Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-02-05T09:00:01.820Z Has data issue: false hasContentIssue false
  • English
  • Français

On strict inclusions of weighted dirichlet spaces of monogenic functions

Published online by Cambridge University Press:  17 April 2009

K. Gürlebeck  and
H. R. Malonek
K. Gürlebeck
Affiliation:
Bauhaus-Universität Weimar Insitute für Mathematik/Physik, Coudray-Str. 13, D-99423 Weimar, GermanyUniversidade de Aveiro, Departamento de Matemática, Campus Universiário de Santiago, P-3810–193 Aveiro, Portugal
H. R. Malonek
Affiliation:
Bauhaus-Universität Weimar Insitute für Mathematik/Physik, Coudray-Str. 13, D-99423 Weimar, GermanyUniversidade de Aveiro, Departamento de Matemática, Campus Universiário de Santiago, P-3810–193 Aveiro, Portugal
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a scale of weighted spaces a quaternion-valued functions of three real variables. This scale generalises the idea of Qp-spaces in complex function theory. The goal of this paper is to prove that the inclusions of spaces from the scale are strict inclusion. As a tool we prove some properties of special monogenic polynomials which have an importance in their own right independently of their use in the scale of Qp-spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 1 , August 2001 , pp. 33 - 50
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Aleman, A., ‘Hilber spaces of analytic functions between he Hardy and the Dirichlet spaces’, Proc. Amer. Math. Soc. 115 (1992), 97–104.CrossRefGoogle Scholar
[2]Aulaskari, R., He, Y., Ristioja, J. and Zhao, R., ‘Qp spaces on Riemann surfaces’, Canad. J. Math. 50 (1998), 449–464.CrossRefGoogle Scholar
[3]Aulaskari, R. and Lappan, P., ‘Criteria for an analytic function to be Bloch and harmonic or meromorphic function to be normal’ in Complex Analysis and its Applications, (Chung-Chun, Y. et al. , Editors), Pitman Research Notes in Mathematics 305 (Longman, 1994), pp. 136–146.Google Scholar
[4]Aulaskari, R., Nowak, M. and Zhao, R., ‘The n-th derivative criterion for Qp’, Bull. Austral. Math. Soc. 58 (1998), 43–56CrossRefGoogle Scholar
[5]Aulaskari, R. and Tovar, L.M., ‘On the function spaces Bp and Qp’, Bull Hong Kong Math. Soc. 1 (1997), 203–208.Google Scholar
[6]Aulaskari, R., Xiao, J. and Zhao, R., ‘On subspaces and subsets of BMOA and UBC’, Analysis 15 (1995), 101–121.CrossRefGoogle Scholar
[7]Baernstein, A. II, ‘Analytic functions of bounded mean oscillation’, in Aspects of contemporary complex analysis (Academic Press, London, 1980), pp. 3–36.Google Scholar
[8]Brackx, F., Delanghe, R., Sommen, F., Clifford analysis, Pitman Research Notes in Math. (Boston, London, Melbourne, 1982).Google Scholar
[9]Choa, J.S., Kim, H.O., Park, Y. Y., ‘A Bergman-Carleson measure characterization of Bloch functions in the unit ball of ℂn’, Bull. Korean Math. Soc. 29 (1992), 285–293.Google Scholar
[10]Cnops, J. and Delanghe, R., ‘Möbius invariant spaces in the unit ball’, Appl. Anal. 73 (2000), 45–64.CrossRefGoogle Scholar
[11]Cnops, J., Delanghe, R., Gürlebeck, K. and Shapiro, M.V., ‘Qp-spaces in Clifford analysis’,in Proceedings of the Conference “Analysis of Dirac Operators” held in Cetraro, October 1997 (to appear).Google Scholar
[12]Delanghe, R., Sommen, F. and Souček, V., Clifford algebra and spinor-valued functions (Kluwer, Dordrecht, 1992).CrossRefGoogle Scholar
[13]Essen, M. and Xiao, J., Qpspaces - a survey, (Department of Mathematics, Report Series No. 3) (University of Joensuu, 2000).Google Scholar
[14]Essen, M., Janson, S., Peng, L. and Xiao, J., Q-spaces of several real variables, (U.U.D.M. Report 1998:17)Google Scholar
[15]Gürlebeck, K., ‘On some weighted spaces of quaternion-valued functions’,in Proceedings of the the Second ISSAC Congress Vol. 2, (Begehr, H.G.W. et al. , Editors) (Kluwer Acad. Publishers, 2000), pp. 1387–1401.CrossRefGoogle Scholar
[16]Gürlebeck, K., Kähler, U., Shapiro, M. and Tovar, L. M., ‘On Qp-spaces of quaternion-valued functions’ Complex Variables Theory Apply. 39 (1999), 115–135.Google Scholar
[17]Gürlebeck, K. and Malonek, H.R., ‘A hypercomplex derivative of monogenic functions in ℝn+1 and its applications’, Complex Variables Theory Appl. 39 (1999), 199–228.Google Scholar
[18]Gürlebeck, K. and Sprößig, W., Quaternionic analysis and elliptic boundary value problems, Int. Ser. Num. Math. 89 (Birkhäuser Verlag, Basel, 1990).CrossRefGoogle Scholar
[19]Gürlebeck, K. and Sproößig, W., Quaternionic and Clifford calculus for engineers and physicists (John Wiley and Sons, Chichester, 1997).Google Scholar
[20]Kravchenko, V.V. and Shapiro, M.V., Integral representations for spatial models of mathematical physics, Pitman Research Notes in Mathematics Series 351 (Longman, Horlow, 1996).Google Scholar
[21]Leutwiler, H., ‘Harmonic functions of bounded mean oscillation’, Math. Ann. 244 (1979), 167–183.CrossRefGoogle Scholar
[22]Malonek, H., ‘Power series representation for monogenic functions in ℝm+1 based on a permutational product”, Complex Variables Theory Appl. 15 (1990), 181–191.Google Scholar
[23]Matelijevic, M. and Pavlovic, M., ‘Lp-behavior of power series with positive coefficients and Hardy spaces’, Proc. Amer. Math. Soc. 87 (1983), 309–316.Google Scholar
[24]Miao, J., ‘A property of analytic functions with Hadamard gaps’, Bull. Austral. Math. Soc. 45 (1992), 105–112.CrossRefGoogle Scholar
[25]Mitelman, I.M. and Shapiro, M.V., “Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyperderivability’, Math. Nachr. 172 (1995), 211–238.CrossRefGoogle Scholar
[26]Ouyang, C., Yang, W. and Zhao, R., ‘Characterizations of Bergman spaces and Bloch space in the unit ball of ℂn’, Trans. Amer. Math. Soc. 347 (1995), 4301–4313.Google Scholar
[27]Ouyang, C., Yang, W. and Zhao, R., “Möbius invariant Qp spaces associated with the Green's function on the unit ball of ℂn’, Pacific J. Math. 182 (1998), 69–99.CrossRefGoogle Scholar
[28]Ryan, J., ‘Conformally covariant operators in Clifford analysis’, J. Anal. Appl. 14 (1995), 677–704.Google Scholar
[29]Stroethoff, K., ‘Besov-Type characterisations for the Bloch space’, Bull. Austral. Math. Soc. 39 (1989), 405–420.CrossRefGoogle Scholar
[30]Sudbery, A., ‘Quaternionic analysis’, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–225.CrossRefGoogle Scholar