# Modified Poisson Integral and Green Potential on a Half-Space

Authors: Lei Qiao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We discuss the behavior at infinity of modified Poisson integral and Green potential on a half-space of then-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.


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## Body

## 1. Introduction and Main Results

LetRn(n≥2) denote the n-dimensional Euclidean space with points x=(x1,x2,…,xn-1,xn)=(x′,xn), where x′∈Rn-1 and xn∈R. The boundary and closure of an open Ω of Rn are denoted by ∂Ω and Ω¯, respectively. The upper half-space is the set H={x=(x′,xn)∈Rn:xn>0}, whose boundary is ∂H. We identify Rn with Rn-1×R and Rn-1 with Rn-1×{0}, writing typical points x,y∈Rn as x=(x′,xn),y=(y′,yn), where x′=(x1,x2,…,xn-1),y′=(y1,y2,…,yn-1)∈Rn-1 and putting x·y=∑j=1n‍xjyj=x′·y′+xnyn,|x|=x·x,|x′|=x′·x′.Forx∈Rn and r>0, let Bn(x,r) denote the open ball with center at x and radius r in Rn.Set(1.1)Eα(x)={-log⁡|x|ifα=n=2,|x|α-nif0<α<n.LetGα(x,y) be the green function of order α for H, that is,
(1.2)Gα(x,y)=Eα(x-y)-Eα(x-y*)x,y∈H¯,x≠y,0<α≤n,
where * denotes reflection in the boundary plane ∂H just as y*=(y1,y2,…,yn-1,-yn).We define the Poisson kernelPα(x,y′) when x∈H and y′∈∂H by
(1.3)Pα(x,y′)=∂Gα(x,y)∂yn|yn=0=Cαxn|x-y′|n-α+2,
where Cα=2(n-α) if 0<α<n and =2 if α=n=2.  It has the expansion
(1.4)P(x,y′)=∑k=0∞‍Cαxn|x|k|y′|n-α+2+kCk(n-α+2)/2(x⋅y′|x||y′|),
where Ck(n-α+2)/2(t) is a Gegenbauer polynomial [1]. The series converges for |y′|>|x|. Each term in the series is a harmonic function of x and vanishes on ∂H.In caseα=n=2, we consider the modified kernel function defined by
(1.5)En,m(x-y)={En(x-y)if|y|<1,En(x-y)+R(log⁡y-∑k=1m-1(xkkyk))if|y|≥1.In case0<α<n, we define
(1.6)Eα,m(x-y)={Eα(x-y)if|y|<1,Eα(x-y)-∑k=0m-1|x|k|y|n-α+kCk(n-α)/2(x⋅y|x||y|)if|y|≥1,
where m is a nonnegative integer and Ckω(t)(ω=(n-α)/2) is also the Gegenbauer polynomials. The expression arises from the generating function for Gegenbauer polynomials
(1.7)(1-2tr+r2)-ω=∑k=0∞‍Ckω(t)rk,
where |r|<1, |t|≤1 and ω>0. Each coefficient Ckω(t) is called the Gegenbauer polynomial of degree k associated with ω, the function Ckω(t) is a polynomial of degree k in t.Then we define the modified Poisson kernelPα,m(x,y′) and Green function Gα,m(x,y) respectively by
(1.8)Pα,m(x,y′)={Pα(x,y′)if|y′|<1,Pα(x,y′)-∑k=0m-1Cαxn|x|k|y′|n-α+2+kCk(n-α+2)/2(x⋅y′|x||y′|)if|y′|≥1;(1.9)Gα,m(x,y)={En,m+1(x-y)-En,m+1(x-y*)ifα=n=2,Eα,m+1(x-y)-Eα,m+1(x-y*)if0<α<n,
where x,y∈H¯ and x≠y.  We remark that the new kernel Pα,m(x,y′) will be of order |y′|-(n+m) as |y′|→∞.Write(1.10)Uα,m(x,ν)=∫∂HPα,m(x,y′)dν(y′),Gα,m(x,μ)=∫HGα,m(x,y)dμ(y),
where ν(resp. μ) is a nonnegative measure on ∂H(resp. H). Here note that U2,0(x,ν)(resp. G2,0(x,μ)) is nothing but the general Poisson integral (resp. Green potential).Letk be a nonnegative Borel measurable function on Rn×Rn, and set
(1.11)k(y,μ)=∫Ek(y,x)dμ(x),k(μ,x)=∫Ek(y,x)dμ(y),
for a nonnegative measure μ on a Borel set E⊂Rn. We define a capacity Ck by
(1.12)Ck(E)=sup⁡μ(Rn),E⊂H,
where the supremum is taken over all nonnegative measures μ such that Sμ(the support of μ) is contained in E and k(y,μ)≤1 for every y∈H.Forβ≤1 and δ≤1, we consider the function kα,β,δ defined by
(1.13)kα,β,δ(y,x)=xn-βyn-δGα(x,y)forx,y∈H.
