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On Hypoellipticity of the Operator \( \exp \left[ { - {{\left| {{x_1}} \right|}^{ - \sigma }}} \right]D_1^2 + x_1^4D_2^2 + 1\)

  • Seiichiro Wakabayashi
  • Nobuo Nakazawa
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 52)

Abstract

Let \( L\left( {x,D} \right) = f\sigma \left( {{x_1}} \right)D_1^2 + x_1^4D_2^2 + 1,\) Where \(x = \left( {{x_1},{x_2}} \right) \in {\mathbb{R}^2},\sigma >0\) And \( {f_\sigma }\left( t \right) = \exp \left[ { - {{\left| t \right|}^{ - \sigma }}} \right]\) and \( {f_\sigma }\left( 0 \right) = 0.\) We shall prove that L(x,D) is hypoelliptic at x=(0,0) if and only if \(\sigma< 2.\)

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References

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Seiichiro Wakabayashi
    • 1
  • Nobuo Nakazawa
    • 1
  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan

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