Subellipticity of some complex vector fields related to the Witten Laplacian

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July & August 2021, 20(7&8): 2709-2724. doi: 10.3934/cpaa.2021047

Subellipticity of some complex vector fields related to the Witten Laplacian

Wei Xi Li ^1, and Chao Jiang Xu ^2,3,,

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 Published July & August 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) $.

Keywords: Subelliptic estimate, maximal estimate, complex system, Witten Laplacian, condition $ (\Psi) $.

Mathematics Subject Classification: Primary: 35H20; Secondary: 35H10.

Citation: Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2709-2724. 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.
[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, SÃ©minaire quations aux dÃ©rivÃ©es 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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