If β=δ=1, then kα=kα,1,1 is extended to be continuous on H¯×H¯ in the extended sense, where H¯=H∪∂H.Recently, Siegel-Talvila [2] proved the following result.Theorem A.
Letf be a measurable function on Rn-1 satisfying ∫Rn-1‍(|f(y′)|/(1+|y′|)(n+m))dy′<∞. Then the function v(x)=∫Rn-1‍P2,m(x,y′)f(y′)dy′ satisfies
(1.14)v∈C2(H)∩C0(H¯),Δv=0,x∈H,lim⁡x→x′⁡v(x)=f(x′)nontangentiallya⋅ex′∈∂H,v(x)=o(xn1-n|x|m+n)as|x|⟶∞,x∈H.
Our first aim is to establish the following theorem.Theorem 1.1.
Letγ>0 and α+γ-n-2≤m<α+γ-n-1. If ν is a nonnegative measure on ∂H satisfying
(1.15)∫∂H1(1+|y′|)γdν(y′)<∞,
then there exists a Borel set E⊂H with properties: (1)
lim|x|→∞,x∈H-Exn-β|x|n-α+β-γ+1Uα,m(x,ν)=0;(2)
∑i=1∞2-i(n-α+β+1)Ckα,β,1(Ei)<∞,where Ei={x∈E:2i≤|x|<2i+1}.Theorem 1.2.
Letγ>δ-1 and α+γ-n-δ-1≤m<α+γ-n-δ. If μ is a nonnegative measure on H satisfying
(1.16)∫Hynδ(1+|y|)γdμ(y)<∞,
then there exists a Borel set F⊂H with properties:(1)
lim|x|→∞,x∈H-Fxn-β|x|n-α+β-γ+δGα,m(x,μ)=0;(2)
∑i=1∞2-i(n-α+β+δ)Ckα,β,δ(Fi)<∞,where Fi={x∈F:2i≤|x|<2i+1}.Remark 1.3.
Ifδ=1, then F=E.
Next we generalize Theorem A to the modifiedα-potentials on H, which is defined by
(1.17)Rα(x)=Uα,m(x,ν)+Gα,m(x,μ),
where 0<α≤n and ν(resp. μ) is a nonnegative measure on ∂H(resp. H) satisfying (1.15) (resp. (1.16) (δ=1)). Clearly, R2(x) is a superharmonic function.The following theorem follows readily from Theorems1.1 and 1.2.Theorem 1.4.
LetRα(x) be defined by (1.17). Then there exists a Borel set E⊂H satisfying Theorem 1.1(2) such that
(1.18)lim⁡|x|→∞,x∈H-Exn-β|x|n-α+β-γ+1Rα(x)=0.Remark 1.5.
