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

  • Timur Akhunov
  • Lyudmila Korobenko
  • Cristian RiosEmail author


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.


Hypoellipticity Infinite vanishing Loss of derivatives Logarithmic estimates 

Mathematics Subject Classification

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



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).


  1. 1.
    Beals, R., Fefferman, C.: On hypoellipticity of second order operators. Commun. Partial Differ. Equ. 1(1), 73–85 (1976)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Christ, M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Partial Differ. Equ. 16(10), 1695–1707 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Christ, M.: Hypoellipticity in the infinitely degenerate regime. In: Complex Analysis and Geometry (Columbus, OH, 1999), Volume 9 of Ohio State University Mathematics Research Institute Publication, pp. 59–84. de Gruyter, Berlin (2001)Google Scholar
  4. 4.
    Fediĭ, V.S.: A certain criterion for hypoellipticity. Mat. Sb. (N.S.) 85(127), 18–48 (1971)MathSciNetGoogle Scholar
  5. 5.
    Fefferman, C., Phong, D.H.: On positivity of pseudo-differential operators. Proc. Natl. Acad. Sci. USA 75(10), 4673–4674 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hörmander, L.: On the theory of general partial differential operators. Acta Math. 94, 161–248 (1955)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hörmander, L.: Hypoelliptic differential operators. Ann. Inst. Fourier Grenoble 11, 477–492 (1961)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators: III. Pseudodifferential Operators. Volume 274 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Springer, Berlin (1985)zbMATHGoogle Scholar
  10. 10.
    Kannai, Y.: An unsolvable hypoelliptic differential operator. Isr. J. Math. 9, 306–315 (1971)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kohn, J.J.: Hypoellipticity and loss of derivatives. Ann. Math. (2) 162(2), 943–986 (2005). (With an appendix by Makhlouf Derridj and David S. Tartakoff) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kohn, J.J.: Hypoellipticity of some degenerate subelliptic operators. J. Funct. Anal. 159(1), 203–216 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Korobenko, L., Rios, C.: Hypoellipticity of certain infinitely degenerate second order operators. J. Math. Anal. Appl. 409(1), 41–55 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kumano-go, H.: Pseudodifferential Operators. MIT Press, Cambridge (1981). (Translated from the Japanese by Rémi Vaillancourt and Michihiro Nagase) zbMATHGoogle Scholar
  15. 15.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus: II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1), 1–76 (1985)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Maz’ja, V.G., Kufner, A.: Variations on the theme of the inequality \((f^{\prime })^2\le 2f{\rm sup}|f^{\prime \prime }|\). Manuscr. Math. 56(1), 89–104 (1986)CrossRefGoogle Scholar
  17. 17.
    Morimoto, Y.: On the hypoellipticity for infinitely degenerate semi-elliptic operators. J. Math. Soc. Jpn. 30(2), 327–358 (1978)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Morimoto, Y.: Nonhypoellipticity for degenerate elliptic operators. Publ. Res. Inst. Math. Sci. 22(1), 25–30 (1986)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Morimoto, Y.: A criterion for hypoellipticity of second order differential operators. Osaka J. Math. 24(3), 651–675 (1987)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Oleĭnik, O.A., Radkevič, E.V.: Second Order Equations with Nonnegative Characteristic Form. Plenum Press, New York (1973). (Translated from the Russian by Paul C. Fife) CrossRefGoogle Scholar
  21. 21.
    Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4), 247–320 (1976)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schwartz, L.: Théorie des distributions. Tome I. Actualités Sci. Ind., no. 1091 = Publ. Inst. Math. Univ. Strasbourg 9. Hermann & Cie., Paris (1950)Google Scholar
  23. 23.
    Schwartz, L.: Some applications of the theory of distributions. In: Lectures on Modern Mathematics, Vol. I, pp. 23–58. Wiley, New York (1963)Google Scholar
  24. 24.
    Taylor, M.E.: Gelfand theory of pseudo differential operators and hypoelliptic operators. Trans. Am. Math. Soc. 153, 495–510 (1971)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Taylor, M.E.: Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Volume 81 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2000)Google Scholar

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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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