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doi: 10.3934/cpaa.2021047

## 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

* Corresponding author

In honor of the 80th birthday of Professor Shuxing CHEN

Received  November 2020 Revised  January 2021 Early access  March 2021

Fund Project: The research of the first author was supported by NSFC (No.11961160716, 11871054, 11771342) and the Fundamental Research Funds for the Central Universities(No.2042020kf0210). The second author is supported by the NSFC (No.12031006) and the Fundamental Research Funds for the Central Universities of China

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 $(\Psi)$.

Citation: Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021047
##### References:
 [1] M. Derridj, Sur une classe d'opérateurs différentiels hypoelliptiques à coefficients analytiques, SÃ©minaire quations aux dÃ©rivÃ©es partielles, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] B. Helffer and J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, MA, 1985.  Google Scholar [9] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar [10] L. Hörmander, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985.  Google Scholar [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.  Google Scholar [12] J. J. Kohn, Lectures on degenerate elliptic problems, In Pseudodifferential Operator with Applications (Bressanone, 1977), Liguori, Naples, (1978), 89-151.  Google Scholar [13] J. J. Kohn, Hypoellipticity and loss of derivatives, Ann. Math., 162 (2005), 943-986.  doi: 10.4007/annals.2005.162.943.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] H. M. Maire, Hypoelliptic overdetermined systems of partial differential equations, Commun. Partial Differ. Equ., 5 (1980), 331-380.  doi: 10.1080/0360530800882142.  Google Scholar [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.  Google Scholar [18] J. Nourrigat, Subelliptic systems, Commun. Partial Differ. Equ., 15 (1990), 341-405.  doi: 10.1080/03605309908820689.  Google Scholar [19] J. Nourrigat, Systèmes sous-elliptiques. II, Invent. Math., 104 (1991), 377-400.  doi: 10.1007/BF01245081.  Google Scholar [20] L. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.  doi: 10.1007/BF02392419.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] M. Derridj, Sur une classe d'opérateurs différentiels hypoelliptiques à coefficients analytiques, SÃ©minaire quations aux dÃ©rivÃ©es partielles, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] B. Helffer and J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, MA, 1985.  Google Scholar [9] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar [10] L. Hörmander, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985.  Google Scholar [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.  Google Scholar [12] J. J. Kohn, Lectures on degenerate elliptic problems, In Pseudodifferential Operator with Applications (Bressanone, 1977), Liguori, Naples, (1978), 89-151.  Google Scholar [13] J. J. Kohn, Hypoellipticity and loss of derivatives, Ann. Math., 162 (2005), 943-986.  doi: 10.4007/annals.2005.162.943.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] H. M. Maire, Hypoelliptic overdetermined systems of partial differential equations, Commun. Partial Differ. Equ., 5 (1980), 331-380.  doi: 10.1080/0360530800882142.  Google Scholar [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.  Google Scholar [18] J. Nourrigat, Subelliptic systems, Commun. Partial Differ. Equ., 15 (1990), 341-405.  doi: 10.1080/03605309908820689.  Google Scholar [19] J. Nourrigat, Systèmes sous-elliptiques. II, Invent. Math., 104 (1991), 377-400.  doi: 10.1007/BF01245081.  Google Scholar [20] L. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.  doi: 10.1007/BF02392419.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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