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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 蕭欽玉(Chin-Yu Hsiao) | |
| dc.contributor.author | Yang-Zhi Lin | en |
| dc.contributor.author | 林揚智 | zh_TW |
| dc.contributor.author | f02221010 | |
| dc.date.accessioned | 2022-11-23T09:11:27Z | - |
| dc.date.available | 2021-11-08 | |
| dc.date.available | 2022-11-23T09:11:27Z | - |
| dc.date.copyright | 2021-11-08 | |
| dc.date.issued | 2021 | |
| dc.date.submitted | 2021-10-06 | |
| dc.identifier.citation | Part I [1] L. Boutet de Monvel and J. Sjöstrand. Sur la singularité des noyaux de Bergman et de Szegő. In Journées: Équations aux Dérivées Partielles de Rennes (1975), pages 123–164. Astérisque, No. 34–35. 1976. [2] SC. Chen and MC. Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics. 19. Providence, RI: American Mathematical Society (AMS). Somerville, MA: International Press, xii, 380 p., (2001), MR1800297, Zbl 0963.32001. [3] JeanPierre Demailly. Sur l’identité de BochnerKodairaNakanoen géométrie hermitienne, Lecture Notes in Math., vol. 1198, pp. 88–97, Springer Verlag, 1985. [4] JeanPierre Demailly. Complex analytic and differential geometry, 2012, published online at https://wwwfourier.ujfgrenoble.fr/demailly/manuscripts/agbook.pdf. [5] G. B. Folland, J. J. Kohn, The Neumann problem for the Cauchy Riemann complex, Annals of Mathematics Studies, No. 75. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. viii+146 pp,MR0461588, Zbl 0247.35093 [6] Alain Grigis and Johannes Sjöstrand. Microlocal analysis for differential operators. An introduction, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. [7] Chin Yu Hsiao. Projections in several complex variables, Mém. Soc. Math. Fr. (N.S.), (123):131, 2010. [8] Chin Yu Hsiao and George Marinescu, Asymptotics of spectral function of lower energy forms and Bergman kernel of semipositive and big line bundles, Comm. Anal. Geom. 22 (2014), no. 1, 1–108. [9] Chin Yu Hsiao, George Marinescu, On the singularities of the Szegő projections on lower energy forms, J. Differential Geometry 107 (2017) 83155. [10] Chin Yu Hsiao, George Marinescu and Huan Wang, Szegő kernel asymptotics on some noncompact complete CR manifolds, arXiv:2012.11457. [11] Bernard Helffer and Johannes Sjöstarnd, Equation de Schrödinger avec champ magnétique et equation de Harper. Schrödinger Operators, H. Holden and A. Jensen (Eds.), Sonderborg, 1988; Lecture Notes in Phys., Vol. 345, Springer Verlag, Berlin, 1989, 118–197. [12] CY. Hsiao and G. Marinescu, Asymptotics of spectral function of lower energy forms and Bergman kernel of semipositive and big line bundles, Commun. Anal. Geom. 22 (2014), No. 1, 1–108, MR3194375, Zbl 1316.32013. [13] Hendrik Herrmann, Chin Yu Hsiao, Xiaoshan Li. An explicit formula for Szegő kernels on the Heisenberg group. Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 8, 1225–1247. [14] Hendrik Herrmann, Chin Yu Hsiao, Xiaoshan Li. Szegő kernels and equivariant embedding theorems for CR manifolds. arXiv:1710.04910, to appear in Mathematical Research letters . [15] Chin Yu Hsiao, George Marinescu and Xiaoshan Li. Equivariant Kodaira embedding for CR manifolds with circle action. Michigan Math. J., doi:10.1307/mmj/1587628815. [16] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry. Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley Interscience Publication. John Wiley Sons, Inc., New York, 1996. xii+329 pp. [17] J. J. Kohn, Boundaries of complex manifolds, Proc. Conf. Complex Analysis (Minneapolis, 1964), pp. 81–94, Springer, Berlin, 1965, MR0175149, Zbl 0166.36003 [18] J. J. Kohn, The range of the tangential Cauchy Riemannoperator, Duke Math. J. 53 (1986), No. 2, 307–562, MR0850548, Zbl 0609.32015. [19] Xiaonan Ma and George Marinescu. Holomorphic Morse inequalities and Bergman kernels, volume 254 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2007. [20] Xiaonan Ma and George Marinescu. Generalized Bergman kernels on symplectic manifolds. Adv. Math., 217(4):1756–1815, 2008. [21] Xiaonan Ma and George Marinescu. Exponential estimate for the asymptotics of Bergman kernels Math. Ann. 362 (2015), no. 34, 13271347 [22] A. Melin and J. Sjöstrand, Fourier integral operators with complexvalued phase functions, Springer Lecture Notes in Math., 459 (1975), 120–223, MR0431289, Zbl 0306.42007. [23] Grigis, A. Sjöstrand, J. (1994), ”Microlocal analysis for differential operators” Cambridge Vol. 196, pp. iv+151. Cambridge University Press. Part II [1] L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Astérisque, 34–35 (1976), 123–164. [2] G. B. Folland, J. J. Kohn, The Neumann problem for the CauchyRiemann complex, Annals of Mathematics Studies, No. 75. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. viii+146 pp. [3] K. Fritsch, H. Herrmann and C.Y. Hsiao, Gequivariant embedding theorems for CR manifolds of high codimension, to appear in Michigan Mathematical Journal, available at arXiv:1810.09629. [4] H. Herrmann, C.Y. Hsiao and X. Li, An Explicit Formula for Szegő Kernels on the Heisenberg Group, Acta Mathematica Sinica, English Series, (2018) 1225–1247. [5] H. Herrmann and X. Li, Morse inequalities and embeddings for CR manifolds with circle action, Bull. Inst. Math. Acad. Sin. (N.S.) 15 (2020), no. 2, 93–122. [6] C.Y. Hsiao, Projections in several complex variables, Mém. Soc. Math. France, Nouv. Sér. 123 (2010), 131 p. [7] CY. Hsiao and G. Marinescu, Asymptotics of spectral function of lower energy forms and Bergman kernel of semipositive and big line bundles, Commun. Anal. Geom. 22 (2014), No. 1, 1–108. [8] C.Y. Hsiao, The second coefficient of the asymptotic expansion of the weighted Bergman kernel for (0, q) forms on Cn, Bull. Inst. Math. Acad. Sin. (N.S.) 11 (2016), no. 3, 521–570. [9] C.Y. Hsiao and G. Marinescu, On the singularities of the Szegő projections on lower energy forms, J. Differential Geom. 107 (2017), no. 1, 83–155. [10] CY. Hsiao and G. Marinescu, Szegő kernel asymptotics and Kodaira embedding theorems of Leviflat CR manifolds, Math. Res. Lett. 24 (2017), no. 5, 1385–1451. [11] C.Y. Hsiao and G. Marinescu, BerezinToeplitz quantization for lower energy forms, Comm. Partial Differential Equations 42 (2017), no. 6, 895–942. [12] C.Y. Hsiao, Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds, Mem. Amer. Math. Soc. 254 (2018), no. 1217, v+142 [13] C.Y. Hsiao and N. Savale, Bergman Szegő kernel asymptotics in weakly pseudoconvex finite type cases, preprint, available at arXiv:2009.07159. [14] J. J. Kohn, Boundaries of complex manifolds, Proc. Conf. Complex Analysis (Minneapolis,1964), pp. 81–94, Springer, Berlin, 1965. [15] X. Ma and G. Marinescu, The first coefficients of the asymptotic expansion of the Bergman kernel of the spinc Dirac operator, Internat. J. Math. 17 (2006), no. 6, 737–759. [16] X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkhäuser, Basel, 2007, 422 pp. [17] X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, Adv. Math. 217 (2008), no. 4, 1756–1815. [18] S. Zelditch, Szegő kernels and a theorem of Tian, Int. Math. Res. Not. 6 (1998), 317–331. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/79790 | - |
| dc.description.abstract | 本論文包含兩個部分，在第一部分中，我們引入了廣義海森堡群的概念，並賦予它一個柯西黎曼結構，並且在我們的均勻有界假設之下，柯恩拉普拉斯算子有部分閉域性質。更進一步的，該結果能幫助我們算出塞格核的漸進展開。在第二部分中，我們引入了高餘為度海森堡群，並且給出了在二次式柯西黎曼結構下的塞格核精確表達式。 | zh_TW |
| dc.description.provenance | Made available in DSpace on 2022-11-23T09:11:27Z (GMT). No. of bitstreams: 1 U0001-1008202110243200.pdf: 1078498 bytes, checksum: fdb90d913aaae26cb2df25b8504e8709 (MD5) Previous issue date: 2021 | en |
