Clifford分析在偏微分方程中的应用
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摘要
本文研究Clifford分析在偏微分方程中的应用。这包括以下四个方面的内容。
(1)针对高维空间中任意阶的Cauchy-Riemann型非线性偏微分方程,建立了局部可解以及整体可解的一般性理论。
(2)在多Clifford变量的分析理论中,建立了非齐次Cauchy-Riemann方程紧支集解的存在性定理。
(3)在关于κ-Cauchy-Fueter算子的分析理论中,建立了非齐次Cauchy-Fueter方程紧支集解的存在性定理。
(4)建立了八元数Hermitian Clifford分析理论,特别研究了Dirichlet边值问题。
本文分为五章,具体内容如下：
第一章是引言部分,给出了Clifford分析在偏微分方程中的应用的研究背景,以及本文的研究方法和主要结论。
第二章研究高维空间中高阶非线性偏微分方程理论,将Nijenhuis-Woolf(Ann. Math.1963)关于一阶非线性偏微分方程可解性理论利用Clifford分析的方法推广到一般情形。这一理论的建立强烈依赖于Teodorescu算子在Holder空间中的有界性。Teodorescu算子是Dirac算子的右逆算子,它是一个奇异积分算子,对于该奇异积分算子的研究,我们引入了行之有效的工具—斜球坐标方法。
第三章研究多Clifford变量的分析理论,它是多复变函数论在非交换领域的推广。对于多Clifford分析中非齐次Cauchy-Riemann方程,我们给出了紧支集解的具体的积分表达式,证明了多Clifford分析理论中存在Hartogs现象,建立了相应的Bochner-Martinelli积分公式,此积分公式统一了单复变、多复变、多四元数理论的相应结果。
第四章研究κ-Cauchy-Fueter算子的分析理论,这是多四元数变量的分析理论。对于非齐次κ-Cauchy-Fueter方程解的研究,古典的方法是代数几何的方法,在低维数时才能给出解的具体的表达式,我们采用的是多复变的方法,其优越性是给出解的具体的构造。我们引入新的技巧,通过提高空间的维数,达到简化κ-Cauchy-Fueter算子的目的。这一技巧使得我们能建立相应的Bochner-Martinelli积分公式。
第五章研究八元数Hermitian Clifford分析。我们构造出了八元数的Witt基,引入了八元数Hermitian Dirac算子,建立了相应的积分理论和边值理论。
The dissertation focuses on the application of Clifford analysis in PDE. This in-cludes the following four aspects.
(1) We establish a general existence theorem for the Cauchy-Riemann type non-linear partial differential systems.
(2) We establish the existence theorem of the solution of the non-homogeneous Cauchy-Riemann equations in the analysis of several Clifford variables.
(3) We establish the existence theorem of the solution of the non-homogeneous K-Cauchy-Futer equations in the analysis of several quternionic variables.
(4) We establish the Octonionic Hermitian Clifford analysis, in particular we deal with the Dirichlet boundary value problem.
This paper is divided into five chapters.
The first chapter is the introduction, given the background of Clifford analysis and its application in partial differential equations. We also outline the main approaches and main conclusions.
The second chapter studies the theory of nonlinear partial differential equations of higher orders in higher dimensional spaces. This is an extension of the result of Nijenhuis-Woolf (Ann. Math.1963) on the theory of first-order nonlinear partial d-ifferential equations based on an approach of Clifford analysis. The theory depends heavily on the boundedness of the Teodorescu operator in Holder spaces. As a right inverse operator of Dirac operator, the Teodorescu operator is a singular integral oper-ator. For the study of this singular integral operator, we introduce an effective tool—approach of oblique spherical coordinates.
The third chapter analyzes the theoretical study of multi-Clifford variables, which is the theory of functions of several complex variables in the promotion of non-commutative realm. For nonhomogenous Cauchy-Riemann equations in multi-Clifford analysis, we give a concrete expression of the solution with compact support. This also yields the Hartogs phenomenon in multi-Clifford analysis. We establish the corresponding Bochner-Martinelli integral formula, which unifies the corresponding result in the the-ory of a single complex variable, several complex variables, and several quaternions.
The fourth chapter studies the theory of κ-Cauchy-Fueter operator, which belongs to the theory of several quaternionic variables. For the study of non-homogeneous κ-Cauchy-Fueter equations, the classical method is based on algebraic geometry, which can produce a specific expression of solution at low dimensions. Our approach comes from the theory of several complex variables, which takes the advantages of explic-it expression of solutions in any dimension. We introduce a new technique through increasing the dimensions of spaces in consideration to simplify the κ-Cauchy-Fueter operator in purpose. This technique allows us to establish the corresponding Bochner-Martinelli integral formula.
In the final chapter, we study the Octonionic Hermitian Clifford analysis. We introduce the Witt basis of Octonions, construct the Octonionic Hermitian Dirac oper-ator, and then establish the corresponding integral theory as well as the boundary value problem.