In the case0≤β≤1, by using Lemma 2.5 below, we can easily show that Ck2,β,1=λβ in the notation of [3]. Thus, Theorem 1.1(2) with α=2 means that E is β-rarefied at infinity in the sense of [3]. In particular, this condition with α=2, β=1(resp. α=2, β=0) means that E is minimally thin at infinity (resp. rarefied at infinity) in the sense of [4].Finally we are concerned with the best possibility of Theorem1.4 as to the size of the exceptional set.Theorem 1.6.
Letγ>0, E⊂H be a Borel set satisfying Theorem 1.1(2) and let Rα(x) be defined by (1.17). Then one can find a nonnegative measure λ defined on H¯ satisfying
(1.19)∫H¯1(1+|y|)γdλ(y)<∞,
such that
(1.20)limsup⁡|x|→∞,x∈Exn-β|x|n-α+β-γ+1Rα(x)=∞,
where dλ(y′)=dν(y′)(y′∈∂H) and dλ(y)=yndμ(y)(y∈H).


## 2. Some Lemmas

Throughout this paper, letM denote various constants independent of the variables in questions, which may be different from line to line.Lemma 2.1 (see [5]).
Letm be a nonnegative integer and M>0.(1)
If1≤|y′|≤|x|/2, then |Pα,m(x,y′)|≤M((xn|x|m-1)/|y′|n+m-α+1).(2)
If|y′|≥2|x| and |y′|≥1, then |Pα,m(x,y′)|≤M((xn|x|m)/|y′|n+m-α+2).Lemma 2.2.
There exists a positive constantM such that Gα(x,y)≤M((xnyn)/|x-y|n-α+2), where 0<α≤n,x=(x1,x2,…,xn) and y=(y1,y2,…,yn) in H.This can be proved by simple calculation.Lemma 2.3.
Gegenbauer polynomials have the following properties:(1)
|Ckω(t)|≤Ckω(1)=Γ(2ω+k)/((Γ(2ω)Γ(k+1)),|t|≤1;(2)
(d/dt)Ckω(t)=2ωCk-1ω+1(t),k≥1;(3)
∑k=0∞‍Ckω(1)rk=(1-r)-2ω;(4)
|Ck(n-α)/2(t)-Ck(n-α)/2(t*)|≤(n-α)Ck-1(n-α+2)/2(1)|t-t*|,|t|≤1,|t*|≤1.Proof.
(1) and (2) can be derived from [1]. (3) follows by taking t=1 in (1.7); (4) follows by (1), (2) and the Mean Value Theorem for Derivatives.Lemma 2.4.
Forx, y∈Rn(α=n=2), one has the following properties:(1)
|ℑ∑k=0m‍(xk/yk+1)|≤∑k=0m-1‍((2kxn|x|k)/|y|k+2);(2)
|ℑ∑k=0∞‍(xk+m+1/yk)|≤2m+1xn|x|m;(3)
|Gn,m(x,y)-Gn(x,y)|≤M∑k=1m‍((kxnyn|x|k-1)/|y|(k+1));(4)
|Gn,m(x,y)|≤M∑k=m+1∞‍((kxnynxk-1)/|y|(k+1)).The following lemma can be proved by using Fuglede (see [6], Théorèm 7.8).Lemma 2.5.
For any Borel setE in H, one has Ckα,β,1(E)=C^kα,β,1(E) and
(2.1)C^kα,β,δ(E)=inf⁡λ(H)(resp.inf⁡λ(H¯))ifδ<1(resp.δ=1),
where the infimum is taken over all nonnegative measures λ on H(resp. H¯) such that kα,β,δ(λ,x)≥1 for every x∈E.


## 3. Proof of Theorems

Proof of Theorem1.1.
For anyε1>0, there exists Rε1>2 such that
(3.1)∫{y′∈∂H,|y′|≥Rε1}1(1+|y′|)γdν(y′)<ϵ1.
