{"title":"两变量平方和的最优Gevrey正则性","authors":"A. Bove, M. Mughetti","doi":"10.2422/2036-2145.202205_011","DOIUrl":null,"url":null,"abstract":"For $ q $, $ a $ integers such that $ a \\geq 1 $, $ 1<q $, $ (x, y) \\in U $, $ U $ a neighborhood of the origin in $ \\mathbb{R}^{2} $, we consider the operator $$ D_{x}^{2} + x^{2(q-1)} D_{y}^{2} + y^{2a} D_{y}^{2} . $$ Slightly modifying the method of proof of \\cite{monom} we can see that it is Gevrey $ s_{0} $ hypoelliptic, where $ s_{0}^{-1} = 1 - a^{-1} (q - 1) q^{-1} $. Here we show that this value is optimal, i.e. that there are solutions to $ P u = f $ with $ f $ more regular than $ G^{s_{0}} $ that are not better than Gevrey $ s_{0} $. The above operator reduces to the M\\'etivier operator (\\cite{metivier81}) when $ a = 1 $, $ q = 2 $. We give a description of the characteristic manifold of the operator and of its relation with the Treves conjecture on the real analytic regularity for sums of squares.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"63 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Optimal Gevrey regularity for certain sums of squares in two variables\",\"authors\":\"A. Bove, M. Mughetti\",\"doi\":\"10.2422/2036-2145.202205_011\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For $ q $, $ a $ integers such that $ a \\\\geq 1 $, $ 1<q $, $ (x, y) \\\\in U $, $ U $ a neighborhood of the origin in $ \\\\mathbb{R}^{2} $, we consider the operator $$ D_{x}^{2} + x^{2(q-1)} D_{y}^{2} + y^{2a} D_{y}^{2} . $$ Slightly modifying the method of proof of \\\\cite{monom} we can see that it is Gevrey $ s_{0} $ hypoelliptic, where $ s_{0}^{-1} = 1 - a^{-1} (q - 1) q^{-1} $. Here we show that this value is optimal, i.e. that there are solutions to $ P u = f $ with $ f $ more regular than $ G^{s_{0}} $ that are not better than Gevrey $ s_{0} $. The above operator reduces to the M\\\\'etivier operator (\\\\cite{metivier81}) when $ a = 1 $, $ q = 2 $. We give a description of the characteristic manifold of the operator and of its relation with the Treves conjecture on the real analytic regularity for sums of squares.\",\"PeriodicalId\":8132,\"journal\":{\"name\":\"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE\",\"volume\":\"63 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2422/2036-2145.202205_011\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2422/2036-2145.202205_011","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
摘要
对于$ q $, $ a $这样的整数,$ a \geq 1 $, $ 1本文章由计算机程序翻译,如有差异,请以英文原文为准。