(1)针对高维空间中任意阶的Cauchy-Riemann型非线性偏微分方程,建立了局部可解以及整体可解的一般性理论。
(2)在多Clifford变量的分析理论中,建立了非齐次Cauchy-Riemann方程紧支集解的存在性定理。
(3)在关于κ-Cauchy-Fueter算子的分析理论中,建立了非齐次Cauchy-Fueter方程紧支集解的存在性定理。
(4)建立了八元数Hermitian Clifford分析理论,特别研究了Dirichlet边值问题。
本文分为五章,具体内容如下：
第一章是引言部分,给出了Clifford分析在偏微分方程中的应用的研究背景,以及本文的研究方法和主要结论。
第二章研究高维空间中高阶非线性偏微分方程理论,将Nijenhuis-Woolf(Ann. Math.1963)关于一阶非线性偏微分方程可解性理论利用Clifford分析的方法推广到一般情形。这一理论的建立强烈依赖于Teodorescu算子在Holder空间中的有界性。Teodorescu算子是Dirac算子的右逆算子,它是一个奇异积分算子,对于该奇异积分算子的研究,我们引入了行之有效的工具—斜球坐标方法。
第三章研究多Clifford变量的分析理论,它是多复变函数论在非交换领域的推广。对于多Clifford分析中非齐次Cauchy-Riemann方程,我们给出了紧支集解的具体的积分表达式,证明了多Clifford分析理论中存在Hartogs现象,建立了相应的Bochner-Martinelli积分公式,此积分公式统一了单复变、多复变、多四元数理论的相应结果。
第四章研究κ-Cauchy-Fueter算子的分析理论,这是多四元数变量的分析理论。对于非齐次κ-Cauchy-Fueter方程解的研究,古典的方法是代数几何的方法,在低维数时才能给出解的具体的表达式,我们采用的是多复变的方法,其优越性是给出解的具体的构造。我们引入新的技巧,通过提高空间的维数,达到简化κ-Cauchy-Fueter算子的目的。这一技巧使得我们能建立相应的Bochner-Martinelli积分公式。
第五章研究八元数Hermitian Clifford分析。我们构造出了八元数的Witt基,引入了八元数Hermitian Dirac算子,建立了相应的积分理论和边值理论。
The dissertation focuses on the application of Clifford analysis in PDE. This in-cludes the following four aspects.
(1) We establish a general existence theorem for the Cauchy-Riemann type non-linear partial differential systems.
(2) We establish the existence theorem of the solution of the non-homogeneous Cauchy-Riemann equations in the analysis of several Clifford variables.
(3) We establish the existence theorem of the solution of the non-homogeneous K-Cauchy-Futer equations in the analysis of several quternionic variables.
(4) We establish the Octonionic Hermitian Clifford analysis, in particular we deal with the Dirichlet boundary value problem.
This paper is divided into five chapters.
The first chapter is the introduction, given the background of Clifford analysis and its application in partial differential equations. We also outline the main approaches and main conclusions.
The second chapter studies the theory of nonlinear partial differential equations of higher orders in higher dimensional spaces. This is an extension of the result of Nijenhuis-Woolf (Ann. Math.1963) on the theory of first-order nonlinear partial d-ifferential equations based on an approach of Clifford analysis. The theory depends heavily on the boundedness of the Teodorescu operator in Holder spaces. As a right inverse operator of Dirac operator, the Teodorescu operator is a singular integral oper-ator. For the study of this singular integral operator, we introduce an effective tool—approach of oblique spherical coordinates.
The third chapter analyzes the theoretical study of multi-Clifford variables, which is the theory of functions of several complex variables in the promotion of non-commutative realm. For nonhomogenous Cauchy-Riemann equations in multi-Clifford analysis, we give a concrete expression of the solution with compact support. This also yields the Hartogs phenomenon in multi-Clifford analysis. We establish the corresponding Bochner-Martinelli integral formula, which unifies the corresponding result in the the-ory of a single complex variable, several complex variables, and several quaternions.
The fourth chapter studies the theory of κ-Cauchy-Fueter operator, which belongs to the theory of several quaternionic variables. For the study of non-homogeneous κ-Cauchy-Fueter equations, the classical method is based on algebraic geometry, which can produce a specific expression of solution at low dimensions. Our approach comes from the theory of several complex variables, which takes the advantages of explic-it expression of solutions in any dimension. We introduce a new technique through increasing the dimensions of spaces in consideration to simplify the κ-Cauchy-Fueter operator in purpose. This technique allows us to establish the corresponding Bochner-Martinelli integral formula.
In the final chapter, we study the Octonionic Hermitian Clifford analysis. We introduce the Witt basis of Octonions, construct the Octonionic Hermitian Dirac oper-ator, and then establish the corresponding integral theory as well as the boundary value problem.
引文
[1]R. Abreu Blaya, J. Bory Reyes, F. Brackx, H. De Schepper, F. Sommen, Cauchy Integral Formulae in Hermitian Quaternionic Clifford Analysis, Complex Anal. Oper. Theory,6 (2012),971-985.
