Abstract
For solenoidal and irrotational vector fields as well as for quaternionic analysis of the Moisil-Théodoresco operator we introduce the notions of the Bergman space and the Bergman reproducing kernel; main properties of them are studied. Among other objects of our interest are: the analogues of the Bergman projections; the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map.
Mathematics Subject Classification (2000)
Keywords
To Nikolai Vasilevski on the occasion of his 60th birthdayframework of COFAA and SIP programs.
The first author was partially supported by CONACYT and by Instituto Politécnico Nacional as Doctoral scholarship and PIFI scholarship recipient.
The second and the third author were partially supported by CONACYT projects as well as by Instituto Politécnico Nacional in the
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. Delanghe, F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, Proc. London Math. Soc. 1978, (3) (37), 545–576.
B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, Modern Methods and Applications, Springer, 1984, Vol. 1.
P.L. Duren and A. Schuster, Bergman Spaces, Mathematical Surveys and Monographs, AMS, 2004, Vol. 100.
J.O. González-Cervantes, M.E. Luna-Elizarrarás, M. Shapiro, On some categories and functors in the theory of quaternionic Bergman spaces, Adv. Appl. Clifford Alg. No. 19 (2009), 325–338.
J.O. González-Cervantes, M. Shapiro, Hyperholomorphic Bergman Spaces and Bergman Operators Associated with Domains in ℂ2. Compl. anal. oper. theory, 2008, Vol. 2, # 2, 361–382.
K. Gürlebeck, W. Sprössig, Quaternionic analysis and elliptic boundary value problems, Birkhäuser Verlag, 1990.
[7] K. Gürlebeck, W. Sprössig, Quaternionic and Clifford calculus for physicists and engineers, John Wiley and Sons, 1997.
[8] X. Ji, T. Qian and J. Ryan Fourier theory under Möbius transformations, in the book Clifford Algebras and their Applications in Mathematical Physics: Clifford Analysis. Birkhäuser, 2000, Vol. 2, 57–80.
A. Meziani, On real analytic planar vector fields near the characteristic set, Contemp. Math., 2000, Vol. 251, 429–438.
I. Porteous, Topological Geometry, Cambridge University, 1981, 2nd ed.
J.R. Retherford, Hilbert Space: Compact Operators and the Trace Theorem. Cambridge Press, 1998.
J. Ryan, The conformal covariance of Huygens’ principle-type integral formulae in Clifford analysis, in the book Clifford Algebras and Spinor Structures, Kluwer Academic Publishers, 1995, 301–310.
J. Ryan, Some applications of conformal covariance in Clifford analysis, in the book Clifford Algebras in Analysis and Related Topics, CRC Press, 1996, 129–155.
M.V. Shapiro, On the conjugate harmonic functions of M. Riesz — E. Stein — G. Weiss. Topics in Complex Analysis, Differential Geometry and Mathematical Physics, World Scientific, 1997, 8–32.
M.V. Shapiro, N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. ψ-hyperholomorphic function theory. Compl. Var. Theory Appls. 1995, Vol. 27, 17–46.
M.V. Shapiro, N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions, singular operators and boundary value problems. II. Algebras of singular integral operators and Riemann type boundary value problems. Compl. Var. Theory Appls., 1995, Vol. 27, 67–96.
M.V. Shapiro, N.L. Vasilevski, On the Bergman Kernel Function in Hyperholomorphic Analysis, Acta Appl. Math., 1997, 1–27.
E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
A. Sudbery, Quaternionic analysis. Math. Proc. Camb. Phil. Soc. 1979, 85, 199–225.
S. Thangavelu, Harmonic analysis on the Heisenberg group, Springer, 1998.
N.L. Vasilevski, Toeplitz operators on the Bergman spaces: inside-the-domain effects, Contemp. Math., 2001, Vol. 289, 79–145.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Basel AG
About this chapter
Cite this chapter
Oscar González-Cervantes, J., Elena Luna-Elizarrarás, M., Shapiro, M. (2010). On the Bergman Theory for Solenoidal and Irrotational Vector Fields, I: General Theory. In: Duduchava, R., Gohberg, I., Grudsky, S.M., Rabinovich, V. (eds) Recent Trends in Toeplitz and Pseudodifferential Operators. Operator Theory: Advances and Applications, vol 210. Springer, Basel. https://doi.org/10.1007/978-3-0346-0548-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0548-9_5
Publisher Name: Springer, Basel
Print ISBN: 978-3-0346-0547-2
Online ISBN: 978-3-0346-0548-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)