{"title":"On a microlocal version of Young’s product theorem","authors":"Claudio Dappiaggi, Paolo Rinaldi, Federico Sclavi","doi":"10.1007/s00229-023-01510-6","DOIUrl":"https://doi.org/10.1007/s00229-023-01510-6","url":null,"abstract":"Abstract A key result in distribution theory is Young’s product theorem which states that the product between two Hölder distributions $$uin mathcal {C}^alpha (mathbb {R}^d)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$vin mathcal {C}^beta (mathbb {R}^d)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>β</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> can be unambiguously defined if $$alpha +beta >0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We revisit the problem of multiplying two Hölder distributions from the viewpoint of microlocal analysis, using techniques proper of Sobolev wavefront set. This allows us to establish sufficient conditions which allow the multiplication of two Hölder distributions even when $$alpha +beta le 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> .","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135924789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weighted isoperimetric problem for spacelike hypersurface in de Sitter space","authors":"Kuicheng Ma","doi":"10.1007/s00229-023-01500-8","DOIUrl":"https://doi.org/10.1007/s00229-023-01500-8","url":null,"abstract":"","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46306443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem","authors":"Prashanta Garain","doi":"10.1007/s00229-023-01504-4","DOIUrl":"https://doi.org/10.1007/s00229-023-01504-4","url":null,"abstract":"","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45467396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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