R E S E A R C H
Open Access
Boundary behaviors of modified Green’s
function with respect to the stationary
Schrödinger operator and its applications
Zhiqiang Li
**Correspondence:
[email protected] Institute of Management Science and Engineering, Henan University, Kaifeng, 475001, China
Abstract
In this paper, we construct a modified Green’s function with respect to the stationary Schrödinger operator on cones. As applications, we not only obtain the boundary behaviors of generalized harmonic functions but also characterize the geometrical properties of the exceptional sets with respect to the Schrödinger operator.
Keywords: boundary behavior; modified Green’s function; stationary Schrödinger
operator; cone
1 Introduction and results
Let R and R+ be the set of all real numbers and the set of all positive real numbers,
re-spectively. We denote by Rn(n≥) then-dimensional Euclidean space. A point in Rnis
denoted byP= (X,xn),X= (x,x, . . . ,xn–). The Euclidean distance between two pointsP
andQin Rnis denoted by|P–Q|. Also|P–O|with the originOof Rnis simply denoted by
|P|. The boundary and the closure of a set S in Rnare denoted by∂Sand S, respectively.
We introduce a system of spherical coordinates (r,),= (θ,θ, . . . ,θn–), in Rnwhich
are related to Cartesian coordinates (x,x, . . . ,xn–,xn) byxn=rcosθ.
The unit sphere and the upper half unit sphere in Rnare denoted by Sn– and Sn– + ,
respectively. For simplicity, a point (,) on Sn– and the set {; (,)∈} for a set ,⊂Sn–, are often identified withand, respectively. For two sets⊂R
+ and
⊂Sn–, the set{(r,)∈Rn;r∈, (,)∈}in Rnis simply denoted by×. In
par-ticular, the half space R+×Sn+–={(X,xn)∈Rn;xn> }will be denoted by Tn.
ForP∈Rnandr> , letB(P,r) denote the open ball with center atPand radiusrin Rn. Sr=∂B(O,r). ByCn(), we denote the set R+×in Rnwith the domainon Sn–. We
call it a cone. ThenTnis a special cone obtained by putting= Sn+–. We denote the sets I×andI×∂with an interval on R byCn(;I) andSn(;I). BySn(;r) we denote
Cn()∩Sr. BySn() we denoteSn(; (, +∞)), which is∂Cn() –{O}.
We shall say that a setE⊂Cn() has a covering{rj,Rj}if there exists a sequence of balls
{Bj}with centers inCn() such thatE⊂
∞
j=Bj, whererjis the radius ofBjandRjis the
distance between the origin and the center ofBj.
LetAadenote the class of non-negative radial potentialsa(P),i.e.≤a(P) =a(r),P=
(r,)∈Cn(), such thata∈Lbloc(Cn()) with someb>n/ ifn≥ and withb= ifn=
orn= .
© 2015 Li. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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This article is devoted to the stationary Schrödinger equation
Schau(P) = –u(P) +a(P)u(P) = forP∈Cn(),
whereis the Laplace operator anda∈Aa. These solutions are called generalized
har-monic functions (associated with the operator Scha). Note that they are (classical) har-monic functions in the casea= . Under these assumptions the operatorSchacan be ex-tended in the usual way from the spaceC∞ (Cn()) to an essentially self-adjoint operator
onL(C
n()) (see []). We will denote itScha as well. The latter has a Green-Sch
func-tionG(;a)(P,Q). HereG(;a)(P,Q) is positive onCn() and its inner normal
deriva-tive∂G(;a)(P,Q)/∂nQ≥. We denote this derivative byPI(;a)(P,Q), which is called
the Poisson kernel with respect to the stationary Schrödinger operator. We remark that G(; )(P,Q) andPI(; )(P,Q) are the Green’s function and Poisson kernel of the Lapla-cian inCn(), respectively.
Let∗be a Laplace-Beltrami operator (spherical part of the Laplace) on⊂Sn–and λj(j= , , . . . , <λ<λ≤λ≤. . .) be the eigenvalues of the eigenvalue problem for∗
on(see,e.g., [], p.)
∗ϕ() +λϕ() = in,
ϕ() = on∂.