[2]R. Abreu Blaya, J. Bory Reyes, F. Brackx, H. De Schepper, F. Sommen, Boundary value problems for the quaternionic Hermitian system in R4n analysis, Bound. Value Probl.,2012, doi:10.1186/1687-2770-2012-74.
[3]W. W. Adams, C. A. Berenstein, P. Loustaunau, I. Sabadini, D. C. Struppa, Reg-ular functions of several quaternionic variables and the Cauchy-Fueter complex, J. Geom. Anal.,9(1)(1999),1-15.
[4]W. W. Adams, P. Loustaunau, V. P. Palamodov, D. C. Struppa, Hartogs'phe-nomenon for polyregular functions and projective dimension of related modulaes over a polynomial ring, Ann. Inst. Fourier,47(1997),623-640.
[5]L. Ahlfors, Complex analysis, An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York,1978.
[6]J. C. Baez, The Octonions, Bull. Amer. Math. Soc.,39 (2002),145-205.
[7]J. C. Baez, J. Huerta, The Strangest Numbers in String Theory, Sci. Am.,304 (5)(2011),60-65.
[8]R. Beals, C. Fefferman, On local solvability of linear partial differential equation-s, Ann. Math.,97 (1973),482-498.
[9]S. Bernstein, S. Ebert, R. Soren KranBhar, On the diffusion equation and diffusion wavelets on flat cylinders and the n-torus, Math. Meth. Appl. Sci.,34(2011),428-441.
[10]S. Bernstein, P. Cerejeiras, U. Kahler, Lax pairs for 3D spatial problems based on the Dirac operator, Complex Var. Elliptic Equ.,2013, doi:10.1080/17476933. 2013.804516.
[11]G. Bossard, Octonionic black holes, arXiv:1203.0530.
[12]F. Brackx, J. Bures, H. De Schepper, D. Eelbode, F. Sommen, V. Soucek, Funda-ments of Hermitean Clifford analysis Part Ⅰ:Complex structure, Complex Anal. Oper. Theory,1(3)(2007),341-365.
[13]F. Brackx, J. Bures, H. De Schepper, D. Eelbode, F. Sommen, V. Soucek, Funda-ments of Hermitean Clifford analysis Part Ⅱ:Splitting of h-monogenic equations, Complex Var. Elliptic Equ.,52(10-11)2007,1063-1079.
[14]F. Brackx, B. De Knock, H. De Schepper, F. Sommen, On Cauchy and Martinelli-Bochner integral formulae in Hermitian Clifford analysis, Bull. Braz. Math. Soc., 40(3)(2009),395-416.
[15]F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Research Notes in Math-ematics,76. Pitman, Boston, MA,1982.
[16]F. Brackx, H. De Schepper, F. Sommen, The Hermitian Clifford analysis toolbox, Adv. Appl. Clifford Algebr.,18(3-4)(2008),451-487.
[17]J. Bures, A. Damiano, I. Sabadini, Explicit resolutions for the complex of several Fueter operators, J. Geom. Phys.,57(3)(2007),765-775.
[18]J. Bures, V. Soucek, Complexes of invariant operators in several quaternionic variables, Complex Var. Elliptic Equ.,51(5-6) (2006),463-485.
[19]F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The Mathematics of Minkowski Space-Time, Birkhauser Verlag, Basel,2008.
[20]P. Cerejeiras, U. Kahler, F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains, Math. Methods Appl. Sci., 28(14)(2005),1715-1724.
[21]陈琳,Hermitean Clifford分析,中国科学技术大学博士学位论文,2013.
[22]S. S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc.,6 (1955),771-782.
[23]W. K. Clifford, Applications of Grassmann's extensive algebra, Amer. Jour. Math.,1 (1878),350-358.
[24]F. Colombo, I. Sabadini, F. Sommen, D. C. Struppa, Analysis of Dirac System-s and Computational Algebra, Progress in Mathematical Physics,39, Boston, Birkhauser,2004.
[25]F. Colombo, V. Soucek, D. C. Struppa, Invariant resolutions for several Fueter operators, J. Geom. Phys.,56(7)(2006),1175-1191.
[26]A. Damiano, D. Eelbode, I. Sabadini, Algebraic analysis of Hermitian monogenic functions, C.R. Acad. Sci. Paris Ser.I,346 (2008),139-142.
[27]A. Damiano, D. Eelbode, I. Sabadini, Invariant syzygies for the Hermitian Dirac operator, Math. Z.,4(2009),929-945.
[28]A. Damiano, D. Eelbode, I. Sabadini, Quaternionic Hermitian spinor systems and compatibility conditions, Adv. Geom.,11 (2011),169-189.
[29]C. A. Deavours, The quaternion calculus, Amer. Math. Monthly 80 (1973),995-1008.
[30]N. Decker, The resolution of the Nirenberg-Treves conjecture, Ann. Math.,163 (2006),405-444.
[31]R. Delanghe, F. Sommen, V. Soucek, Clifford Algebra and Spinor-Valued Func-tions, Kluwer, Dordrecht,1992.
[32]C. Doran, A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, Cambridge,2003.
