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Quaternionic and Clifford Analysis in Several Variables

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Abstract

This article discusses how the theory of Fueter regular functions on quaternions can be extended to the case of several variables. This can be done in two different (complementary) ways. One can follow the traditional approach to several complex variables developed in the first part of the twentieth century, and construct suitable generalizations of the Cauchy–Fueter formula to the setting of several variables. In this way one obtains an analog of the Bochner–Martinelli formula for regular functions of several quaternionic variables, and from that starting point one can develop most of the fundamental results of the theory. On the other hand, one can take a more algebraic point of view, in line with the general ideas of Ehrenpreis on solutions to systems of linear constant coefficients partial differential equations, and exploit the fact that regular functions in several variables are infinitely differentiable functions that satisfy a reasonably simple overdetermined system of differential equations. By using this characterization, and the fundamental ideas pioneered by Ehrenpreis and Palamodov, one can construct a sheaf theoretical approach to regular functions of several quaternionic variables that rather immediately allows one to discover important global properties of such functions, and indeed to develop a rigorous theory of hyperfunctions in the quaternionic domain. This article further shows how this process can be adapted to variations of Fueter regularity such as biregularity and Moisil–Theodorescu regularity, as well as to the case of monogenic functions of several vector variables. Finally the article considers the notion of slice monogeneity and slice regularity, and shows how they can also be extended to several variables. The theories in these cases are very recent, and rapidly developing.

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References

  1. Adams, W.W., Berenstein, C.A., Loustaunau, P., Sabadini, I., Struppa, D.C.: Regular functions of several quaternionic variables and the Cauchy-Fueter complex. J. Geom. Anal. 9, 1–15 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society Providence (1994)

    Google Scholar 

  3. Adams, W.W., Loustaunau, P.: Analysis of the module determining the properties of regular functions of several quaternionic variables. Pacific J. 196, 1–15 (2001)

    Article  MathSciNet  Google Scholar 

  4. Adams, W.W., Loustaunau, P., Palamodov, V.P., Struppa, D.C.: Hartogs’ phenomenon for polyregular functions and projective dimension of related modules over a polynomial sing. Ann. Inst. Fourier 47, 623–640 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ahlfors, L.: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill, New York (1979)

    Google Scholar 

  6. Alpay, D., Luna Elizarraras, M.E., Shapiro, M.V., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer Briefs in Mathematics. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  7. Baston, R.J.: Quaternionic complexes. J. Geom. Phys. 8, 29–52 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Berenstein, C.A., Sabadini, I., Struppa, D.C.: Boundary Values of Regular Functions of Quaternionic Variables. Pitman Research Notes in Mathematics, vol 347, pp. 220–232. Longman, Harlow (1996)

    Google Scholar 

  9. Brackx, F., Pincket, W.: A Bochner-Martinelli formula for the biregular functions of Clifford analysis. Complex Var. 4, 39–48 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brackx, F., Pincket, W.: Two Hartogs theorems for nullsolutions of overdetermined systems in Euclidean Space. Complex Var. 4, 205–222 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brackx, F., Pincket, W.: Domains of biregularity in Clifford analysis. Rend. Circ. Mat. Palermo 2(9), 21–35 (1985/1986)

    Google Scholar 

  12. Brackx, F., Pincket, W.: Series expansions for the biregular functions of Clifford analysis. Simon Stevin Quat. J. Pure Appl. Math. 60, 41–55 (1986)

    MATH  MathSciNet  Google Scholar 

  13. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol, 76. Pitman, Boston (1982)

    Google Scholar 

  14. Bredon, G.E.: Sheaf Theory. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  15. Bures, J., Damiano, A., Sabadini, I.: Explicit resolutions for the complex of several Fueter operators. J. Geom. Phys. 57(3), 765–775 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Colombo, F., Gentili, G., Sabadini, I., Struppa, D.C.: Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222(5), 1793–1808 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Colombo, F., Gonzales Cervantes, J.O., Sabadini, I.: A nonconstant coefficients differential operator associated to slice monogenic functions. Trans. Am. Math. Soc. 365(1), 303–318 (2013)

    Article  MATH  Google Scholar 

  18. Colombo, F., Luna-Elizarraras, M.E., Sabadini, I., Shapiro, M., Struppa, D.C.: A quaternionic treatment of the inhomogeneous div-rot system. Moscow Math. J. 12(1), 37–48 (2012)

    MATH  MathSciNet  Google Scholar 

  19. Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra. Progress in Mathematical Physics, vol. 39. Birkhauser, Boston (2004)

    Google Scholar 

  20. Colombo, F., Sabadini, I., Struppa, D.C.: Dirac equation in the octonionic algebra. In: Contemp. Math. Analysis, Geometry, Number Theory: the Mathematics of Leon Ehrenpreis Philadelphia, vol. 251, pp. 117–134. American Mathematical Society, Providence (1998/2000)

    Google Scholar 

  21. Colombo, F., Sabadini, I., Struppa, D.C.: Slice monogenic functions. Isr. J. Math. 171, 385–403 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, vol. 289. Birkhauser, Boston (2011)

