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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 825240, 9 pages
http://dx.doi.org/10.1155/2012/825240
Research Article

Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone

Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China

Received 17 May 2012; Accepted 8 July 2012

Academic Editor: DachunΒ Yang

Copyright Β© 2012 Lei Qiao. This 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.

Abstract

Our aim in this paper is to deal with non-tangential limits for modified Poisson integrals of boundary functions in a cone, which generalized results obtained by Brundin and Mizuta-Shimomura.

1. Introduction and Main Results

Let 𝐑 and 𝐑+ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by 𝐑𝑛(𝑛β‰₯2) the 𝑛-dimensional Euclidean space. A point in 𝐑𝑛 is denoted by 𝑃=(𝑋,π‘₯𝑛), where 𝑋=(π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1). The Euclidean distance of two points 𝑃 and 𝑄 in 𝐑𝑛 is denoted by |π‘ƒβˆ’π‘„|. Also |π‘ƒβˆ’π‘‚| with the origin 𝑂 of 𝐑𝑛 is simply denoted by |𝑃|. The boundary, the closure, and the complement of a set 𝐒 in 𝐑𝑛 are denoted by πœ•π’, 𝐒, and 𝐒𝑐, respectively.

We introduce a system of spherical coordinates (π‘Ÿ,Θ), β€‰Ξ˜=(πœƒ1,πœƒ2,…,πœƒπ‘›βˆ’1), in 𝐑𝑛 which are related to cartesian coordinates (π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1,π‘₯𝑛) by π‘₯𝑛=π‘Ÿcosπœƒ1.

For positive functions β„Ž1 and β„Ž2, we say that β„Ž1β‰²β„Ž2 if β„Ž1β‰€π‘€β„Ž2 for some constant 𝑀>0. If β„Ž1β‰²β„Ž2 and β„Ž2β‰²β„Ž1, we say that β„Ž1β‰ˆβ„Ž2.

For π‘ƒβˆˆπ‘π‘› and 𝑅>0, let 𝐡(𝑃,𝑅) denote the open ball with center at 𝑃 and radius 𝑅 in 𝐑𝑛. The unit sphere and the upper half unit sphere are denoted by π’π‘›βˆ’1 and 𝐒+π‘›βˆ’1, respectively. For simplicity, a point (1,Θ) on π’π‘›βˆ’1 and the set {Θ;(1,Θ)∈Ω} for a set Ξ©, Ξ©βŠ‚π’π‘›βˆ’1 are often identified with Θ and Ξ©, respectively. For two sets ΞžβŠ‚π‘+ and Ξ©βŠ‚π’π‘›βˆ’1, the set {(π‘Ÿ,Θ)βˆˆπ‘π‘›;π‘ŸβˆˆΞž,(1,Θ)∈Ω} in 𝐑𝑛 is simply denoted by ΞžΓ—Ξ©. In particular, the half space 𝐑+×𝐒+π‘›βˆ’1={(𝑋,π‘₯𝑛)βˆˆπ‘π‘›;π‘₯𝑛>0} will be denoted by 𝐓𝑛.

By 𝐢𝑛(Ξ©), we denote the set 𝐑+Γ—Ξ© in 𝐑𝑛 with the domain Ξ© on π’π‘›βˆ’1. We call it a cone. Then 𝑇𝑛 is a special cone obtained by putting Ξ©=𝐒+π‘›βˆ’1. We denote the sets 𝐼×Ω and πΌΓ—πœ•Ξ© with an interval on 𝐑 by 𝐢𝑛(Ξ©;𝐼) and 𝑆𝑛(Ξ©;𝐼). By 𝑆𝑛(Ξ©) we denote 𝑆𝑛(Ξ©;(0,+∞)) which is πœ•πΆπ‘›(Ξ©)βˆ’{𝑂}.

Let Ξ© be a domain on π’π‘›βˆ’1 with smooth boundary. Consider the Dirichlet problem: Λ𝑛+πœ†πœ‘=0onΞ©,πœ‘=0onπœ•Ξ©,(1.1) where Λ𝑛 is the spherical part of the Laplace operator Ξ”π‘›βˆΆΞ”π‘›=π‘›βˆ’1π‘Ÿπœ•+πœ•πœ•π‘Ÿ2πœ•π‘Ÿ2+Ξ›π‘›π‘Ÿ2.(1.2) We denote the least positive eigenvalue of this boundary value problem by πœ†Ξ© and the normalized positive eigenfunction corresponding to πœ†Ξ© by πœ‘Ξ©(Θ), ξ€œΞ©πœ‘2Ξ©(Θ)π‘‘πœŽΞ˜=1,(1.3) where π‘‘πœŽΞ˜ is the surface area on π‘†π‘›βˆ’1. We denote the solutions of the equation 𝑑2+(π‘›βˆ’2)π‘‘βˆ’πœ†Ξ©=0 by 𝛼Ω,βˆ’π›½Ξ© (𝛼Ω,𝛽Ω>0). If Ξ©=𝐒+π‘›βˆ’1, then 𝛼Ω=1,𝛽Ω=π‘›βˆ’1, and πœ‘1(Θ)=(2π‘›π‘ π‘›βˆ’1)1/2cosπœƒ1, where 𝑠𝑛 is the surface area 2πœ‹π‘›/2(Ξ“(𝑛/2))βˆ’1 of 𝐒1.