[33]J. Y. Du, N. Xu, Z. X. Zhang, Boundary behavior of Cauchy-type integrals in Clifford analysis, Acta Math. Sci. Ser. B Engl. Ed.,29(1)(2009),210-224.
[34]M. Eastwood, R. Penrose, R. Wells, Cohomology and massless fields, Comm. Math. Phys.,78 (3) (1980),305-351.
[35]D. Eelbode, Arbitrary complex powers of the Dirac operator on the hyperbolic unit ball, Ann. Acad. Sci. Fenn. Ser. Al-Math.,29 (2004),367-381.
[36]D. Eelbode, F. Sommen, The inverse Radon transform and the fundamental solu-tion of the hyperbolic Dirac equation, Math. Z.,247(2004),733-745.
[37]R. Fueter, Die Funktionentheorie der Differentialgleichungen (?)u=0 und (?)(?)u=0 mit vier reellen Variablen, Comment. Math. Helv.,7(1)(1934),307-330.
[38]R. Fueter, uber die analytische Darstellung der regularen Funktionen einer Quaternionenvariablen, Comment. Math. Helv.,8(1)(1935),371-378.
[39]D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Or-der, Classics in Mathematics, Springer-Verlag, Berlin,2001.
[40]J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Anal-ysis, Cambridge University Press, Cambridge,1991.
[41]H. Grassmann, Die Lineale Ausdehnungslehre:Ein neuer Zweig der Mathematik. Leipzig, Verlag von Otto Wigand,1844.
[42]K. Giirlebeck, K. Habetha, W. Sprossig, Holomorphic Functions in the Plane and n-Dimensional Space, Birkhauser Verlag, Basel,2007.
[43]K. Giirlebeck, K. Habetha, W. Sprossig, Quaternionic and Clifford Calculus for Physicists and Engineers, Mathematical Methods in Practice, Wiley, Chichester, 1998.
[44]K. Giirlebeck, W. Sprossig, Quaternionic analysis and elliptic boundary value problems, International Series of Numerical Mathematics,89, Birkhauser Verlag, Basel,1990.
[45]H. Haefeli, Hyperkomplexe Differentiale, Comment. Math. Helv.,20 (1947), 382-420.
[46]W. R. Hamilton, On quaternions, or on a new system of imaginaries in algebra, Philosophical Magazine,25(3)(1844),489-495.
[47]D. Hestenes, Space-Time Algebra, Gordon Breach,1966.
[48]D. Hestenes, Quantum mechanics from self-interaction, Found. Phys.,15 (1) (1985),63-87.
[49]D. Hestenes, G. Sobczyk, Clifford algebra to geometric calculus, A unified lan-guage for mathematics and physics, Fundamental Theories of Physics, D. Reidel Publishing Co., Dordrecht,1984.
[50]L. Hormander, The Analysis of Linear Partial Differential Operators I, Classics in Mathematics, Springer-Verlag, Berlin,2003.
[51]L. Hormander, An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library,7, North-Holland Publishing Co., Amsterdam, 1990.
[52]S. Huang, Y. Y. Qiao, G. C. Wen, Real and Complex Clifford Analysis, Springer, New York,2005.
[53]V. Ifimie, Functions Hypercomplex, Bull Math de la Soc Sci Math de la R S R, 9(57)(1965),279-332.
[54]康倩倩,Penrose变换κ-Cauchy Fueter算子,浙江大学博士学位论文.
[55]Q. Q. Kang, W. Wang, On Penrose integral formula and series expansion of k-regular functions on the quaternionic space Hn, J. Geom. Phys.,64(2013),192-208.
[56]S. G. Krantz, Function Theory of Several Complex Variables, Providence, RI, AMS Chelsea Publishing,2001.
[57]N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Holder Spaces, Graduate Studies in Mathematics,12, American Mathematical Society, Provi-dence, RI,1996.
[58]A. Kytmanov, The Bochner-Martinelli Integral and its Applications, Birkhauser, Basel-Boston-Berlin,1995.
[59]H. Lawson, Jr. Blaine, Michelsohn Marie-Louise, Spin Geometry, Princeton Mathematical Series 38, Princeton NJ, Princeton University Press,1989.
[60]N. Lerner, Solving pseudo-differential equations, Proceedings of the ICM 2002 in Beijing, vol. Ⅱ, Higher Education Press,2002,711-720.
[61]H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math.,66 (1957),155-158.
[62]H. B. Li, Invariant Algebras and Geometric Reasoning, World Scientific Publish- ing Co. Pte. Ltd., Hackensack, NJ,2008.
[63]X. M. Li, L. Z. Peng, The Cauchy integral formulas on the octonions, Bull. Belg. Math. Soc. Simon Stevin 9(1)(2002),47-64.
[64]L. Liu, G. B. Ren, Normalized system for wave and Dunkl operators, Taiwanese J. Math.,14 (2010),675-683.
[65]X. N. Ma, G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics,254, Basel-Boston-Berlin, Birkhauser,2007.
