RETRACTED ARTICLE: A note on the boundary behavior for a modified Green function in the upper-half space

Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential.


Introduction and main results
Let R n (n ≥ ) denote the n-dimensional Euclidean space. The upper half-space H is the set H = {x = (x  , x  , . . . , x n ) ∈ R n : x n > }, whose boundary and closure are ∂H and H respectively.
For x ∈ R n and r > , let B(x, r) denote the open ball with center at x and radius r. Set E α (x) = -log |x| if α = n = , |x| α-n if  < α < n.
Let G α be the Green function of order α for H, that is, where * denotes reflection in the boundary plane ∂H just as y * = (y  , y  , . . . , -y n ). In case α = n = , we consider the modified kernel function, which is defined by In case  < α < n, we define where m is a non-negative integer, C ω k (t) (ω = n-α  ) is the ultraspherical (or Gegenbauer) polynomial (see []). The expression arises from the generating function for Gegenbauer polynomials where |r| < , |t| ≤  and ω > . The coefficient C ω k (t) is called the ultraspherical (or Gegenbauer) polynomial of degree k associated with ω, the function C ω k (t) is a polynomial of degree k in t.
Then we define the modified Green function G α,m (x, y) by where x, y ∈ H and x = y. We remark that this modified Green function is also used to give unique solutions of the Neumann and Dirichlet problem in the upper-half space where μ is a non-negative measure on H. Here note that G , (x, μ) is nothing but the general Green potential. Let k be a non-negative Borel measurable function on R n × R n , and set for a non-negative measure μ on a Borel set E ⊂ R n . We define a capacity C k by where the supremum is taken over all non-negative measures μ such that S μ (the support of μ) is contained in E and k(y, μ) ≤  for every y ∈ H. For β ≤ , δ ≤  and β ≤ δ, we consider the kernel function Now we prove the following result. For related results in a smooth cone and tube, we refer the reader to the papers by Qiao (see [, ]) and Liao-Su (see []), respectively. The readers may also find some related interesting results with respect to the Schrödinger operator in the papers by Su (see []), by Polidoro and Ragusa (see []) and the references therein.
then there exists a Borel set E ⊂ H with properties: Remark By using Lemma  below, condition () in Theorem with α = , β = , δ =  means that E is -thin at ∂H in the sense of [].

Some lemmas
Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line.

This can be proved by a simple calculation.
Lemma  Gegenbauer polynomials have the following properties: Proof () and () can be derived from [], p.. Equality () follows from expression (.) by taking t = ; property () is an easy consequence of the mean value theorem, () and also ().
Lemma  For x, y ∈ R n (α = n = ), we have the following properties: The following lemma can be proved by using Fuglede (see [], Théorèm .).

Proof of Theorem
We write We distinguish the following two cases. Case .  < α < n.

Consider the sets
Then G ⊂ {y ∈ H :  -i- < y n <  -i+ }. Let ν be a non-negative measure on H such that S ν ⊂ E i , where S ν is the support of ν. Then we have k α,β,δ (y, ν) ≤  for y ∈ H and

R E T R
Setting E = ∞ i= E i , we see that () in Theorem is satisfied and Note that C ω  (t) ≡ . By () and () in Lemma , we take t = x·y |x||y| , t * = x·y * |x||y * | in Lemma () and obtain x n y n |x||y|