Research Article Green’s Function for a Slice of the Korányi Ball in the Heisenberg Group H  Shivani Dubey, 1 Ajay Kumar, 1 and Mukund Madhav Mishra 2 1 Department of Mathematics, University of Delhi, Delhi 110007, India 2 Department of Mathematics, Hans Raj College, University of Delhi, Delhi 110007, India Correspondence should be addressed to Ajay Kumar; akumar@maths.du.ac.in Received 25 June 2015; Revised 18 August 2015; Accepted 24 August 2015 Academic Editor: Heinrich Begehr Copyright © 2015 Shivani Dubey et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a representation formula for solution of the inhomogeneous Dirichlet problem on the upper half Kor´ anyi ball and for the slice of the Kor´ anyi ball in the Heisenberg group H  by obtaining explicit expressions of Green-like kernel when the given data has certain radial symmetry. 1. Introduction Te Heisenberg groups, in discrete and continuous versions, appear in several streams of mathematics, including Fourier analysis, several complex variables, geometry, and topology. Te well known concepts of Green, Neumann, and Robin functions are important for representing solutions to certain boundary value problems for elliptic equations and to realize their smoothness properties. While the existence of these particular fundamental solutions for admissible domains are presented in all textbooks (see, e.g., [1–3]), explicit expressions for these functions are rare. In case of the Laplace operator mostly just the unit ball serves as an example. Te objective of this paper is to continue the search for explicit Green’s functions for domains other than the Kor´ anyi ball in the Heisenberg group. Tere is little hope to get explicit kernels that work for arbitrary continuous boundary data, but it is possible to fnd some if one is restricted to boundary data having certain symmetry properties. Tis line of investigation was started by Gaveau et al. in [4] where they dealt with the case of the unit ball in the 3-dimensional Heisenberg group H 1 and functions invariant under a circle action. Tis result was extended in [5] to the general Heisenberg group H  with its natural metric, for functions invariant under the unitary group (). Further, in the case of H  , it has been shown that the method of [5] works for the much larger class of circular functions, that is, functions invariant under a circle action [6]. Te Dirichlet problem on the Heisenberg group and the existence of unique solution was discussed in [7]. Green’s function for circular data in the Heisenberg group has been studied for various domains, for example, for half space in [6], for quarter space in [8], and for annulus in [9]. In following sections, we obtain the circular Green’s function for the upper half Kor´ anyi ball and a slice of the Kor´ anyi ball by two parallel planes. We apply the method of infnitely many refections along the boundaries of the domain, which was introduced by Courant and Hilbert in [2], generalized for the annulus in the Heisenberg group [10]. 2. Analysis on the Heisenberg Group We begin by recalling some notions from [11], which have laid the foundation for harmonic analysis on the Heisenberg group. Defnition 1. Te Heisenberg group H  (of degree ) is the Lie group structure on C  × R whose group law is given by [,]⋅ [  ,  ]=[+  ,+  +2Im (⋅   )], ⋅   =  ∑ =1   ⋅    . (1) Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2015, Article ID 460461, 7 pages http://dx.doi.org/10.1155/2015/460461