Corresponding eigenfunctions are denoted byϕjv(≤v≤vj), wherevjis the multiplicity
ofλj. We setλ= , normalize the eigenfunctions inL(), andϕ=ϕ> .
In order to ensure the existence ofλj(j= , , . . .), we put a rather strong assumption
on: ifn≥, thenis aC,α-domain ( <α< ) on Sn–surrounded by a finite number
of mutually disjoint closed hypersurfaces (e.g., see [], pp.-, for the definition ofC,α
-domain). Then ϕjv∈C() (j= , , , . . . , ≤v≤vj) and∂ϕ/∂n> on ∂(here and
below,∂/∂ndenotes differentiation along the interior normal).
Hence the well-known estimates (see,e.g., [], p.) imply the following inequality:
vj
v= ϕjv()
∂ϕjv()
∂n ≤
M(n)jn–, (.)
where the symbolM(n) denotes a constant depending only onn.
LetVj(r) (j= , , , . . .) andWj(r) (j= , , , . . .) stand, respectively, for the increasing and
non-increasing, asr→+∞, solutions of the equation
–Q(r) –n– r Q
(r) +λj
r+a(r)
Q(r) = , <r<∞, (.)
normalized under the conditionVj() =Wj() = (see [, ]).
We shall also consider the classBa, consisting of the potentialsa∈Aasuch that there
exists a finite limitlimr→∞ra(r) =k∈[,∞), moreover,r–|ra(r) –k| ∈L(,∞). Ifa∈
Ba, then the g.h.f.s. are continuous (see []).
In the rest of this paper, we assume thata∈Baand we shall suppress this assumption
for simplicity. Further, we use the standard notationsu+=max(u, ),u–= –min(u, ), [d]
is the integer part ofdandd= [d] +{d}, wheredis a positive real number.
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Denote
ι±j,k= –n±
(n– )+ (k+λ
j)
(j= , , , . . .).
It is well known (see []) that in the case under consideration the solutions to equation (.) have the asymptotics
Vj(r)∼dr
ι+j,k
, Wj(r)∼dr
ι–j,k
asr→ ∞, (.)
wheredanddare some positive constants.
Ifa∈Aa, it is well known that the following expansion holds for the Green’s function
G(;a)(P,Q) (see [], Chapter ):
G(;a)(P,Q) =
∞
j=
χ()Vj
min(r,t)Wj
max(r,t)
vj
v=
ϕjv()ϕjv()
, (.)
whereP= (r,),Q= (t,),r=tandχ(s) =w(W(r),V(r))|r=s is their Wronskian. The
series converges uniformly if eitherr≤stort≤sr( <s< ). The expansion (.) can also be rewritten in terms of the Gegenbauer polynomials.
For a non-negative integermand two pointsP= (r,),Q= (t,)∈Cn(), we put
K(;a,m)(P,Q) =
if <t< ,
K(;a,m)(P,Q) if ≤t<∞,
where
K(;a,m)(P,Q) =
m
j=
χ()Vj(r)Wj(t)
vj
v=
ϕjv()ϕjv()
.
If we modify the Green’s function with respect to the stationary Schrödinger operator on cones as follows:
G(;a,m)(P,Q) =G(;a)(P,Q) –K(;a,m)(P,Q)
for two pointsP= (r,),Q= (t,)∈Cn(), then the modified Poisson kernel with respect
to the stationary Schrödinger operator on cones can be defined by
PI(;a,m)(P,Q) =∂G(;a,m)(P,Q) ∂nQ
.
We remark that
PI(;a, )(P,Q) =PI(;a)(P,Q).
In this paper, we shall use the modified Poisson integrals with respect to the stationary Schrödinger operator defined by
PIa
(m,u)(P) =
Sn()PI
(;a,m)(P,Q)u(Q)dσQ,
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whereu(Q) is a continuous function on∂Cn() anddσQis the surface area element on
Sn().
