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Subellipticity of some complex vector fields related to the Witten Laplacian
1. | School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China |
2. | Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China |
3. | Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France |
We consider some system of complex vector fields related to the semi-classical Witten Laplacian, and establish the local subellipticity of this system basing on condition $ (\Psi) $.
References:
[1] |
M. Derridj, Sur une classe d'opérateurs différentiels hypoelliptiques à coefficients analytiques, Séminaire quations aux dérivées partielles, 1971. |
[2] |
M. Derridj, Subelliptic estimates for some systems of complex vector fields, in Hyperbolic Problems and Regularity Questions, Birkhäuser, Basel, (2007), 101-108.
doi: 10.1007/978-3-7643-7451-8_11. |
[3] |
M. Derridj, On some systems of real or complex vector fields and their related Laplacians, in Analysis and Geometry in Several Complex Variables, Amer. Math. Soc., Providence, RI, (2017), 85-124. |
[4] |
M. Derridj and B. Helffer,
Subelliptic estimates for some systems of complex vector fields: quasihomogeneous case, Trans. Amer. Math. Soc., 361 (2009), 2607-2630.
doi: 10.1090/S0002-9947-08-04601-1. |
[5] |
M. Derridj and B. Helffer, On the subellipticity of some hypoelliptic quasihomogeneous systems of complex vector fields, in Complex Analysis, Birkhäuser/Springer Basel AG, Basel, (2010), 109-123.
doi: 10.1007/978-3-0346-0009-5_6. |
[6] |
M. Derridj and B. Helffer, Subellipticity and maximal hypoellipticity for two complex vector fields in $(2+2)$-variables, In Geometric Analysis of Several Complex Variables and Related Topics, volume 550 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2011), 15-56.
doi: 10.1090/conm/550/10865. |
[7] |
B. Helffer and F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Springer-Verlag, Berlin, 2005.
doi: 10.1007/b104762. |
[8] |
B. Helffer and J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, MA, 1985. |
[9] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[10] |
L. Hörmander, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985. |
[11] |
J. L. Journé and J. M. Trépreau, Hypoellipticité sans sous-ellipticité: le cas des systèmes de $n$ champs de vecteurs complexes en $(n+1)$ variables, In Seminaire: Equations aux Dérivées Partielles, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2006. |
[12] |
J. J. Kohn, Lectures on degenerate elliptic problems, In Pseudodifferential Operator with Applications (Bressanone, 1977), Liguori, Naples, (1978), 89-151. |
[13] |
J. J. Kohn,
Hypoellipticity and loss of derivatives, Ann. Math., 162 (2005), 943-986.
doi: 10.4007/annals.2005.162.943. |
[14] |
N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Birkhäuser Verlag, Basel, 2010.
doi: 10.1007/978-3-7643-8510-1. |
[15] |
W. X. Li,
Compactness of the resolvent for the Witten Laplacian, Ann. Henri Poincaré, 19 (2018), 1259-1282.
doi: 10.1007/s00023-018-0659-5. |
[16] |
H. M. Maire,
Hypoelliptic overdetermined systems of partial differential equations, Commun. Partial Differ. Equ., 5 (1980), 331-380.
doi: 10.1080/0360530800882142. |
[17] |
F. Nier, Hypoellipticity for Fokker-Planck operators and Witten Laplacians, in Lectures on The Analysis of Nonlinear Partial Differential Equations, Int. Press, Somerville, MA, (2012), 31-84. |
[18] |
J. Nourrigat,
Subelliptic systems, Commun. Partial Differ. Equ., 15 (1990), 341-405.
doi: 10.1080/03605309908820689. |
[19] |
J. Nourrigat,
Systèmes sous-elliptiques. II, Invent. Math., 104 (1991), 377-400.
doi: 10.1007/BF01245081. |
[20] |
L. Rothschild and E. M. Stein,
Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.
doi: 10.1007/BF02392419. |
[21] |
F. Trèves,
A new method of proof of the subelliptic estimates, Commun. Pure Appl. Math., 24 (1971), 71-115.
doi: 10.1002/cpa.3160240107. |
[22] |
F. Treves,
Study of a model in the theory of complexes of pseudodifferential operators, Ann. Math., 104 (1976), 269-324.