[66]H. R. Malonek, G. Ren, Almansi-type theorems in Clifford analysis, Clifford analysis in applications, Math. Methods Appl. Sci.,25(16-18)(2002),1541-1552.
[67]E. Marmolejo-Olea, M. Mitrea, Harmonic analysis for general first order differen-tial operators in Lipschitz domains, Clifford Algebras,91-114, Prog. Math. Phys., 34, Boston, Birkhauser,2004.
[68]S. Maxson, Generalized Wavefunctions for Correlated Quantum Oscillators IV: Bosonic and Fermionic Gauge Fields,2008.
[69]A. McIntosh, M. Mitrea, Clifford algebras and Maxwell's equations in Lipschitz domains, Math. Methods Appl. Sci.,22(18)(1999),1599-1620
[70]M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics,1575, Springer-Verlag, Berlin,1994.
[71]T. Mohaupt, Special Geometry, Black Holes and Euclidean Supersymmetry, Handbook of pseudo-Riemannian geometry and supersymmetry,149-181, IRMA Lect. Math. Theor. Phys,16, Eur. Math. Soc., Zurich,2010.
[72]A. Nijenhuis, W. Woolf, Some integration problems in almost-complex manifold-s, Ann. Math.,77 (1963),424-489.
[73]L. Nirenberg, F. Treves, On local solvability of linear partial differential equa-tions. I. Necessary conditions, Comm. Pure Appl. Math.,23 (1970),1-38.
[74]L. Nirenberg, F. Treves, On local solvability of linear partial differential equa-tions. II. Sufficient conditions, Comm. Pure Appl. Math.,23 (1970),459-509.
[75]S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press,1995.
[76]V. P. Palamodov, Hartogs phenomenon for systems of differential equations, J. Geom. Anal., DOI 10.1007/s12220-012-9350-0.
[77]Y. Pan, On solvability of nonlinear partial differential systems of any order in the complex plane, arXiv:1109.3325v2,2011.
[78]D. Pena-Pena, I. Sabadini, F. Sommen, Quaternionic Clifford analysis:the Her-mitian setting, Complex Anal. Oper. Theory,1 (2007),97-13.
[79]D. Pertici, Regular functions of several quaternionic variables,(Italian. English summary), Ann. Mat. Pura Appl.151 (1988),39-65.
[80]G. B. Ren, Harmonic Bergman spaces with small exponents in the unit ball, Col-lect. Math.,53 (2002),83-98.
[81]G. B. Ren, L. Chen, H. Y. Wang, Split quaternion analysis of cyclic Hermitian Dirac operator, submitted.
[82]G. B. Ren, H. Y. Wang, L. Chen, Paracomplex Hermitean Clifford analysis, Com-plex Anal. Oper. Theory., doi:10.1007/s11785-013-0341-3.
[83]G. B. Ren, H. Y. Wang, Oblique spherical coordinates and potential estimates in metaharmonic analysis, preprint.
[84]B. Rosenfeld, Geometry of Lie groups, Mathematics and its Applications,393, Kluwer Academic Publishers Group, Dordrecht,1997.
[85]W. Rudin, Function theory in the unit ball of Cn, Classics in Mathematics, Springer-Verlag, Berlin,2008.
[86]J. Ryan, Complexified Clifford analysis, Complex Var. Theory Appl.,1(1982), 119-149.
[87]I. Sabadini, F. Sommen, Hermitian Clifford analysis and resolutions, Math. Meth-ods Appl. Sci.,25(16-18)(2002),1395-1413.
[88]I. Sabadini, D. C. Struppa, F. Sommen, P. Van Lancker, Complexes of Dirac operators in Clifford algebras, Math. Z.,239(2)(2002),293-320.
[89]史济怀,多复变函数论基础,高等教育出版社,北京,1996.
[90]E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series,32, Princeton University Press, Princeton, N.J., 1971.
[91]A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Math. Soc.,85 (1979), 199-225.
[92]H. Y. Wang, G. B. Ren, Holder continuity Teodorescu operator in Clifford analy-sis, preprint.
[93]H. Y. Wang, G. B. Ren, The boundedness of volume potential for κ-Cauchy Fueter operator, preprint.
[94]H. Y. Wang, G. B. Ren, The boundary behaviour for κ-Cauchy Fueter operator, preprint.
[95]W. Wang, On non-homogeneous Cauchy-Fueter equations and Hartogs'phe-nomenon in several quaternionic variables, J. Geom. Phys.,58(2008),1203-1210.
[96]W. Wang, The κ-Cauchy-Fueter complex, Penrose transformation and Hartogs phenomenmenon for quaternionic κ-regular functions, J. Geom. Phys.,60(2010), 513-530.
[97]王增玺,Gilbarg-Trudinger Holder内估计理论的改进,中国科学技术大学硕士学位论文,2012.
[98]R. Wells, J. Wolf, Complex manifolds and mathematical physics, Bull. Amer. Math. Soc.,1 (2)(1979),296-336.
[99]S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc.,12 (1999),445-495.