    Google Scholar 

  23. Colombo, F., Sabadini, I., Struppa, D.C.: Sheaves of slice regular functions. Math. Nachr. 285(8–9), 949–958 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Colombo, F., Sabadini, I., Struppa, D.C.: Slice regular functions in several variables. Indiana Univ. Math. J. 61(4), 1581–1602 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Colombo, F., Soucek, V., Struppa, D.C.: Invariant resolutions for several Fueter operators. J. Geom. Phys. 56(7), 1175–1191 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Damiano, A., Eelbode, D., Sabadini, I.: Algebraic analysis of Hermitian monogenic functions. CRAS 346(3), 139–142 (2008)

    MATH  MathSciNet  Google Scholar 

  27. Damiano, A., Eelbode, D., Sabadini, I.: Invariant sygyzies for the Hermitian Dirac operator. Math. Z. 262(4), 929–945 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Damiano, A., Eelbode, D., Sabadini, I.: Quaternionic Hermitian spinor systems and compatibility conditions. Adv. Geom. 11(1), 169–189 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Damiano, A., Sabadini, I., Struppa, D.C.: New algebraic properties of biregular functions in 2 n quaternionic variables. Complex Var. Elliptic Equ. 51(5–6), 497–510 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Deavours, C.A.: The quaternion calculus. Am. Math. Mon. 80, 995–1008 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  31. De Schepper, N., Sommen, F.: Introductory Clifford analysis. Springer References (2014)

    Google Scholar 

  32. Ehrenpreis, L.: A new proof and an extension of Hartogs’ theorem. Bull. Am. Math. Soc. 67, 507–509 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ehrenpreis, L.: Fourier Analysis in Several Complex Variables. Wiley, New York (1970)

    MATH  Google Scholar 

  34. Fabiano, A., Gentili, G., Struppa, D.C.: Sheaves of quaternionic hyperfunctions and microfunctions. Complex Var. 24, 161–184 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Fueter, R.: Analytische Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 4(1), 9–20 (1932)

    Article  MathSciNet  Google Scholar 

  36. Gentili, G., Stoppato, C., Struppa, D.C.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  37. Gentili, G., Struppa, D.C.: A new approach to Cullen-regular functions of a quaternionic variable. CRAS Paris 342(10), 741–744 (2006)

    MATH  MathSciNet  Google Scholar 

  38. Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Gentili, G., Struppa, D.C.: Regular functions on the space of Cayley numbers. Rocky Mountain J. Math. 40(1), 225–241 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  40. Ghiloni, R., Perotti, A.: Slice regular functions on real alternative algebras. Adv. Math. 226, 1662–1691 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ghiloni, R., Perotti, A.: Slice regular functions of several Clifford variables. In: Proceedings of ICNPAA. AIP Conference Proceedings, vol. 1493, pp. 734–738 (2012)

    Google Scholar 

  42. Guerlebeck, K., Sproessig, W.: Clifford Analysis and Elliptic Boundary Value Problems. Mathematics and Its Applications, vol. 321. Kluwer, Dordrecht (1995)

    Google Scholar 

  43. Guerlebeck, K., Sproessig, W.: Quaternionic analysis: general aspects. Springer References (2014)

    Google Scholar 

  44. Hartogs, F.: Einege Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrere Veranderlichen. Sitzungber. Kongl. Bayer. Akad. Wissen 36, 223–241 (1906)

    Google Scholar 

  45. Imaeda, K., Imaeda, M.: Sedenions: algebra and analysis. Appl. Math. Comp. 115(2), 77–88 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  46. Kang, Q., Wang, W.: On Radon-Penrose transformation and k −Cauchy-Fueter operator. Sci. China Math. 55(9), 1921–1936 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  47. Kang, Q., Wang, W.: On Penrose integral formula and series expansion of k −regular functions on the quaternionic space \(\mathbb{H}^{n},\) J. Geom. Phys. 64, 192–208 (2013)

    Google Scholar 

  48. Kato, G., Struppa, D.C.: Fundamentals of Microlocal Algebraic Analysis. Marcel Dekker, New York (1999)

    MATH  Google Scholar 

  49. Komatsu, H.: Resolution by hyperfunctions of sheaves of solutions of differential equations with constant coefficients. Math. Ann. 176, 77–86 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  50. Komatsu, H.: Relative Cohomology of Sheaves of Solutions of Differential Equations. Springer Lecture Notes in Mathematics, vol. 287, pp. 192–261. Springer, New York (1973)

    Google Scholar 

  51. Krantz, S.: Theory of Several Complex Variables. Belmont, California (1992)

    MATH  Google Scholar 

  52. Kravchenko, V.V.: Applied Quaternionic Analysis. Research and Exposition in Mathematics, vol. 28. Heldermann, Leipzig (2003)

    Google Scholar 

  53. Kravchenko, V.V., Shapiro, M.V.: Integral Representation for Spatial Models of Mathematical Physics. Pitman Research Notes in Mathematics, vol. 351. Longman, Harlow (1996)