For fixed x∈H and |x|≥2Rε1, we write
(3.2)Uα,m(x,ν)=∫G1‍Pα,m(x,y′)dν(y′)+∫G2‍Pα,m(x,y′)dν(y′)+∫G3‍[Pα,m(x,y′)-Pα(x,y′)]dν(y′)+∫G4‍Pα(x,y′)dν(y′)+∫G5‍Pα,m(x,y′)dν(y′)=U1(x)+U2(x)+U3(x)+U4(x)+U5(x),
where
(3.3)G1={y′∈∂H:|y′|<1},G2={y′∈∂H:1≤|y′|<|x|2},G3=G4={y′∈∂H:|x|2≤|y′|<2|x|},G5={y′∈∂H:|y′|≥2|x|}.
First note that
(3.4)|U1(x)|≤xn(|x|2)α-n-2∫G1‍dν(y′)≤Mxn|x|α-n-2.
By Lemma 2.1(1), we have
(3.5)|U2(x)|≤Mxn|x|m-1∫G21|y′|n+m-α+1dν(y′)≤Mxn|x|m-1∫G2|y′|α+γ-n-m-11|y′|γdν(y′)≤Mxn|x|α+γ-n-2∫G21|y′|γdν(y′).
Write(3.6)U2(x)=U21(x)+U22(x),
where
(3.7)U21(x)=∫G2⋂‍Bn-1(0,Rϵ1)Pα,m(x,y′)dν(y′),U22(x)=∫G2-Bn-1(0,Rϵ1)Pα,m(x,y′)dν(y′).
If|x|>2Rϵ1,  then we have
(3.8)|U21(x)|≤MRϵ1α+γ-n-m-1xn|x|m-1,|U22(x)|≤Mϵ1xn|x|α+γ-n-2
from (3.5).
So(3.9)|U2(x)|≤Mϵ1xnβ|x|α+γ-n-β-1.
We have by Lemma2.3(3)
(3.10)|U3(x)|≤M∫G3‍∑k=0m-1‍xn|x|k|y′|n-α+2+kCk(n-α+2)/2(1)dν(y′)≤Mxn|x|m∑k=0m-1‍12kCk(n-α+2)/2(1)∫G3|y′|α+γ-n-m-21|y′|γdν(y′)≤Mε1xn|x|α+γ-n-2.
By Lemma2.1(2), we obtain
(3.11)|U5(x)|≤Mxn|x|m∫G5|y′|α+γ-n-m-21|y′|γdν(y′)≤Mϵ1xn|x|α+γ-n-2.
Note thatU4(x)=xnβ∫G4‍kα,β,1(y′,x)dν(y′). In view of (1.15), we can find a sequence {ai} of positive numbers such that limi→∞ai=∞ and ∑i=1∞‍aibi<∞, where
(3.12)bi=∫{y′∈∂H:2i-1<|y′|<2i+2}1|y′|γdν(y′).
Consider the sets(3.13)Ei={x∈H:2i≤|x|<2i+1,xn-βU4(x)≥ai-12-i(n-α+β-γ+1)},
for i=1,2,…. If μ is a nonnegative measure on H such that Sμ⊂Ei and kα,β,1(y′,μ)≤1 for y′∈∂H, then we have
(3.14)∫Hdμ≤ai2i(n-α+β-γ+1)∫xn-βU4(x)dμ(X)≤Mai2i(n-α+β-γ+1)∫{y′∈∂H:2i-1<|y′|<2i+2}kα,β,1(y′,μ)dν(y′)≤Mai2i(n-α+β-γ+1)∫{y′∈∂H:2i-1<|y′|<2i+2}dν(y′)≤M2i(n-α+β+1)aibi
so that
(3.15)Ckα,β,1(Ei)≤M2i(n-α+β+1)aibi
which yields
(3.16)∑i=1∞‍2-i(n-α+β+1)Ckα,β,1(Ei)<∞.
Setting E=⋃i=1∞‍Ei, we see that Theorem 1.1(2) is satisfied and
(3.17)limsup⁡|x|→∞,x∈H-Exn-β|x|n-α+β-γ+1U4(x)≤limsup⁡i→∞2|n-α+β-γ+1|ai-1=0.
