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Hypoellipticity of Fediĭ’s type operators under Morimoto’s logarithmic condition

  • Timur Akhunov
  • Lyudmila Korobenko
  • Cristian RiosEmail author
Article
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Abstract

We prove hypoellipticity of second order linear operators on \(\mathbb {R}^{n+m}\) of the form \(L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)\), where \(L_j\), \(j=1,2\), satisfy Morimoto’s super-logarithmic estimates \(||\log \!\left<\xi \right>^2 \hat{u}(\xi )||^2 \le \varepsilon (L_j u,u) + C_{\varepsilon ,K} ||u||^2\), and g is smooth, nonnegative, and vanishes only at the origin in \(\mathbb {R}^n\) to any arbitrary order. We also show examples in which our hypotheses are necessary for hypoellipticity.

Keywords

Hypoellipticity Infinite vanishing Loss of derivatives Logarithmic estimates 

Mathematics Subject Classification

35H10 35S05 35A18 35G05 35B65 35H20 35A22 

Notes

Funding

Funding was provided by Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN/04872-2017).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesBinghampton UniversityBinghamtonUSA
  2. 2.Mathematics DepartmentReed CollegePortlandUSA
  3. 3.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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