Ifγ is a real number andγ ≥ (resp.γ < ), we assume in addition that ≤p<∞,
ι+[γ],k+{γ}>–ι+,k–n+ p+n–
resp. –ι+[–γ],k–{–γ}>–ι+,k–n+ p+n– , in the casep> ,
ι+[γ],k+{γ}–n+ p <ι
+
m+,k<
ι+[γ],k+{γ}–n+
p +
resp. –ι
+
[–γ],k–{–γ}–n+
p <ι +
m+,k<
–ι+[–γ],k–{–γ}–n+
p +
,
and in the casep= ,
ι+[γ],k+{γ}–n+ ≤ιm++,k<ι+[γ],k+{γ}–n+
resp. –ι+[–γ],k–{–γ}–n+ ≤ι+m+,k< –ι+[–γ],k–{–γ}–n+ .
If these conditions all hold, we writeγ ∈C(k,p,m,n) (resp.γ∈D(k,p,m,n)).
Letγ ∈C(k,p,m,n) (resp.γ ∈D(k,p,m,n)) andube functions on∂Cn() satisfying
Sn()
|u(t,)|p +tι+[γ],k+{γ}
dσQ<∞
resp.
Sn()
u(t,)p +tι+[–γ],k+{–γ}dσ
Q<∞
.
(.)
Forγ andu, we define the positive measureμ(resp.ν) on Rnby
dμ(Q) =
|u(t,)|pt–ι+[γ],k–{γ}dσ
Q, Q= (t,)∈Sn(; (, +∞)),
, Q∈Rn–S
n(; (, +∞))
resp.dν(Q) =
|u(t,)|ptι+[–γ],k+{–γ}
dσQ, Q= (t,)∈Sn(; (, +∞)),
, Q∈Rn–Sn(; (, +∞))
.
We remark that the total masses ofμandνare finite.
Let p> –, > , ≤ζ ≤np and μ be any positive measure on Rn having finite mass. For eachP= (r,)∈Rn–{O}, the maximal function with respect to the station-ary Schrödinger operator is defined by (see [])
M(P;μ,ζ) = sup
<ρ<r
μB(P,ρ)V(ρ)W(ρ)
p
ρζ–p.
The set
P= (r,)∈Rn–{O};M(P;μ,ζ)V(ρ)W(ρ)
–p
ρp–ζ> is denoted byE(;μ,ζ).
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Recently, Yoshida-Miyamoto (cf. [], Theorem ) gave the asymptotic behavior of
PI
(m,u)(P) at infinity on cones.
Theorem A If u is a continuous function on∂Cn()satisfying
∂Cn()
|u(t,)|
+tι+n,+mdQ<∞,
then
lim
r→∞,P=(r,)∈TnPI
(m,u)(P) =o
ι+m+,ϕ–n().
Now we have the following.
Theorem If p> –,γ ∈C(k,p,m,n) (resp. γ ∈D(k,p,m,n))and u is a measurable function on∂Cn()satisfying(.),then there exists a covering{rj,Rj}of E(;μ,ζ) (resp.
E(;ν,ζ)) (⊂Cn())satisfying
∞
j=
rj
Rj
p–ζ
Vj(Rj)
Vj(rj)
Wj(Rj)
Wj(rj)
p
<∞ (.)
such that
lim
r→∞,P=(r,)∈Cn()–E(;μ,ζ)
r
–ι+[γ],k–{γ}+n– p ϕ
ζ p–
()PIa(m,u)(P) = (.)
resp. lim
r→∞,P=(r,)∈Cn()–E(;ν,ζ)r
ι+[–γ],k+{–γ}+n– p ϕ
ζ p–
()PIa(m,u)(P) =
. (.)
Remark In the case thata= ,p= ,γ =n+mandζ=n, then (.) is a finite sum, the set E(;μ,n) is a bounded set and (.)-(.) hold inCn(). This is just the result of Theorem A.
As an application of modified Green’s function with respect to the stationary Schrö-dinger operator and Theorem , we give the solutions of the Dirichlet problem for the Schrödinger operator onCn().
Theorem If u is a continuous function on∂Cn()satisfying
Sn()
|u(t,)| +Vm+(t)tn–
dσQ<∞, (.)
then the functionPIa(m,u)(P)satisfies
PIa
(m,u)∈C
C
n()
∩CCn()
,
SchaPIa(m,u) = in Cn(),
PIa
(m,u) =u on∂Cn(),
lim
r→∞,P=(r,)∈Cn()r –ι+m+,k
ϕn–()PIa(m,u)(P) = .