| dc.description.tableofcontents | "Contents 摘要 iii Abstract v Contents vii I Szegő kernels and Partial Closed Range properties for Kohn Laplacians on some class of on generalized Heisenberg Groups 1 Chapter 1 Introduction 3 Chapter 2 Preliminaries 13 2.1 Standard notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3 CR manifold and Generalized Heisenberg group 19 3.1 Generalized Heisenberg group . . . . . . . . . . . . . . . . . . . . . 19 3.2 Positivilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Hermitian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 4 The tangential CauchyRiemann operator 25 4.1 CauchyRiemann complex . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 The Kohn Laplacain . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Representation of ∂bu and ∂∗bu . . . . . . . . . . . . . . . . . . . . . 29 Chapter 5 Main Results 37 5.1 Local Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Global Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3 The Partial Inverse of b . . . . . . . . . . . . . . . . . . . . . . . 48 References 55 II Explicit Formula For Szegö Kernel 59 Chapter 6 Introduction 61 Chapter 7 Preliminaries 67 7.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.2 CauchyRiemann Manifold . . . . . . . . . . . . . . . . . . . . . . 69 7.3 CauchyRiemann complex . . . . . . . . . . . . . . . . . . . . . . . 73 7.4 The Kohn Laplacain for functions . . . . . . . . . . . . . . . . . . . 76 Chapter 8 Partial Fourier Transformation 79 8.1 Definition and Basic Propositions . . . . . . . . . . . . . . . . . . . 79 8.2 The Space with Weight . . . . . . . . . . . . . . . . . . . . . . . . . 81 Chapter 9 Weighted Holomorphic space and Bergman Kernel 83 Chapter 10 The Proof of Main Theorem 91 Chapter 11 A review of the construction of Herrmann, Hsiao and Li 97 11.1 Complex Fourier integral operators . . . . . . . . . . . . . . . . . . 103 11.2 Proof of Theorem 11.0.2 . . . . . . . . . . . . . . . . . . . . . . . . 107 11.3 Proof of Theorem 11.0.3 . . . . . . . . . . . . . . . . . . . . . . . . 112 References 117 Appendix A — 121 A.1 The Sobolev space . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.3 Further Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Appendix B — Introduction 125 B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.2 Further Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 125" | |
| dc.language.iso | en | |
| dc.subject | 廣義海森堡群 | zh_TW |
| dc.subject | 柯西─黎曼流型 | zh_TW |
| dc.subject | 閉域性質 | zh_TW |
| dc.subject | 次橢圓估計 | zh_TW |
| dc.subject | 柯恩─拉普拉斯算子 | zh_TW |
| dc.subject | 塞格核 | zh_TW |
| dc.subject | Szegő kernel | en |
| dc.subject | generalized Heisenberg group | en |
| dc.subject | CR-manifold | en |
| dc.subject | closed range properties | en |
| dc.subject | subelliptic estimate | en |
| dc.subject | Kohn Laplacian operator | en |
| dc.title | 柯恩─拉普拉斯算子在廣義海森堡群上的部分閉域性質 | zh_TW |
| dc.title | The Partial Closed Range Properties of Kohn Laplacian on Generalize Heisenberg Group | en |
| dc.date.schoolyear | 109-2 | |
| dc.description.degree | 博士 | |
| dc.contributor.oralexamcommittee | 邱鴻麟(Hsin-Tsai Liu),黃榮宗(Chih-Yang Tseng),鄭日新,蔡宜洵 | |
| dc.subject.keyword | 柯西─黎曼流型,閉域性質,次橢圓估計,柯恩─拉普拉斯算子,塞格核,廣義海森堡群, | zh_TW |
| dc.subject.keyword | CR-manifold,closed range properties,subelliptic estimate,Kohn Laplacian operator,,Szegő kernel,generalized Heisenberg group, | en |
| dc.relation.page | 125 | |
| dc.identifier.doi | 10.6342/NTU202102236 | |
| dc.rights.note | 同意授權(全球公開) | |
| dc.date.accepted | 2021-10-07 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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