Abstract
Formulas for the Euler vector fields, the Neumann derivatives, and the Euler as well as Dirichlet product are derived. Extensions to a Riemann domain of the Gauss operator, the Gauss’ lemma and the related jump formulas are given, and the Gauss–Helmholtz representation with ramifications proved. Examples of elementary solutions to certain modified Laplace operators, applications to pseudospherical harmonics, and characterizations of pseudoradial, pseudospherical, nearly holomorphic, and holomorphic functions, are obtained, and constancy criterion for locally Lipschitz, semiharmonic, respectively, weakly holomorphic functions are given.
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References
Andreotti, A., Stoll, W.: Analytic and algebraic dependence of meromorphic functions. In: Lecture Notes in Mathematics, vol. 234. Springer, Berlin (1971)
Arnold V.I.: Lectures on Partial Differential Equations. Springer, Berlin (2004)
Cialdea, A.: The simple- and multiple-layer potential approachin n-dimensional problems. In: Tutschke, W., Mshimba, A. (eds.) Functional analytic methods in complex analysis and applications to partial differential equations, pp. 375–378. World Scientific, Singapore (1995)
Folland G.B.: Introduction to Partial Differential Equations. In: Mathematical Notes, vol. 17. Princeton University Press, Princeton (1976)
Grauert, H., Remmert, R.: Theory of stein spaces. In: Grundlehren der Mathematischen Wissenschaften, vol. 236. Springer, Berlin (1979)
Kytmanov A.M.: The Bochner–Martinelli integrals and its applications. Birkhäuser-Verlag, Basel (1995)
Martinelli, E.: Qualche riflessione sulla rapresentazione integrale di massima dimensione per le funzioni di più variabili complesse, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 76 (1984), no. 4, 235–242 (1985)
Maz’ya, V.G., Nikol’skii, S.M. (eds): Analysis IV, Encyclopaedia of Math. Sciences, vol. 27. Springer, Berlin (1991)
Mikhlin S.G.: Mathematical physics, an advanced course. North-Holland, Amsterdam (1970)
Steinbach O.: Numerical approximation methods for elliptic boundary value problems. Springer, NewYork (2008)
Tung, C.: The first main theorem of value distribution on complex spaces. In: Memoire dell’Accademia Nazionale dei Lincei, Serie VIII, vol. 15, Sez.1, pp. 91–263 (1979)
Tung C.: Semi-harmonicity, integral means and Euler typevector fields. Adv. Appl. Clifford Algebras 17, 555–573 (2007)
Tung C.: Integral products, Bochner–Martinelli transforms and applications. Taiwanese J. Math. 13(5), 1583–1608 (2009)
Tung C.: On semisubmedian functions and weak plurisubharmonicity. Cubo Math. J. 12(2), 239–263 (2010)
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Communicated by Lucian Beznea.
Supports by the “Globale Methoden in der komplexen Geometrie” Grant of the German research society DFG and the Faculty Improvement Grant of Minnesota State University, Mankato, are gratefully acknowledged.
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Tung, Cc. On Generalized Integral Means and Euler Type Vector Fields. Complex Anal. Oper. Theory 5, 701–730 (2011). https://doi.org/10.1007/s11785-010-0086-1
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DOI: https://doi.org/10.1007/s11785-010-0086-1
Keywords
- Euler vector fields
- Neumann vector fields
- Admissible potential
- Gauss mean
- Bochner–Martinelli mean
- a-Pseudospherical
- a-Pseudoradial