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2 Lemmas
Throughout this paper, letMdenote various constants independent of the variables in questions, which may be different from line to line.
Lemma
(i) PI(;a)(P,Q)≤Mrι–,ktι+,k–ϕ
()
(ii) resp.PI(;a)(P,Q)≤Mrι+,ktι–,k–ϕ
()
for any P= (r,)∈Cn()and any Q= (t,)∈Sn()satisfying <rt ≤(resp. <rt ≤);
(iii) PI(; )(P,Q)≤Mϕ() tn– +M
rϕ() |P–Q|n
for any P= (r,)∈Cn()and any Q= (t,)∈Sn(; (r,r)).
Proof (i) and (ii) are obtained by Levin (see [], Chapter ). (iii) follows from the work of
Azarin (see [], Lemma and Remark).
Lemma (see [], p.) For a non-negative integer m,we have
PI(;a,m)(P,Q)≤M(n,m,s)Vm+(r) Wm+(t)
t ϕ() ∂ϕ()
∂n
(.)
for any P = (r,)∈Cn() and Q= (t,)∈Sn() satisfying r≤st ( <s< ), where
M(n,m,s)is a constant dependent on n,m,and s.
Lemma Let p> –andμbe any positive measure onRnhaving finite total mass.Then
E(;μ,ζ)has a covering{rj,Rj}(j= , , . . .)satisfying
∞
j=
rj
Rj
p–ζ
Vj(Rj)
Vj(rj)
Wj(Rj)
Wj(rj)
p
<∞.
Proof Set
Ej(;μ,ζ) =
P= (r,)∈E(;μ,ζ) : j≤r< j+ (j= , , , . . .). IfP= (r,)∈Ej(;μ,ζ), then there exists a positive numberρ(P) such that
ρ(P) r
p–ζ
Vj(r)
Vj(ρ(P))
Wj(r)
Wj(ρ(P))
p
∼
ρ(P) r
np–ζ
≤μ(B(P,ρ(P)))
.
Here Ej(;μ,ζ) can be covered by the union of a family of balls (B(Pj,i,ρj,i) :Pj,i ∈
Ej(;μ,ζ)) (ρj,i =ρ(Pj,i)). By the Vitali lemma (see []), there exists j⊂ Ej(;μ,ζ),
which is at most countable, such that (B(Pj,i,ρj,i) :Pj,i∈j) are disjoint andEj(;μ,ζ)⊂
Pj,i∈jB(Pj,i, ρj,i).
So
∞
j=
Ej(;μ,ζ)⊂
∞
j=
Pj,i∈j
B(Pj,i, ρj,i).
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On the other hand, note thatPj,i∈jB(Pj,i,ρj,i)⊂(P= (r,) :
j–≤r< j+), so that
Pj,i∈j
ρj,i
|Pj,i|
p–ζ
Vj(|Pj,i|)
Vj(ρj,i)
Wj(|Pj,i|)
Wj(ρj,i)
p
∼
Pj,i∈j
ρj,i
|Pj,i|
np–ζ
≤np–ζ
Pj,i∈j
μ(B(Pj,i,ρj,i))
≤ np–ζ μ
Cn
;j–, j+. Hence we obtain
∞
j=
Pj,i∈j
ρj,i
|Pj,i|
p–ζ
Vj(|Pj,i|)
Vj(ρj,i)
Wj(|Pj,i|)
Wj(ρj,i)
p
∼
∞
j=
Pj,i∈j
ρj,i
|Pj,i|
np–ζ
≤
∞
j=
μ(Cn(; [j–, j+)))
≤ μ(Rn) .