    Google Scholar 

  54. Luna-Elizarraras, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex numbers and their elementary functions. Cubo 14(2), 61–80 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  55. Luna-Elizarraras, M.E., Shapiro, M.V., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: the Algebra, Geometry, and Analysis of Bicomplex Numbers (submitted)

    Google Scholar 

  56. Malonek, H.: Historical notes on quaternionic and Clifford analysis. Springer References (2014)

    Google Scholar 

  57. Moisil, G., Theodorescu, N.: Functions holomorphes dans l’ espace. Math. (Cluj) 5, 142–159 (1931)

    Google Scholar 

  58. Oka, K.: Domaines d’holomorphie. J. Sci. Hiroshima Univ. 7, 115–130 (1937)

    Google Scholar 

  59. Oka, K.: Deuxieme probleme de Cousin. J. Sci. Hiroshima Univ. 9, 7–19 (1939)

    MATH  MathSciNet  Google Scholar 

  60. Oka, K.: L’integrale de Cauchy. Jpn. J. Math. 17, 523–531 (1941)

    MATH  MathSciNet  Google Scholar 

  61. Palamodov, V.P.: Linear Partial Differential Operators with Constant Coefficients. Springer, Berlin (1970)

    Book  Google Scholar 

  62. Palamodov, V.P.: Holomorphic synthesis of monogenic functions of several quaternionic variables. J. Anal. Math. 78, 177–204 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  63. Pena-Pena, D., Sabadini, I., Sommen, F.: Quaternionic Clifford analysis: the Hermitian setting. Complex Anal. Oper. Theory 1(1), 97–113 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  64. Pertici, D.: Funzioni regolari di piú variabili quaternioniche. Ann. Mat. Pura Appl. Ser. IV CLI, 39–65 (1988)

    Google Scholar 

  65. Price, G.B.: An Introduction to Multicomplex Spaces and Functions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 140. Marcel Dekker, New York (1991)

    Google Scholar 

  66. Range, R.M.: Extension phenomena in multidimensional complex analysis: correction of the historical record. Math. Intell. 24, 4–12 (2002)

    Article  MathSciNet  Google Scholar 

  67. Sabadini, I.: Verso una teoria delle iperfunzioni quaternioniche. Ph.D. Dissertation, Milano (1995)

    Google Scholar 

  68. Sabadini, I., Shapiro, M.V., Struppa, D.C.: Algebraic analysis of the Moisil-Theodorescu system. Complex Var. Theory Appl. 40(4), 333–357 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  69. Sabadini, I., Sommen, F.: Hermitian Clifford analysis and resolutions. Math. Meth. Appl. Sci. 25(16–18), 1395–1413 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  70. Sabadini, I., Sommen, F.: Hermitian Clifford analysis. Springer References (2014)

    Google Scholar 

  71. Sabadini, I., Sommen, F., Struppa, D.C.: The Dirac complex on abstract vector variables: megaforms. Exp. Math. 12, 351–364 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  72. Sabadini, I., Sommen, F., Struppa, D.C., Van Lancker, P.: Complexes of Dirac operators in Clifford algebras. Math. Z. 239, 215–240 (2002)

    Article  MathSciNet  Google Scholar 

  73. Shapiro, M., Struppa, D.C., Vajiac, A., Vajiac, M.B.: Hyperbolic numbers and their functions. An. Univ. Oradea Fasc. Mat. 19(1), 265–283 (2012)

    MATH  MathSciNet  Google Scholar 

  74. Somberg, P.: Quaternionic Complexes in Clifford Analysis. NATO Science Series II, vol. 25, pp. 203–301. Kluwer, Dordrecht (2001)

    Google Scholar 

  75. Sommen, F.: Clifford analysis in two and several vector variables. Appl. Anal. 73, 225–253 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  76. Soucek, V.: Representation theory in Clifford analysis. Springer References (2014)

    Google Scholar 

  77. Struppa, D.C.: The first eighty years of Hartogs’ theorem. Sem. Geom. Dip. Mat. Bologna 127–209 (1987)

    Google Scholar 

  78. Struppa, D.C.: Slice monogenic functions with values in some real algebras. Springer References (2014)

    Google Scholar 

  79. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85(2), 199–224 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  80. Wang, W.: On non-homogeneous Cauchy-Fueter equations and Hartogs’ phenomenon in several quaternionic variables. J. Geom. Phys. 58(9), 1203–1210 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  81. Wang, W.: The k −Cauchy-Fueter complex, Penrose transformation and Hartogs’ phenomenon for quaternionic k -regular functions. J. Geom. Phys. 60(3), 513–530 (2010)

    Google Scholar 

  82. Wang, W.: The tangential Cauchy-Fueter complex on the quaternionic Heisenberg group. J. Geom. Phys. 61(1), 363–380 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Struppa, D.C. (2014). Quaternionic and Clifford Analysis in Several Variables. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_26-1

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