[2]R. Abreu Blaya, J. Bory Reyes, F. Brackx, H. De Schepper, F. Sommen, Boundary value problems for the quaternionic Hermitian system in R4n analysis, Bound. Value Probl.,2012, doi:10.1186/1687-2770-2012-74.
[3]W. W. Adams, C. A. Berenstein, P. Loustaunau, I. Sabadini, D. C. Struppa, Reg-ular functions of several quaternionic variables and the Cauchy-Fueter complex, J. Geom. Anal.,9(1)(1999),1-15.
[4]W. W. Adams, P. Loustaunau, V. P. Palamodov, D. C. Struppa, Hartogs'phe-nomenon for polyregular functions and projective dimension of related modulaes over a polynomial ring, Ann. Inst. Fourier,47(1997),623-640.
[5]L. Ahlfors, Complex analysis, An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York,1978.
[6]J. C. Baez, The Octonions, Bull. Amer. Math. Soc.,39 (2002),145-205.
[7]J. C. Baez, J. Huerta, The Strangest Numbers in String Theory, Sci. Am.,304 (5)(2011),60-65.
[8]R. Beals, C. Fefferman, On local solvability of linear partial differential equation-s, Ann. Math.,97 (1973),482-498.
[9]S. Bernstein, S. Ebert, R. Soren KranBhar, On the diffusion equation and diffusion wavelets on flat cylinders and the n-torus, Math. Meth. Appl. Sci.,34(2011),428-441.
[10]S. Bernstein, P. Cerejeiras, U. Kahler, Lax pairs for 3D spatial problems based on the Dirac operator, Complex Var. Elliptic Equ.,2013, doi:10.1080/17476933. 2013.804516.
[11]G. Bossard, Octonionic black holes, arXiv:1203.0530.
[12]F. Brackx, J. Bures, H. De Schepper, D. Eelbode, F. Sommen, V. Soucek, Funda-ments of Hermitean Clifford analysis Part Ⅰ:Complex structure, Complex Anal. Oper. Theory,1(3)(2007),341-365.
[13]F. Brackx, J. Bures, H. De Schepper, D. Eelbode, F. Sommen, V. Soucek, Funda-ments of Hermitean Clifford analysis Part Ⅱ:Splitting of h-monogenic equations, Complex Var. Elliptic Equ.,52(10-11)2007,1063-1079.
[14]F. Brackx, B. De Knock, H. De Schepper, F. Sommen, On Cauchy and Martinelli-Bochner integral formulae in Hermitian Clifford analysis, Bull. Braz. Math. Soc., 40(3)(2009),395-416.
[15]F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Research Notes in Math-ematics,76. Pitman, Boston, MA,1982.
[16]F. Brackx, H. De Schepper, F. Sommen, The Hermitian Clifford analysis toolbox, Adv. Appl. Clifford Algebr.,18(3-4)(2008),451-487.
[17]J. Bures, A. Damiano, I. Sabadini, Explicit resolutions for the complex of several Fueter operators, J. Geom. Phys.,57(3)(2007),765-775.
[18]J. Bures, V. Soucek, Complexes of invariant operators in several quaternionic variables, Complex Var. Elliptic Equ.,51(5-6) (2006),463-485.
[19]F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The Mathematics of Minkowski Space-Time, Birkhauser Verlag, Basel,2008.
[20]P. Cerejeiras, U. Kahler, F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains, Math. Methods Appl. Sci., 28(14)(2005),1715-1724.
[21]陈琳,Hermitean Clifford分析,中国科学技术大学博士学位论文,2013.
[22]S. S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc.,6 (1955),771-782.
[23]W. K. Clifford, Applications of Grassmann's extensive algebra, Amer. Jour. Math.,1 (1878),350-358.
[24]F. Colombo, I. Sabadini, F. Sommen, D. C. Struppa, Analysis of Dirac System-s and Computational Algebra, Progress in Mathematical Physics,39, Boston, Birkhauser,2004.
[25]F. Colombo, V. Soucek, D. C. Struppa, Invariant resolutions for several Fueter operators, J. Geom. Phys.,56(7)(2006),1175-1191.
[26]A. Damiano, D. Eelbode, I. Sabadini, Algebraic analysis of Hermitian monogenic functions, C.R. Acad. Sci. Paris Ser.I,346 (2008),139-142.
[27]A. Damiano, D. Eelbode, I. Sabadini, Invariant syzygies for the Hermitian Dirac operator, Math. Z.,4(2009),929-945.
[28]A. Damiano, D. Eelbode, I. Sabadini, Quaternionic Hermitian spinor systems and compatibility conditions, Adv. Geom.,11 (2011),169-189.
[29]C. A. Deavours, The quaternion calculus, Amer. Math. Monthly 80 (1973),995-1008.
[30]N. Decker, The resolution of the Nirenberg-Treves conjecture, Ann. Math.,163 (2006),405-444.
[31]R. Delanghe, F. Sommen, V. Soucek, Clifford Algebra and Spinor-Valued Func-tions, Kluwer, Dordrecht,1992.