SinceE(;μ,ζ)∩ {P= (r,)∈Rn;r≥}=∞j=Ej(;μ,ζ),E(;μ,ζ) is finally covered by
a sequence of balls (B(Pj,i,ρj,i),B(P, )) (j= , , . . . ;i= , , . . .) satisfying
j,i
ρj,i
|Pj,i|
p–ζ
Vj(|Pj,i|)
Vj(ρj,i)
Wj(|Pj,i|)
Wj(ρj,i)
p
∼
j,i
ρj,i
|Pj,i|
np–ζ
≤μ(Rn) +
np–ζ < +∞,
whereB(P, ) (P= (, , . . . , )∈Rn) is the ball which covers{P= (r,)∈Rn;r< }.
3 Proof of Theorem 1
We only prove the casep> – andγ ≥, the remaining cases can be proved similarly. For any> , there existsR> such that
Sn(;(R,∞))
|u(Q)|p
+tι+[γ],k+{γ}
dσQ<. (.)
The relationG(;a)(P,Q)≤G(; )(P,Q) implies the inequality (see [])
PI(;a)(P,Q)≤PI(; )(P,Q). (.)
For <s<and any fixed pointP= (r,)∈Cn() –E(;μ,ζ) satisfyingr>R, letI= Sn(; (, )),I=Sn(; [,R]),I=Sn(; (R,r]),I=Sn(; (r,r)),I=Sn(; [r,rs)),
I=Sn(; [rs,∞)) andI=Sn(; [,rs)), we write
PIa
(m,u)(P) =
i=
Ii
PI(;a,m)(P,Q)u(Q)dσQ
=
i=
Ii
PI(;a)(P,Q)u(Q)dσQ
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–
I
∂K(;a,m)(P,Q) ∂nQ
u(Q)dσQ
+
I
PI(;a,m)(P,Q)u(Q)dσQ,
which yields
PIa
(m,u)(P)≤
i= Ui(P),
where
Ui(P) =
Ii
PI(;a)(P,Q)u(Q)dσQ (i= , , , , ),
U(P) =
I
PI(;a,m)(P,Q)u(Q)dσQ,
U(P) =
I
∂K(;a∂n,mQ)(P,Q)u(Q)dσQ.
Ifι+
[γ],k+{γ}> (–ι+,k–n+ )p+n– , then (ι+,k– +
ι+[γ],k+{γ}
p )q+n– > . By (.), (.),
Lemma (i), and Hölder’s inequality, we have the following growth estimates:
U(P)≤Mrι
– ,kϕ
()
I
tι+,k–u(Q)dσ
Q
≤Mrι–,kϕ
()
I
|u(Q)|p tι+[γ],k+{γ}
dσQ
p
I
t(ι+,k–+ ι+
[γ],k+{γ} p )qdσ
Q
q
≤Mrι–,kRι + ,k+n–+
ι+[ γ],k+{γ}–n+
p
ϕ(), (.)
U(P)≤Mrι
–
,kϕ(), (.)
U(P)≤Mr
ι+[γ],k+{γ}–n+ p ϕ
(). (.)
Ifι+m+,k>ι
+
[γ],k+{γ}–n+
p , then (ι
– ,k– +
ι+[γ],k+{γ}
p )q+n– < . We obtain by (.), Lemma (ii),
and Hölder’s inequality
U(P)≤Mrι
+ ,kϕ
()
Sn(;[r,∞))
tι–,k–u(Q)dσ
Q
≤Mrι+,kϕ
()
Sn(;[r,∞)) |u(Q)|p
tι+[γ],k+{γ}
dσQ
p
×
Sn(;[r,∞)) t(ι–,k–+
ι+[ γ],k+{γ}
p )qdσ
Q
q
≤Mr
ι+[γ],k+{γ}–n+ p ϕ
(). (.)
By (.) and Lemma (iii), we consider the inequality
U(P)≤U(P) +U(P),
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where
U(P) =Mϕ()
I
t–nu(Q)dσQ, U(P) =Mrϕ()
I
|u(Q)|
|P–Q|ndσQ.
We first have
U(P) =Mϕ()
I
tι+,k+ι–,k–u(Q)dσQ
≤Mrι+,kϕ
()
Sn(;(r,∞))
tι–,k–u(Q)dσ
Q
≤Mr
ι+[
γ],k+{γ}–n+ p ϕ
(), (.)
which is similar to the estimate ofU(P).
