11"></event><event type="manuscriptRevised" date="2010-04-09"></event><event type="manuscriptAccepted" date="2010-05-04"></event><event type="xmlConverted" agent="Converter:JWSART34_TO_WML3G version:3.1.3 standalone mode:FullText mathml2tex" date="2012-02-29"></event><event type="publishedOnlineEarlyUnpaginated" date="2011-11-03"></event><event type="publishedOnlineFinalForm" date="2012-02-21"></event><event type="firstOnline" date="2011-11-03"></event><event type="xmlConverted" agent="Converter:WILEY_ML3G_TO_WILEY_ML3GV2 version:3.8.8" date="2014-02-02"></event><event type="xmlConverted" agent="Converter:WML3G_To_WML3G version:4.3.4 mode:FullText" date="2015-02-24"></event></eventGroup><numberingGroup><numbering type="pageFirst">411</numbering><numbering type="pageLast">425</numbering></numberingGroup><correspondenceTo><lineatedText><line>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410</line><line>D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494</line><line>A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</line></lineatedText></correspondenceTo><linkGroup><link type="toTypesetVersion" href="file:MANA.MANA201010039.pdf"></link></linkGroup></publicationMeta><contentMeta><countGroup><count type="figureTotal" number="0"></count><count type="tableTotal" number="0"></count><count type="referenceTotal" number="23"></count></countGroup><titleGroup><title type="main" xml:lang="en">Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title><title type="short" xml:lang="en">Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title></titleGroup><creators><creator xml:id="au1" creatorRole="author" affiliationRef="#af1" corresponding="yes"><personName><givenNames>A. A.</givenNames><familyName>Albanese</familyName></personName><contactDetails><email>angela.albanese@unisalento.it</email></contactDetails></creator><creator xml:id="au2" creatorRole="author" affiliationRef="#af2" corresponding="yes"><personName><givenNames>D.</givenNames><familyName>Jornet</familyName></personName><contactDetails><email>djornet@mat.upv.es</email></contactDetails></creator><creator xml:id="au3" creatorRole="author" affiliationRef="#af3" corresponding="yes"><personName><givenNames>A.</givenNames><familyName>Oliaro</familyName></personName><contactDetails><email>alessandro.oliaro@unito.it</email></contactDetails></creator></creators><affiliationGroup><affiliation xml:id="af1" countryCode="IT" type="organization"><unparsedAffiliation>Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Per Arnesano, P. O. Box 193, I‐73100 Lecce, Italy</unparsedAffiliation></affiliation><affiliation xml:id="af2" countryCode="ES" type="organization"><unparsedAffiliation>Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain</unparsedAffiliation></affiliation><affiliation xml:id="af3" countryCode="IT" type="organization"><unparsedAffiliation>Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I‐10123 Torino, Italy</unparsedAffiliation></affiliation></affiliationGroup><keywordGroup xml:lang="en" type="author"><keyword xml:id="kwd1">Non‐quasianalytic weight function</keyword><keyword xml:id="kwd2">pseudodifferential operators</keyword><keyword xml:id="kwd3">linear partial differential operators</keyword><keyword xml:id="kwd4">wave front set</keyword><keyword xml:id="kwd5">propagation of singularities</keyword><keyword xml:id="kwd6"><b>MSC (2010)</b> Primary: 46F05</keyword><keyword xml:id="kwd7">35A18</keyword><keyword xml:id="kwd8">35A21</keyword></keywordGroup><abstractGroup><abstract type="main" xml:lang="en"><title type="main">Abstract</title><p>We prove the following inclusion <displayedItem xml:id="di-ueqn-1" type="mathematics" numbered="no"><mediaResourceGroup><mediaResource alt="equation image" href="urn:x-wiley:0025584X:media:MANA201010039:tex2gif-ueqn-1"></mediaResource><mediaResource alt="TeX code" mimeType="application/x-latex"><data>\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ WF_* (u)\subset WF_*(Pu)\cup \Sigma , \quad u\in \mathcal {E}^\prime _\ast (\Omega ), $$ \end{document}</data></mediaResource></mediaResourceGroup></displayedItem>where <i>WF</i><sub>*</sub> denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of <span type="latex">\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}</span>, <i>P</i> is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of <i>P</i>. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.</p></abstract></abstractGroup></contentMeta></header></component></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title></titleInfo><titleInfo type="abbreviated" lang="en"><title>Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes</title></titleInfo><name type="personal"><namePart type="given">A. A.</namePart><namePart type="family">Albanese</namePart><affiliation>Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Per Arnesano, P. O. Box 193, I‐73100 Lecce, Italy</affiliation><affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">D.</namePart><namePart type="family">Jornet</namePart><affiliation>Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain</affiliation><affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Oliaro</namePart><affiliation>Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I‐10123 Torino, Italy</affiliation><affiliation>A. A. Albanese, Phone: +39 832 297426, Fax: +39 832 297410D. Jornet, Instituto Universitario de Matemática Pura y Aplicada IUMPA‐UPV, Universidad Politécnica de Valencia, C/Camino de Vera, s/n, E‐46022 Valencia, Spain, Phone: +34 963 877 000, Fax: +34 963 879 494A. Oliaro, Phone: +39 011 6702912, Fax: +39 011 6702878.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="article" displayLabel="article"></genre><originInfo><publisher>WILEY‐VCH Verlag</publisher><place><placeTerm type="text">Germany</placeTerm></place><dateIssued encoding="w3cdtf">2012-03</dateIssued><dateCaptured encoding="w3cdtf">2010-02-11</dateCaptured><dateValid encoding="w3cdtf">2010-05-04</dateValid><copyrightDate encoding="w3cdtf">2012</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType><extent unit="references">23</extent></physicalDescription><abstract lang="en">We prove the following inclusion \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ WF_* (u)\subset WF_*(Pu)\cup \Sigma , \quad u\in \mathcal {E}^\prime _\ast (\Omega ), $$ \end{document} where WF* denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.</abstract><subject lang="en"><genre>keywords</genre><topic>Non‐quasianalytic weight function</topic><topic>pseudodifferential operators</topic><topic>linear partial differential operators</topic><topic>wave front set</topic><topic>propagation of singularities</topic><topic>MSC (2010) Primary: 46F05</topic><topic>35A18</topic><topic>35A21</topic></subject><relatedItem type="host"><titleInfo><title>Mathematische Nachrichten</title></titleInfo><titleInfo type="abbreviated"><title>Math. Nachr.</title></titleInfo><genre type="journal">journal</genre><subject><genre>article-category</genre><topic>Research Article</topic></subject><identifier type="ISSN">0025-584X</identifier><identifier type="eISSN">1522-2616</identifier><identifier type="DOI">10.1002/(ISSN)1522-2616</identifier><identifier type="PublisherID">MANA</identifier><part><date>2012</date><detail type="volume"><caption>vol.</caption><number>285</number></detail><detail type="issue"><caption>no.</caption><number>4</number></detail><extent unit="pages"><start>411</start><end>425</end><total>15</total></extent></part></relatedItem><identifier type="istex">31A1B00698B95DB79E29436E8E2611B7B2F46E70</identifier><identifier type="DOI">10.1002/mana.201010039</identifier><identifier type="ArticleID">MANA201010039</identifier><accessCondition type="use and reproduction" contentType="copyright">Copyright © 2012 WILEY‐VCH Verlag GmbH & Co. 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