[32]C. Doran, A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, Cambridge,2003.
[33]J. Y. Du, N. Xu, Z. X. Zhang, Boundary behavior of Cauchy-type integrals in Clifford analysis, Acta Math. Sci. Ser. B Engl. Ed.,29(1)(2009),210-224.
[34]M. Eastwood, R. Penrose, R. Wells, Cohomology and massless fields, Comm. Math. Phys.,78 (3) (1980),305-351.
[35]D. Eelbode, Arbitrary complex powers of the Dirac operator on the hyperbolic unit ball, Ann. Acad. Sci. Fenn. Ser. Al-Math.,29 (2004),367-381.
[36]D. Eelbode, F. Sommen, The inverse Radon transform and the fundamental solu-tion of the hyperbolic Dirac equation, Math. Z.,247(2004),733-745.
[37]R. Fueter, Die Funktionentheorie der Differentialgleichungen (?)u=0 und (?)(?)u=0 mit vier reellen Variablen, Comment. Math. Helv.,7(1)(1934),307-330.
[38]R. Fueter, uber die analytische Darstellung der regularen Funktionen einer Quaternionenvariablen, Comment. Math. Helv.,8(1)(1935),371-378.
[39]D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Or-der, Classics in Mathematics, Springer-Verlag, Berlin,2001.
[40]J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Anal-ysis, Cambridge University Press, Cambridge,1991.
[41]H. Grassmann, Die Lineale Ausdehnungslehre:Ein neuer Zweig der Mathematik. Leipzig, Verlag von Otto Wigand,1844.
[42]K. Giirlebeck, K. Habetha, W. Sprossig, Holomorphic Functions in the Plane and n-Dimensional Space, Birkhauser Verlag, Basel,2007.
[43]K. Giirlebeck, K. Habetha, W. Sprossig, Quaternionic and Clifford Calculus for Physicists and Engineers, Mathematical Methods in Practice, Wiley, Chichester, 1998.
[44]K. Giirlebeck, W. Sprossig, Quaternionic analysis and elliptic boundary value problems, International Series of Numerical Mathematics,89, Birkhauser Verlag, Basel,1990.
[45]H. Haefeli, Hyperkomplexe Differentiale, Comment. Math. Helv.,20 (1947), 382-420.
[46]W. R. Hamilton, On quaternions, or on a new system of imaginaries in algebra, Philosophical Magazine,25(3)(1844),489-495.
[47]D. Hestenes, Space-Time Algebra, Gordon Breach,1966.
[48]D. Hestenes, Quantum mechanics from self-interaction, Found. Phys.,15 (1) (1985),63-87.
[49]D. Hestenes, G. Sobczyk, Clifford algebra to geometric calculus, A unified lan-guage for mathematics and physics, Fundamental Theories of Physics, D. Reidel Publishing Co., Dordrecht,1984.
[50]L. Hormander, The Analysis of Linear Partial Differential Operators I, Classics in Mathematics, Springer-Verlag, Berlin,2003.
[51]L. Hormander, An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library,7, North-Holland Publishing Co., Amsterdam, 1990.
[52]S. Huang, Y. Y. Qiao, G. C. Wen, Real and Complex Clifford Analysis, Springer, New York,2005.
[53]V. Ifimie, Functions Hypercomplex, Bull Math de la Soc Sci Math de la R S R, 9(57)(1965),279-332.
[54]康倩倩,Penrose变换κ-Cauchy Fueter算子,浙江大学博士学位论文.
[55]Q. Q. Kang, W. Wang, On Penrose integral formula and series expansion of k-regular functions on the quaternionic space Hn, J. Geom. Phys.,64(2013),192-208.
[56]S. G. Krantz, Function Theory of Several Complex Variables, Providence, RI, AMS Chelsea Publishing,2001.
[57]N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Holder Spaces, Graduate Studies in Mathematics,12, American Mathematical Society, Provi-dence, RI,1996.
[58]A. Kytmanov, The Bochner-Martinelli Integral and its Applications, Birkhauser, Basel-Boston-Berlin,1995.
[59]H. Lawson, Jr. Blaine, Michelsohn Marie-Louise, Spin Geometry, Princeton Mathematical Series 38, Princeton NJ, Princeton University Press,1989.
[60]N. Lerner, Solving pseudo-differential equations, Proceedings of the ICM 2002 in Beijing, vol. Ⅱ, Higher Education Press,2002,711-720.
[61]H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math.,66 (1957),155-158.
[62]H. B. Li, Invariant Algebras and Geometric Reasoning, World Scientific Publish- ing Co. Pte. Ltd., Hackensack, NJ,2008.
[63]X. M. Li, L. Z. Peng, The Cauchy integral formulas on the octonions, Bull. Belg. Math. Soc. Simon Stevin 9(1)(2002),47-64.
[64]L. Liu, G. B. Ren, Normalized system for wave and Dunkl operators, Taiwanese J. Math.,14 (2010),675-683.
[65]X. N. Ma, G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics,254, Basel-Boston-Berlin, Birkhauser,2007.