Next, we shall estimateU(P).
Take a sufficiently small positive numbercsuch thatI⊂B(P,r) for anyP= (r,)∈ (c) (see []), where
(c) =
P= (r,)∈Cn(); inf
z∈∂(,) – (,z)<c, <r<∞
,
and divideCn() into two sets(c) andCn() –(c).
IfP= (r,)∈Cn() –(c), then there exists a positivecsuch that|P–Q| ≥crfor any
Q∈Sn(), and hence
U(P)≤Mϕ()
I
t–nu(Q)dσQ
≤Mr
ι+[γ],k+{γ}–n+ p ϕ
(), (.)
which is similar to the estimate ofU(P).
We shall consider the caseP= (r,)∈(c). Now put
Hi(P) =
Q∈I; i–δ(P)≤ |P–Q|< iδ(P)
,
whereδ(P) =infQ∈∂Cn()|P–Q|.
SinceSn()∩ {Q∈Rn:|P–Q|<δ(P)}=∅, we have
U(P) =M
i(P)
i=
Hi(P)
rϕ() | u(Q)|
|P–Q|ndσQ,
wherei(P) is a positive integer satisfying i(P)–δ(P)≤r
< i(P)δ(P).
Sincerϕ()≤Mδ(P) (P= (r,)∈Cn()), similar to the estimate ofU(P), we obtain
Hi(P)
rϕ() |u(Q)|
|P–Q|ndσQ
≤(–i)nϕ()δ(P)
ζ–np p
Hi(P)
δ(P)ζp–nu(Q)dσ
Q
RETRACTED
≤Mϕ–
ζ p
()δ(P)
ζ–np p
Hi(P)
r–ζpu(Q)dσ
Q
≤Mrn–ζpϕ– ζ p
()δ(P)
ζ–np p
Hi(P)
t–nu(Q)dσQ
≤Mr
ι+
[γ],k+{γ}–n–ζ+ p +nϕ–
ζ p
()
μ(Hi(P))
(iδ(P))ζ
p
fori= , , , . . . ,i(P).
SinceP= (r,) /∈E(;μ,ζ), we have
μ(Hi(P))
{iδ(P)}np–ζ μ
BP, iδ(P)V
iδ(P)W
iδ(P)piδ(P)ζ–p
M(P;μ,ζ)
≤V(r)W(r)
p
rζ–p
≤rζ–np i= , , , . . . ,i(P) –
and
μ(Hi(P)(P)) {iδ(P)}np–ζ μ
B
P,r
V
r
W
r
p
r
ζ–p
≤rζ–np.
So
U(P)≤Mr
ι+
[γ],k+{γ}–n+ p ϕ–
ζ p
(). (.)
We only considerU(P) in the casem≥, sinceU(P)≡ form= . By the definition
ofK(;a,m), (.), and Lemma , we see (see [])
U(P)≤ M χ()
m
j=
jn–qj(r),
where
qj(r) =Vj(r)ϕ()
I
Wj(t)|u(Q)|
t dσQ. To estimateqj(r), we write
qj(r)≤qj(r) +qj(r),
where
qj(r) =Vj(r)ϕ()
I
Wj(t)|u(Q)|
t dσQ, qj(r) =Vj(r)ϕ()
Sn(;(R,rs))
Wj(t)|u(Q)|
t dσQ.
RETRACTED
Ifι+m+,k<ι
+
[γ],k+{γ}–n+
p + , then (–ι
+
m+,k–n+ +
ι+[ γ],k+{γ}
p )q+n– > . Notice that
Vj(r)
Vm+(t) Vj(t)t
≤MVm+(r) r ≤Mr
ι+m+,k–
t≥,R<
r s
.
Thus, by (.), (.), and Hölder’s inequality we conclude
qj(r) =Vj(r)ϕ()
I
|u(Q)| Vj(t)tn–
dσQ
≤MVj(r)ϕ()
I
Vm+(t) tι+m+,k
|u(Q)| Vj(t)tn–
dσQ
≤rι+m+,k–ϕ
()
I
|u(Q)|p
tι+[γ],k+{γ}
dσQ
p
I
t(–ι+m+,k–n++ ι+[γ],k+{γ}
p )qdσ
Q
q
≤Mrι+m+,k–R–ι + m+,k++
ι+[γ],k+{γ}–n+ p
ϕ().
