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Associated Spaces in Clifford Analysis

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Abstract

In this paper we give a characterization of all linear first order partial differential operators with Clifford-algebra-valued coefficients that are associated to the generalized Cauchy-Riemann operator. In order to achieve our goal, we make use of a rule for D(u·v), where u and v are Clifford-algebra-valued functions. As an application, the solvability of initial value problems involving these operators is shown.

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References

  1. U. Y. Abbas, U. Yüksel, Necessary and Sufficient Conditions for First Order Differential Operators to be Associated with a Disturbed Dirac Operator in Quaternionic Analysis. Adv. Appl. Clifford Algebras, DOI 10.1007/s00006-014-0464-2, (2014).

  2. Bolívar Y., Vanegas C. J.: Initial value problems in Clifford-type analysis. Complex Variables and Elliptic Equations 58, 557–569 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Pitman Research Notes in Mathematics, Boston 76, (1982).

  4. Çelebi O., Yüksel U.: Solution of Initial Value Problems of Cauchy-Kovalevsky Type in the Space of Generalized Monogenic Functions. Adv. App. Clifford Alg. 20, 427–444 (2010)

    Article  MATH  Google Scholar 

  5. J. Conway, A course in functional analysis vol. 96, Springer, 1990.

  6. P. Enflo, V. Lomonosov, Some aspects of the invariant subspace problem. Handbook of the geometry of Banach Spaces I, Amsterdam, North Holland, (2001).

  7. K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists an Engineers. John Wiley & Sons., Chichester-New York, 1997.

  8. K. Gürlebeck, K. Habetha and W. Sprössig, Holomorphic Functions in the Plane and n-dimensional Space. Birkhäuser, Basel-Boston-Berlin, 2008.

  9. Lewy H.: An example of a smooth linear partial differential equation without solution. Ann. of Math. 66, 155–158 (1957)

    Article  MathSciNet  Google Scholar 

  10. Nagumo M.: Über das Anfangswertproblem Partieller Differentialgleichungen. Japan Journal Math. 18, 41–47 (1941)

    MathSciNet  Google Scholar 

  11. N. Q. Hung, N. C. Luong, First order differential operators associated to the Dirac Operator of quaternionic analysis. Contained in Methods of Complex and Clifford Analysis. Proceedings of ICAM Hanoi 2004. SAS International Publications (2006), 369-378.

  12. Hung N. Q., Son L. H.: Initial value problems with regular initial functions in quaternionic analysis. Complex Variables and Elliptic Equations 54, 1163–1170 (2009)

    Article  MathSciNet  Google Scholar 

  13. E. Obolashvili, Partial Differential Equations in Clifford Analysis. Pitman Monographs and Surveys in Pure and AppliedMathematics 96, Longman, 1998.

  14. Peña D., Sommen F.: A Note on the Fueter Theorem. Adv. Appl. Clifford Algebras 20, 379–391 (2010)

    Article  MATH  Google Scholar 

  15. D. Peña and F. Sommen, Some Power Series Expansions for Monogenic Functions. Computational Methods and Function Theory vol. 7 No. 1 (2007), 265-275.

  16. L. H. Son, N. Q. Hung, The initial value problems in Clifford and quaternion analysis. Procedings of the 15th ICFIDCAA, Osaka 2007, OCAMI Studies Vol. 2 (2008), 317-323.

  17. L. H. Son, N. T. Van, Differential Associated Operators in Clifford Analysis and Their Applications. Procedings of the 15th ICFIDCAA, Osaka, OCAMI Studies Vol. 2 (2008), 325-332.

  18. L.H. Son, W. Tutschke, Complex methods in higher dimensions- recent trends for solving boundary value and initial value problems. Complex Variables Theory and Applications 50, no. 7–11 (2005), 673–679.

  19. W. Tutschke,Solution of initial value problems in classes of generalized analytic functions. Teubner Leipzig and Springer-Verlag, 4, (1989).

  20. W. Tutschke Interior estimates in the theory of partial differential equations and their applications to initial value problems. Contained in Mem. Diff. Equat. Math. Phys. Vol. 12 (1997), 204-209.

  21. W. Tutschke, Associated spaces - a new tool of real and complex analysis. Natl. Univ. Publ. Hanoi, (2008).

  22. Tutschke W., Vanegas C., Métodos del análisis complejo en dimensiones superiores. XXI Escuela Venezolana de Matemáticas. Ediciones IVIC, (2008).

  23. Yüksel U.: Solution of Initial Value Problems with Monogenic Functions in Banach Spaces with L p -Norm. Adv. Appl. Clifford Algebras 20, 201–209 (2010)

    Article  MATH  Google Scholar 

  24. N.T. Van, First order differential operators transforming regular functions of quaternionic analysis into themselves. Contained in Methods of Complex and Clifford Analysis. Proceedings of ICAM Hanoi 2004. SAS International Publications (2006), 363-367.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad de Oriente, Cumaná, Venezuela

    Yanett Bolívar & Licet Lezama

  2. Departamento de Matemática, Universidad Simón Bolívar, Valle de Sartenejas, Caracas, Venezuela

    Luis Gerardo Mármol & Judith Vanegas

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  1. Yanett Bolívar
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  2. Licet Lezama
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  3. Luis Gerardo Mármol
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  4. Judith Vanegas
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Correspondence to Yanett Bolívar.

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Bolívar, Y., Lezama, L., Mármol, L.G. et al. Associated Spaces in Clifford Analysis. Adv. Appl. Clifford Algebras 25, 539–551 (2015). https://doi.org/10.1007/s00006-015-0528-y

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