To simplify our consideration in the following, we will assume that if 𝑛β‰₯3, then Ξ© is a 𝐢2,𝛼-domain (0<𝛼<1) on π’π‘›βˆ’1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see ([1], pages 88-89) for the definition of 𝐢2,𝛼-domain). Then by modifying Miranda’s method ([2], pages 7-8), we can prove the following inequality: πœ‘Ξ©(Θ)β‰ˆdist(Θ,πœ•Ξ©)(Θ∈Ω).(1.4)

For any (1,Θ)∈Ω, we have (see [3]) πœ‘Ξ©ξ€·(Θ)β‰ˆdist(1,Θ),πœ•πΆπ‘›ξ€Έ(Ξ©),(1.5) which yields that 𝛿(𝑃)β‰ˆπ‘Ÿπœ‘Ξ©(Θ),(1.6) where 𝛿(𝑃)=dist(𝑃,πœ•πΆπ‘›(Ξ©)) and 𝑃=(π‘Ÿ,Θ)βˆˆπΆπ‘›(Ξ©).

Let 𝐺Ω(𝑃,𝑄)(𝑃=(π‘Ÿ,Θ),𝑄=(𝑑,Ξ¦)βˆˆπΆπ‘›(Ξ©)) be the Green function of 𝐢𝑛(Ξ©). We define the Poisson kernel 𝐾Ω(𝑃,𝑄) by 𝐾Ω1(𝑃,𝑄)=π‘π‘›πœ•πœ•π‘›π‘„πΊΞ©(𝑃,𝑄),(1.7) where 𝑐𝑛=ξ‚»(2πœ‹π‘›=2,π‘›βˆ’2)𝑠𝑛𝑛β‰₯3,(1.8)π‘„βˆˆπ‘†π‘›(Ξ©) and πœ•/πœ•π‘›π‘„ denotes the differentiation at 𝑄 along the inward normal into 𝐢𝑛(Ξ©).

In this paper, we consider functions π‘“βˆˆπΏπ‘(πœ•πΆπ‘›(Ξ©)), where 1≀𝑝<∞. Then the Poisson integral π‘ŠΞ©π‘“(𝑃)(π‘ƒβˆˆπΆπ‘›(Ξ©)) is defined by π‘ŠΞ©ξ€œπ‘“(𝑃)=𝑆𝑛(Ξ©)𝐾Ω(𝑃,𝑄)𝑓(𝑄)π‘‘πœŽπ‘„,(1.9) where π‘‘πœŽπ‘„ is the surface area element on 𝑆𝑛(Ξ©).

Remark 1.1. Let Ξ©=𝐒+π‘›βˆ’1. Then 𝐺𝐒+π‘›βˆ’1ξ‚»||(𝑃,𝑄)=logπ‘ƒβˆ’π‘„βˆ—||||||||||βˆ’logπ‘ƒβˆ’π‘„π‘›=2,π‘ƒβˆ’π‘„2βˆ’π‘›βˆ’||π‘ƒβˆ’π‘„βˆ—||2βˆ’π‘›π‘›β‰₯3,(1.10) where π‘„βˆ—=(π‘Œ,βˆ’π‘¦π‘›), that is, π‘„βˆ— is the mirror image of 𝑄=(π‘Œ,𝑦𝑛) with respect to πœ•π‘‡π‘›. Hence, for the two points 𝑃=(𝑋,π‘₯𝑛)βˆˆπ‘‡π‘› and 𝑄=(π‘Œ,𝑦𝑛)βˆˆπœ•π‘‡π‘›, we have 𝑐𝑛𝐾𝐒+π‘›βˆ’1πœ•(𝑃,𝑄)=πœ•π‘›π‘„πΊπ’+π‘›βˆ’1ξ‚»2||||(𝑃,𝑄)=π‘ƒβˆ’π‘„βˆ’2π‘₯𝑛||||𝑛=2,2(π‘›βˆ’2)π‘ƒβˆ’π‘„βˆ’π‘›π‘₯𝑛𝑛β‰₯3.(1.11)

We fix an open, nonempty, and bounded set 𝐺(Ξ©)βŠ‚πœ•πΆπ‘›(Ξ©). In 𝐢𝑛(Ξ©), we normalise the extension, with respect to 𝐺(Ξ©), by π’«β„Ξ©π‘Šπ‘“(𝑃)=Ω𝑓(𝑃)π‘ŠΞ©πœ’πΊ(Ξ©)(,𝑃)(1.12) where πœ’πΊ(Ξ©) denotes the characteristic function of 𝐺(Ξ©).