Analogous to the estimate ofqj(r), we have
qj(r)≤Mr
ι+[γ],k+{γ}–n+ p ϕ
().
Thus we can conclude that
qj(r)≤Mr
ι+[γ],k+{γ}–n+ p ϕ
(),
which yields
U(P)≤Mr
ι+[ γ],k+{γ}–n+
p ϕ
(). (.)
Ifι+m+,k>ι
+
[γ],k+{γ}–n+
p , then (–ι
+
m+,k–n+ +
ι+[γ],k+{γ}
p )q+n– < . By (.), Lemma , and
Hölder’s inequality we have
U(P)≤MVm+(r)ϕ()
I
|u(Q)| Vm+(t)tn–
dσQ
≤MVm+(r)ϕ()
I
|u(Q)|p
tι+[γ],k+{γ}
dσQ
p
I
t(–ι+m+,k–n++ ι+
[γ],k+{γ} p )qdσ
Q
q
≤Mr
ι+[γ],k+{γ}–n+ p ϕ
(). (.)
Combining (.)-(.), we see that ifR is sufficiently large and is sufficiently small,
then
PIa
(m,u)(P) =o
r
ι+ [γ],k+{γ}–n+
p ϕ– ζ p
()
asr→ ∞, whereP= (r,)∈Cn(; (R, +∞)) –E(;μ,ζ). Finally, there exists an additional
finite ballBcoveringCn(; (,R]), which, together with Lemma , gives the conclusion
of Theorem .
RETRACTED
4 Proof of Theorem 2
For any fixedP= (r,)∈Cn(), take a number satisfyingR>max(,rs) ( <s<).
By (.) and Lemma , we have
Sn(;(R,∞))
PI(;a,m)(P,Q)u(Q)dσQ
≤Vm+(r)ϕ()
Sn(;(R,∞))
|u(Q)| Vm+(t)tn–
dσQ
≤MVm+(r)ϕ()
<∞.
ThenPIa(m,u)(P) is absolutely convergent and finite for anyP∈Cn(). ThusPIa(m,
u)(P) is a generalized harmonic function onCn().
Now we study the boundary behavior ofPIa(m,u)(P). LetQ= (t,)∈∂Cn() be any
fixed point andlbe any positive number satisfyingl>max(t+ ,R).
SetχS(l), the characteristic function ofS(l) ={Q= (t,)∈∂Cn(),t≤l}and write
PIa
(m,u)(P) =
Sn(;(,))
+
Sn(;[,l])
+
Sn(;(l,∞))
PI(;a,m)(P,Q)u(Q)dσQ
=U(P) –U(P) +U(P),
where
U(P) =
Sn(;(,l])
PI(;a)(P,Q)u(Q)dσQ,
U(P) =
Sn(;[,l])
∂K(;a,m)(P,Q) ∂nQ
u(Q)dσQ,
U(P) =
Sn(;(l,∞))
PI(;a,m)(P,Q)u(Q)dσQ.
Notice thatU(P) is the Poissona-integral ofu(Q)χS(
l), we havelimP→Q,P∈Cn()U (P) =
u(Q). Sincelim→ϕjv() = (j= , , . . . ; ≤v≤vj) as P= (r,)→Q= (t,)∈
Sn(), we have limP→Q,P∈Cn()U(P) = from the definition of the kernel function K(;a,m)(P,Q).U(P) =O(Vm+(r)ϕ()) and therefore it tends to zero.
So the functionPIa(m,u)(P) can be continuously extended toCn() such that
lim
P→Q,P∈Cn()PI
a
(m,u)(P) =u
Q
for anyQ= (t,)∈∂Cn() from the arbitrariness ofl, which with Theorem gives the
conclusion of Theorem .
Competing interests
The author declares that there is no conflict of interests regarding the publication of this article.
Acknowledgements
The author is thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper.
Received: 29 May 2015 Accepted: 26 November 2015
RETRACTED
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