[66]H. R. Malonek, G. Ren, Almansi-type theorems in Clifford analysis, Clifford analysis in applications, Math. Methods Appl. Sci.,25(16-18)(2002),1541-1552.
[67]E. Marmolejo-Olea, M. Mitrea, Harmonic analysis for general first order differen-tial operators in Lipschitz domains, Clifford Algebras,91-114, Prog. Math. Phys., 34, Boston, Birkhauser,2004.
[68]S. Maxson, Generalized Wavefunctions for Correlated Quantum Oscillators IV: Bosonic and Fermionic Gauge Fields,2008.
[69]A. McIntosh, M. Mitrea, Clifford algebras and Maxwell's equations in Lipschitz domains, Math. Methods Appl. Sci.,22(18)(1999),1599-1620
[70]M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics,1575, Springer-Verlag, Berlin,1994.
[71]T. Mohaupt, Special Geometry, Black Holes and Euclidean Supersymmetry, Handbook of pseudo-Riemannian geometry and supersymmetry,149-181, IRMA Lect. Math. Theor. Phys,16, Eur. Math. Soc., Zurich,2010.
[72]A. Nijenhuis, W. Woolf, Some integration problems in almost-complex manifold-s, Ann. Math.,77 (1963),424-489.
[73]L. Nirenberg, F. Treves, On local solvability of linear partial differential equa-tions. I. Necessary conditions, Comm. Pure Appl. Math.,23 (1970),1-38.
[74]L. Nirenberg, F. Treves, On local solvability of linear partial differential equa-tions. II. Sufficient conditions, Comm. Pure Appl. Math.,23 (1970),459-509.
[75]S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press,1995.
[76]V. P. Palamodov, Hartogs phenomenon for systems of differential equations, J. Geom. Anal., DOI 10.1007/s12220-012-9350-0.
[77]Y. Pan, On solvability of nonlinear partial differential systems of any order in the complex plane, arXiv:1109.3325v2,2011.
[78]D. Pena-Pena, I. Sabadini, F. Sommen, Quaternionic Clifford analysis:the Her-mitian setting, Complex Anal. Oper. Theory,1 (2007),97-13.
[79]D. Pertici, Regular functions of several quaternionic variables,(Italian. English summary), Ann. Mat. Pura Appl.151 (1988),39-65.
[80]G. B. Ren, Harmonic Bergman spaces with small exponents in the unit ball, Col-lect. Math.,53 (2002),83-98.
[81]G. B. Ren, L. Chen, H. Y. Wang, Split quaternion analysis of cyclic Hermitian Dirac operator, submitted.
[82]G. B. Ren, H. Y. Wang, L. Chen, Paracomplex Hermitean Clifford analysis, Com-plex Anal. Oper. Theory., doi:10.1007/s11785-013-0341-3.
[83]G. B. Ren, H. Y. Wang, Oblique spherical coordinates and potential estimates in metaharmonic analysis, preprint.
[84]B. Rosenfeld, Geometry of Lie groups, Mathematics and its Applications,393, Kluwer Academic Publishers Group, Dordrecht,1997.
[85]W. Rudin, Function theory in the unit ball of Cn, Classics in Mathematics, Springer-Verlag, Berlin,2008.
[86]J. Ryan, Complexified Clifford analysis, Complex Var. Theory Appl.,1(1982), 119-149.
[87]I. Sabadini, F. Sommen, Hermitian Clifford analysis and resolutions, Math. Meth-ods Appl. Sci.,25(16-18)(2002),1395-1413.
[88]I. Sabadini, D. C. Struppa, F. Sommen, P. Van Lancker, Complexes of Dirac operators in Clifford algebras, Math. Z.,239(2)(2002),293-320.
[89]史济怀,多复变函数论基础,高等教育出版社,北京,1996.
[90]E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series,32, Princeton University Press, Princeton, N.J., 1971.
[91]A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Math. Soc.,85 (1979), 199-225.
[92]H. Y. Wang, G. B. Ren, Holder continuity Teodorescu operator in Clifford analy-sis, preprint.
[93]H. Y. Wang, G. B. Ren, The boundedness of volume potential for κ-Cauchy Fueter operator, preprint.
[94]H. Y. Wang, G. B. Ren, The boundary behaviour for κ-Cauchy Fueter operator, preprint.
[95]W. Wang, On non-homogeneous Cauchy-Fueter equations and Hartogs'phe-nomenon in several quaternionic variables, J. Geom. Phys.,58(2008),1203-1210.
[96]W. Wang, The κ-Cauchy-Fueter complex, Penrose transformation and Hartogs phenomenmenon for quaternionic κ-regular functions, J. Geom. Phys.,60(2010), 513-530.
[97]王增玺,Gilbarg-Trudinger Holder内估计理论的改进,中国科学技术大学硕士学位论文,2012.
[98]R. Wells, J. Wolf, Complex manifolds and mathematical physics, Bull. Amer. Math. Soc.,1 (2)(1979),296-336.
[99]S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc.,12 (1999),445-495.