Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1191-1206

ON THE CYLINDRICAL GREEN'S FUNCTION FOR REPRESENTATION THEORY AND ITS APPLICATIONS

Lei Qiao

Henan University of Economics and Law, School of Mathematics and Information Science
Zhengzhou 450046, P.R. China; qiaocqu 'at' huel.edu.cn

Abstract. It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been applied to the representation theory. In this paper, we extend the notion of conformal invariance of Green's function to superharmonic functions, and use this extension to construct and estimate the cylindrical Green's function for Dirichlet boundary value problem. These considerations lead to a new proof of the representation theory of superharmonic functions defined in cylinders. We also show this estimation can be used to obtain a covering property of rarefied sets at infinity. Finally, by giving an example, we show that the reverse of this property is not true.

2010 Mathematics Subject Classification: Primary 31B10, 35J05, 35J10, 35J40, 65N38, 65N55.

Key words: Cylindrical Green's function, cylindrical Green's potential, representation theory, covering property.

Reference to this article: L. Qiao: On the cylindrical Green's function for representation theory and its applications. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1191-1206.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4466