Abstract
This paper is devoted to the compactness of the hypercomplex commutator S γ M a − M a S γ, where S γ is the Cauchy singular integral operator (in the Douglis sense), a is a Hölder continuous hypercomplex function and M a is the multiplication operator given by M a f = a f. We extend a known compactness sufficient condition for the commutator of the Cauchy singular integral operator to the frame of the hypercomplex analysis, where γ is merely required to be an arbitrary regular closed Jordan curve.
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Abreu Blaya R., Bory Reyes J.: La compacidad del conmutador en los espacios generalizados de Hölder. Rev. Integr. Temas Mat. 15(2), 1–5 (1997)
Abreu Blaya R., Peña Peña D., Bory Reyes J.: Conjugate hyperharmonic functions and Cauchy type integrals in Douglis analysis. Complex Var. Theory Appl. 48(12), 1023–1039 (2003)
Abreu Blaya R., Bory Reyes J., Peña Peña D.: Riemann boundary value problem for hyperanalytic functions. Int. J. Math. Math. Sci. 17, 2821–2840 (2005)
Abreu Blaya R., Bory Reyes J.: Commutators and singular integral operators in Clifford analysis. Complex Var. Theory Appl. 50(4), 265–281 (2005)
Begehr H.: Complex Analytic Methods for Partial Differential Equations. An Introductory Text. World Scientific Publishing Co. Inc., River Edge (1994)
Bojarski, B.: Old and new on Beltrami equations. In: Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations. Proceedings of the ICTP, pp. 173–187, Trieste, Italy, 8–19 Feb (1988)
Douglis A.: A function-theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6, 259–289 (1953)
Gilbert, R.P., Buchanan, J.L.: First order elliptic systems. In: Mathematics in Science and Engineering, Series of Monographs and Textbooks, vol. 163 (1989)
Gilbert R.P., Hile G.N.: Generalized hypercomplex function theory. Trans. Am. Math. Soc. 195, 1–29 (1974)
Iwaniec, T., Martin, G.: What’s new for the Beltrami equation? Geometric analysis and applications (Canberra, 2000). In: Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 39, pp. 132–148, Austral. Nat. Univ., Canberra (2001)
Pogorzelski, W.: Integral equations and theirs applications (translated from the Polish), vol. 1. In: International Series of Monographs in Pure and Applied Mathematics, vol. 88. Pergamon Press, Oxford-New York-Frankfurt, PWN-Polish Scientific Warszawa (1966)
Rolewicz, D.P., Rolewicz, S.: Equations in linear spaces (translated from the Polish by Julian Musielak). In: Monografie Matematyczne, Tom 47. Monographs in Mathematics, vol. 47. PWN—Polish Scientific Publishers, Warsaw, 380 pp. (1968) (errata insert)
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Communicated by Daniel Alpay.
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Abreu-Blaya, R., Bory-Reyes, J. & Vilaire, JM. On the Compactness of the Hypercomplex Commutator in Hölder Continuous Functions Spaces. Complex Anal. Oper. Theory 4, 133–143 (2010). https://doi.org/10.1007/s11785-009-0013-5
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DOI: https://doi.org/10.1007/s11785-009-0013-5