doi: 10.2307/1971048. |
show all references
References:
[1] |
M. Derridj, Sur une classe d'opérateurs différentiels hypoelliptiques à coefficients analytiques, Séminaire quations aux dérivées partielles, 1971. |
[2] |
M. Derridj, Subelliptic estimates for some systems of complex vector fields, in Hyperbolic Problems and Regularity Questions, Birkhäuser, Basel, (2007), 101-108.
doi: 10.1007/978-3-7643-7451-8_11. |
[3] |
M. Derridj, On some systems of real or complex vector fields and their related Laplacians, in Analysis and Geometry in Several Complex Variables, Amer. Math. Soc., Providence, RI, (2017), 85-124. |
[4] |
M. Derridj and B. Helffer,
Subelliptic estimates for some systems of complex vector fields: quasihomogeneous case, Trans. Amer. Math. Soc., 361 (2009), 2607-2630.
doi: 10.1090/S0002-9947-08-04601-1. |
[5] |
M. Derridj and B. Helffer, On the subellipticity of some hypoelliptic quasihomogeneous systems of complex vector fields, in Complex Analysis, Birkhäuser/Springer Basel AG, Basel, (2010), 109-123.
doi: 10.1007/978-3-0346-0009-5_6. |
[6] |
M. Derridj and B. Helffer, Subellipticity and maximal hypoellipticity for two complex vector fields in $(2+2)$-variables, In Geometric Analysis of Several Complex Variables and Related Topics, volume 550 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2011), 15-56.
doi: 10.1090/conm/550/10865. |
[7] |
B. Helffer and F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Springer-Verlag, Berlin, 2005.
doi: 10.1007/b104762. |
[8] |
B. Helffer and J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, MA, 1985. |
[9] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[10] |
L. Hörmander, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985. |
[11] |
J. L. Journé and J. M. Trépreau, Hypoellipticité sans sous-ellipticité: le cas des systèmes de $n$ champs de vecteurs complexes en $(n+1)$ variables, In Seminaire: Equations aux Dérivées Partielles, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2006. |
[12] |
J. J. Kohn, Lectures on degenerate elliptic problems, In Pseudodifferential Operator with Applications (Bressanone, 1977), Liguori, Naples, (1978), 89-151. |
[13] |
J. J. Kohn,
Hypoellipticity and loss of derivatives, Ann. Math., 162 (2005), 943-986.
doi: 10.4007/annals.2005.162.943. |
[14] |
N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Birkhäuser Verlag, Basel, 2010.
doi: 10.1007/978-3-7643-8510-1. |
[15] |
W. X. Li,
Compactness of the resolvent for the Witten Laplacian, Ann. Henri Poincaré, 19 (2018), 1259-1282.
doi: 10.1007/s00023-018-0659-5. |
[16] |
H. M. Maire,
Hypoelliptic overdetermined systems of partial differential equations, Commun. Partial Differ. Equ., 5 (1980), 331-380.
doi: 10.1080/0360530800882142. |
[17] |
F. Nier, Hypoellipticity for Fokker-Planck operators and Witten Laplacians, in Lectures on The Analysis of Nonlinear Partial Differential Equations, Int. Press, Somerville, MA, (2012), 31-84. |
[18] |
J. Nourrigat,
Subelliptic systems, Commun. Partial Differ. Equ., 15 (1990), 341-405.
doi: 10.1080/03605309908820689. |
[19] |
J. Nourrigat,
Systèmes sous-elliptiques. II, Invent. Math., 104 (1991), 377-400.
doi: 10.1007/BF01245081. |
[20] |
L. Rothschild and E. M. Stein,
Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.
doi: 10.1007/BF02392419. |
[21] |
F. Trèves,
A new method of proof of the subelliptic estimates, Commun. Pure Appl. Math., 24 (1971), 71-115.
doi: 10.1002/cpa.3160240107. |
[22] |
F. Treves,
Study of a model in the theory of complexes of pseudodifferential operators, Ann. Math., 104 (1976), 269-324.
doi: 10.2307/1971048. |
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