On a new method of proving Gevrey hypoellipticity for certain sums of squares

详细信息    查看全文

• 作者：Antonio Bove ; ^{bove@bo.infn.it" class="auth_mail" title="E-mail the corresponding author} ; Marco Mughetti ^{marco.mughetti@unibo.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35H10 ; 35B65 ; secondary ; 35A27 ; 35A20
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：30 April 2016
• 年：2016
• 卷：293
• 期：Complete
• 页码：146-220
• 全文大小：925 K

文摘

We consider an operator being a sum of squares of vector fields. It has the form, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000645&_mathId=si1.gif&_user=111111111&_pii=S0001870816000645&_rdoc=1&_issn=00018708&md5=f30b36947b72db6b2eaeaffb1f8de5a7" title="Click to view the MathML source">p,r∈Nclass="mathContainer hidden">class="mathCode">$p,r\in \mathbb{N}$, This type of operator is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000645&_mathId=si3.gif&_user=111111111&_pii=S0001870816000645&_rdoc=1&_issn=00018708&md5=1d6b789af9bddd7fb8fefb8f6fb05976" title="Click to view the MathML source">C^∞class="mathContainer hidden">class="mathCode">$C^{\infty }$ hypoelliptic by Hörmander's theorem, [18]. Its analytic or Gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the Treves conjecture. The Poisson–Treves stratification of P includes both symplectic and non-symplectic strata.

In this paper we show that P   is Gevrey class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000645&_mathId=si4.gif&_user=111111111&_pii=S0001870816000645&_rdoc=1&_issn=00018708&md5=8e9d46440d8eb36254f688e3880626bd" title="Click to view the MathML source">(p+r)/pclass="mathContainer hidden">class="mathCode">$(p+r)/p$ hypoelliptic, by constructing a parametrix whose symbol belongs to some exotic classes. One can also show that this number is optimal.