Combining (3.4) and (3.9)–(3.17), Theorem 1.1(1) holds.
Then we complete the proof of Theorem1.1.Proof of Theorem1.2.
For anyε2>0, there exists Rε2>2 such that
(3.18)∫{y∈H,|y|≥Rε2}ynδ(1+|y|)γdμ(y)<ϵ2.
For fixed x∈H and |x|≥2Rε2, we write
(3.19)Gα,m(x,μ)=∫H1Gα(x,y)dμ(y)+∫H2Gα(x,y)dμ(y)+∫H3[Gα,m(x,y)-Gα(x,y)]dμ(y)+∫H4Gα,m(x,y)dμ(y)+∫H5‍Gα(x,y)dμ(y)+∫H6‍[Gα,m(x,y)-Gα(x,y)]dμ(y)+∫H7Gα,m(x,y)dμ(y)=V1(x)+V2(x)+V3(x)+V4(x)+V5(x)+V6(x)+V7(x),
where
(3.20)H1={y∈H:|y|≥Rϵ2,|x-y|≤|x|2},H2={y∈H:|y|≥Rϵ2,|x|2<|x-y|≤3|x|},H3={y∈H:|y|≥Rϵ2,|x-y|≤3|x|},H4={y∈H:|y|≥Rϵ2,|x-y|>3|x|},H5=H6{y∈H:1≤|y|<Rε2},H7={y∈H:|y|<1}.
We distinguish the following two cases.
Case  1.
0
<
α
<
n.
Note that(3.21)V1(x)=xnβ∫H1kα,β,δ(y,x)dλ(y),
where dλ(y)=ynδdμ(y).
By the lower semicontinuity ofkα,β,δ(y,x), then we can prove the following fact in the same way as U4(x) in the proof of Theorem 1.1:
(3.22)lim⁡|x|→∞,x∈H-Fxn-β|x|n-α+β-γ+δV1(x)=0,
where F=⋃i=1∞‍Fi,Fi={x∈F:2i≤|x|<2i+1} and ∑i=1∞‍2-i(n-α+β+δ)Ckα,β,δ(Fi)<∞.
Moreover, by Lemma2.2(3.23)|V2(x)|≤Mxn∫H2‍yn|x-y|n-α+2dμ(y)≤Mxn|x|α-n-2∫H2|y|γ-δ+1ynδ|y|γdμ(y)≤Mϵ2xn|x|α+γ-n-δ-1.
Note thatC0ω(t)≡1. By (3) and (4) in Lemma 2.3, we take t=x·y/|x||y|, t*=x·y*/|x||y*| in Lemma 2.3(4) and obtain
(3.24)V3(x)|≤∫H3∑k=1m‍|x|k|y|n-α+k2(n-α)Ck-1(n-α+2)/2(1)xnyn|x||y|dμ(y)≤Mxn|x|m-1∑k=1m14k-1Ck-1(n-α+2)/2(1)∫H3|y|α+γ-n-m-δynδ|y|γdμ(y)≤Mε2xn|x|α+γ-n-δ-1.
Similarly, we have by (3) and (4) in Lemma2.3(3.25)|V4(x)|≤∫H4∑k=m+1∞‍|x|k|y|n-α+k2(n-α)Ck-1(n-α+2)/2(1)xnyn|x||y|dμ(y)≤Mxn|x|m∑k=m+1∞‍12k-1Ck-1(n-α+2)/2(1)∫H4|y|α+γ-n-m-δ-1ynδ|y|γdμ(y)≤Mε2xn|x|α+γ-n-δ-1.
By Lemma2.2, we get
(3.26)|V5(x)|≤Mxn|x|m-1∫H5yn|x-y|n-α+2dμ(y)|V5(x)|≤Mxn|x|α-n-2∫H2|y|γ-δ+1ynδ|y|γdμ(y)|V5(x)|≤MRϵ2γ-δ+1xn|x|α-n-2,|V7(x)|≤Mxn|x|α-n-2.
