Abstract
We consider a second order differential operator \(P\) in \({\mathbb {R}}^3\) and we study the effect of the complex lower order terms on its \(C^\infty \)-hypoellipticity.
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Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Kluwer Academic Publishers, New York (1991)
Boutet de Monvel, L.: Hypoelliptic operators with double characteristics and related pseudodifferential operators. Commun. Pure Appl. Math. 27, 585–639 (1974)
Boutet de Monvel, L., Grigis, A., Helffer, B.: Paramétrixes d’opérateurs pseudo-différentiels à caractéristiques multiples. Astérisque 34–35, 93–121 (1976)
Bove, A., Derridj, M., Kohn, J., Tartakoff, D.: Sums of squares of complex vector fields and (analytic-) hypoellipticity. Math. Res. Lett. 13(5–6), 683–701 (2006)
Bove, A., Mughetti, M., Tartakoff, D.S.: Gevrey Hypoellipticity for an Interesting Variant of Kohn’s Operator, Complex Analysis, Trends in Mathematics. Springer, Basel (2010)
Bove, A., Mughetti, M.: Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable. J. Differ. Equ. 255(4), 728–758 (2013). MR 3056589
Bove, A., Mughetti, M., Tartakoff, D.S.: Hypoellipticity and non hypoellipticity for sums of squares of complex vector fields. Anal. PDE 6(2), 371–445 (2013). doi:10.2140/apde.2013.6.371
Egorov, YuB: Subelliptic operators. Uspekhi Mat. Nauk. Russ. Math. Surv 30(2), 59–118 (1975)
Gilioli, A., Treves, F.: An example in the local solvability theory of linear PDE’s. Am. J. Math. 24, 366–384 (1974)
Grigis, A., Rothschild, L.: A criterion for analytic hypoellipticity of a class of differential operators with polynomial coefficients. Ann. Math. 2 118(3), 443–460 (1983)
Grushin, V.V.: On a class of elliptic pseudodifferential operators degenerate on a submanifold. Mat. Sbornik 84(126), 155–185 (1971)
Helffer, B.: Sur l’hypoellipticité des opérateurs pseudodifférentiels à caractéristiques multiples (perte de 3/2 dérivées). Bull. Soc. Math. Fr. Mém. 51, 13–61 (1977)
Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques. Astérisque 112 (1984)
Hörmander, L.: Subelliptic operators. Seminar on Singularities of Solutions of Linear Partial Differential Equations, vol. 91, pp. 127–208. Princeton University Press, Princeton (1979)
Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. III–IV. Springer-Verlag, Berlin (1983)
Kato, T.: Perturbation Theory of Linear Operators, vol. 132. Springer-Verlag, Berlin (1966)
Kogoj, A.E., Lanconelli, E.: On semilinear \(\Delta _\lambda \)-Laplace equation. Nonlinear Anal. 75(12), 4637–4649 (2012)
Kohn, J.J.: Hypoellipticity and loss of derivatives. With an appendix by Makhlouf Derridj and David S. Tartakoff. Ann. Math. 2 16(2), 943–986 (2005)
Mascarello, M., Rodino, L.: A class of pseudodifferential operators with multiple non-involutive characteristics. Ann. Scuola Norm. Sup. Pisa Ser. IV 8, 575–603 (1981)
Mascarello, M., Rodino, L.: Partial Differential Equations with Multiple Characteristics. Akademie Verlag, Berlin (1997)
Menikoff, A.: Some examples of hypoelliptic partial differential equations. Math. Ann. 221, 167–181 (1976)
Mughetti, M., Parenti, C., Parmeggiani, A.: Lower bound estimates without transversal ellipticity. Commun. Partial Differ. Equ. 32(7–9), 1399–1438 (2007)
Mughetti, M., Nicola, F.: Hypoellipticity for a class of operators with multiple characteristics. J. Anal. Math. 103, 377–396 (2007)
Mughetti, M., Nicola, F.: On the positive parts of second order symmetric pseudodifferential operators. Integr. Eqn. Oper. Theory 64, 553–572 (2009)
Mughetti, M.: A problem of transversal anisotropic ellipticity. Rend. Sem. Mat. Univ. Padova 106, 111–142 (2001)
Mughetti, M.: On the spectrum of an anharmonic oscillator. Trans. Am. Math. Soc. (2014, to appear)
Mughetti, M.: Hypoellipticity and Higher Order Levi Conditions. http://arxiv.org/abs/1401.3124 (2013, Submitted)
Parenti, C., Parmeggiani, A.: On the hypoellipticity with a big loss of derivatives. Kyushu J. Math. 59, 155–230 (2005)
Popivanov, P.R.: Hypoellipticity, solvability and construction of solutions with prescribed singularities for several classes of pde having symplectic characteristics. Rend. Sem. Mat. Univ. Pol. Torino 66(4), 321–337 (2008)
Reed, M., Simon, B.: Methods of modern mathematical physics, vol. IV. Academic Press Inc., London (1978)
Shubin, M.: Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin (1987)
Sjöstrand, J.: Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat. 12, 85–130 (1974)
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Mughetti, M. Regularity properties of a double characteristics differential operator with complex lower order terms. J. Pseudo-Differ. Oper. Appl. 5, 343–358 (2014). https://doi.org/10.1007/s11868-014-0093-5
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DOI: https://doi.org/10.1007/s11868-014-0093-5