Abstract
We survey some results concerning Clifford analysis and the L2 theory of boundary value problems on domains with Lipschitz boundaries. Some novelty is introduced when using Rellich inequalities to invert boundary operators.
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References
P. Auscher, P. Tchamitchian, Bases d’ondelettes sur les courbes corde-arc, noyau de Cauchy et espaces de Hardy associés, Rev. Mat. Iber. 5 (1989), 139–170.
F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman Advanced Publ. Program, London, 1982.
A.P. Caldér on, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (1977), 1324–1327.
R.R. Coifman, P. Jones, S. Semmes, Two elementary proofs of the L2 bounded-ness of Cauchy integrals on Lipschitz curves, J. of Amer. Math. Soc. 2 (1989), 553–564.
R.R. Coifman, A. Mcintosh, Y. Meyer, L’intégral de Cauchy définit un opérateur borné sur L2 pour les courbes Lipschitziennes, Annals of Math., 116 (1982), 361–387.
G. David, J.-L. Journé, S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iber. 1 (1985), 1–57.
L. Escauriaza, E.B. Fabes, G. Verchota, On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proc. Amer. Math. Soc. 115 (1992), 1069–1076.
G.I, Gaudry, R.L. Long, T. Qian, A martingale proof of L2-boundedness of Clifford-valued singular integrals, Annali Matematica, Pura Appl. 165 (1993), 369–394.
J. Gilbert, M. A. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 1991.
K. Gürlebeck, W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhäuser Verlag, Basel, 1990.
K. Gürlebeck, W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley, Chichester, 1997.
B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics, World Sci. Publ., Singapore, 1988.
C. Li, A. Mcintosh, T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iber., 10 (1994), 665–721.
C. Li, A. Mcintosh, S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), no. 3, 455–481.
A. Mcintosh, Clifford algebras and the higher dimensional Cauchy integral, Approximation and Function Spaces, Banach Center Publ., Warsaw, vol. 22 (1989), 253–267.
A. Mcintosh, Clifford algebras, Fourier theory, singular integrals and harmonic functions on Lipschitz domains, in: Clifford Algebras in Analysis and Related Topics (J. Ryan ed.), 33–88, CRC Press, Boca Raton, FL, 1996.
A. Mcintosh, M. Mitrea, Clifford algebras and Maxwell’s equations in Lipschitz domains, Math. Meth. Appl. Sci., 22 (1999), 1599–1620.
A. Mcintosh, D. Mitrea, M. Mitrea, Rellich type estimates for one-sided monogenic functions in Lipschitz domains and applications, in: Analytical and Numerical Methods in Quaternionic and Clifford Algebras, K. Gürlebeck and W. Sprössig eds, the Proceedings of the Seiffen Conference, Germany, 1996, pp. 135–143.
D. Mitrea, The method of layer potentials for non-smooth domains with arbitrary topology, Integral Equations Operator Theory 29 (1997), 320–338.
M. Mitrea, Clifford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics, no. 1575, Springer, Berlin, 1994.
M. Mitrea, The method of layer potentials in electromagnetic scattering theory on nonsmooth domains, Duke Math. J. 77 (1995), no. 1, 111–133.
M. Mitrea, R.H. Torres, G.V. Weiland, Regularity and approximation results for the Maxwell problem on C1 and Lipschitz domains, Clifford Algebras in Analysis and Related Topics (J. Ryan ed.), 297–308, CRC Press, Boca Raton, FL, 1996.
M. Murray, The Cauchy integral, Calderón commutators and conjugations of singular integrals in Rn, Trans, of the Amer. Math. Soc. 289 (1985), 497–518.
J. Ryan (editor), Clifford Algebras in Analysis and Related Topics, CRC Press, Boca Raton, FL, 1996.
T. Tao, Harmonie convolution operators on Lipschitz graphs, Adv. Appl. Clifford Alg. 6 (1996), 207–218.
G. Verchota, Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572–611.
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Axelsson, A., Grognaxd, R., Hogan, J., McIntosh, A. (2001). Harmonic Analysis of Dirac Operators on Lipschitz Domains. In: Brackx, F., Chisholm, J.S.R., Souček, V. (eds) Clifford Analysis and Its Applications. NATO Science Series, vol 25. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0862-4_22
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DOI: https://doi.org/10.1007/978-94-010-0862-4_22
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