Similar to the estimate ofV3(x), we obtain
(3.27)|V6(x)|≤∫H6∑k=1m‍|x|k|y|n-α+k2(n-α)Ck-1(n-α+2)/2(1)xnyn|x||y|dμ(y)≤Mxn∑k=1m‍Ck-1(n-α+2)/2(1)xk-1Rεm-k∫H6|y|α+γ-n-m-δynδ|y|γdμ(y)≤MRϵ2α+γ-n-δ-1xn|x|m-1.
Combining (3.22)–(3.27), we see that Theorem 1.2(1) holds. Then we prove Case 1.
Case  2. α=n=2.
Since the estimates ofV1(x), V2(x), V5(x) and V7(x) are similar to those of Case 1, we omit them. (3.22), (3.23), and (3.26) still hold in Case 2.
Moreover, by Lemma2.4(3), we find
(3.28)|V3(x)|≤M∫H3∑k=1m‍kxnyn|x|k-1|y|k+1dμ(y)≤Mxn|x|m-1∑k=1mk4k-1∫H3|y|γ-m-δynδ|y|γdμ(y)≤Mε2xn|x|γ-δ-1.
By Lemma2.4(4), we have
(3.29)|V4(x)|≤M∫H4∑k=m+1∞‍kxnyn|x|k-1|y|k+1dμ(y)≤Mxn|x|m∑k=m+1∞‍k2k-1∫H4|y|γ-m-δ-1ynδ|y|γdμ(y)≤Mε2xn|x|γ-δ-1.
Similar to the estimate of V3(x), we have
(3.30)|V6(x)|≤MRϵ2γ-δ-1xn|x|m-1.
Combining (3.22), (3.23), (3.26) and (3.28)–(3.30), we see that Theorem 1.2(1) holds. Then we prove Case 2.
Hence we complete the proof of Theorem1.2.Proof of Theorem1.6.
We prove the case0<α<n, because the case α=n=2 can be proved similarly. Further, we only need prove
(3.31)lim⁡sup⁡|x|→∞,x∈Exn1-β|x|n-α+β-γ+1kα(λ,x)=∞.
By Lemma 2.5, for each i we can find λi on H¯ such that λi(H¯)<Ckα,β,1(Ei)+1 and kα,β,1(λi,x)≥1 on Ei. Denote by λi′ the restriction of λi to the set {y∈H¯:2i-1<|y|<2i+2}.
Setλ=∑i=1∞‍ai2-i(n-α+β-γ+1)λi′, where {ai} is a sequence of positive numbers such that limi→∞ai=∞ but ∑i=1∞‍ai2-i(n-α+β+1){Ckα,β,1(Ei)+1}<∞. Then
(3.32)∫H¯‍1(1+|y|)γdλ(y)=∑i=1∞‍ai2-i(n-α+β-γ+1)∫H¯‍1(1+|y|)γdλi′(y)≤M∑i=1∞‍ai2-i(n-α+β-γ+1){Ckα,β,1(Ei)+1}<∞.
If x∈Ei, then
(3.33)kα,β,1(λi′,x)≥1-∫{y∈H¯:|y|≤2i-1}∩{y∈H¯:|y|≥2i+2}kα,β,1(y,x)dλi(y)≥1-M2-i(n-α+β+1){Ckα,β,1(Ei)+1}.
We also have(3.34)xn1-β|x|n-α+β-γ+1kα(λ,x)=|x|n-α+β-γ+1kα,β,1(λ,x)≥aikα,β,1(λi′,x),
which implies that
(3.35)limsup⁡|x|→∞,x∈Exn1-β|x|n-α+β-γ+1kα(λ,x)=∞.
Thusλ satisfies all the conditions in the Theorem 1.6.



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