Let Ξ“ξ€½(Ξ©,𝜁)=𝑃=(π‘Ÿ,Θ)βˆˆπΆπ‘›||||ξ€Ύ(Ξ©)∢(π‘Ÿ,Θ)βˆ’πœβ‰²π›Ώ(𝑃)(1.13) be a nontangential cone in 𝐢𝑛(Ξ©) with vertex πœβˆˆπœ•πΆπ‘›(Ξ©).

We define ℡𝑝1(𝑓,𝑙,𝑃)=π‘™π‘›βˆ’1ξ€œπ΅(𝑃,𝑙)||||𝑓(𝑄)π‘π‘‘πœŽπ‘„ξ‚Ά1/𝑝,𝔼𝑝𝑓(𝐺(Ξ©))=π‘ƒβˆˆπΊ(Ξ©)βˆΆβ„΅π‘ξ€Ύ.(π‘“βˆ’π‘“(𝑃),𝑙,𝑃)⟢0asπ‘™βŸΆ0(1.14)

Note that, if π‘“βˆˆπΏπ‘(πœ•πΆπ‘›(Ξ©)), then |𝐺(Ξ©)⧡𝔼𝑝𝑓(𝐺(Ξ©))|=0 (a.e. point is a Lebesgue point).

In 𝑇𝑛, the following conclusion was proved by Brundin (see ([4], pages 11–16)) and Mizuta and Shimomura (see ([5], Theorem 3)), respectively. In the unit disc, about related results, we refer the readers to the papers by SjΓΆgren (see [6, 7]), RΓΆnning (see [8]), and Brundin (see [9]).

Theorem A. For a.e. 𝜁∈𝐺(𝐒+π‘›βˆ’1), 𝒫ℐ𝐒+π‘›βˆ’1𝑓(𝑃)→𝑓(𝜁) (see Remark 1.1 for the definition of 𝒫ℐ𝐒+π‘›βˆ’1𝑓(𝑃)) as π‘ƒβ†’πœ along Ξ“(𝐒+π‘›βˆ’1,𝜁).

Our aim is to generalize Theorem A to the conical case.

Theorem 1.2. For any πœβˆˆπ”Όπ‘π‘“(𝐺(Ξ©)) (in particular, for a.e. 𝜁∈𝐺(Ξ©)) one has that 𝒫ℐΩ𝑓(𝑃)→𝑓(𝜁) as π‘ƒβ†’πœ along Ξ“(Ξ©,𝜁).

2. Some Lemmas

Lemma 2.1. One has 𝐾Ω(𝑃,𝑄)β‰ˆπ‘Ÿβˆ’π›½Ξ©π‘‘π›ΌΞ©βˆ’1πœ‘Ξ©ξ€·(Θ),resp.𝐾Ω(𝑃,𝑄)β‰ˆπ‘Ÿπ›ΌΞ©π‘‘βˆ’π›½Ξ©βˆ’1πœ‘Ξ©ξ€Έ,(Θ)(2.1) for any 𝑃=(π‘Ÿ,Θ)βˆˆπΆπ‘›(Ξ©) and any 𝑄=(𝑑,Ξ¦)βˆˆπ‘†π‘›(Ξ©) satisfying 0<𝑑/π‘Ÿβ‰€4/5(resp.0<π‘Ÿ/𝑑≀4/5); 𝐾Ω(𝑃,𝑄)β‰ˆπ‘Ÿπœ‘Ξ©(Θ)||||π‘ƒβˆ’π‘„π‘›,(2.2) for any 𝑃=(π‘Ÿ,Θ)βˆˆπΆπ‘›(Ξ©) and any 𝑄=(𝑑,Ξ¦)βˆˆπ‘†π‘›(Ξ©;(4π‘Ÿ/5,5π‘Ÿ/4)).

Proof. These immediately follow from ([10], Lemma 2), ([11], Lemma 4 and Remark), and (1.4).

Lemma 2.2. One has π‘ŠΞ©1(𝑃)=𝑂(1)asπ‘ƒβŸΆπœβˆˆπΊ.(2.3)

Proof. Write π‘ŠΞ©ξ€œ1(𝑃)=𝐸1+ξ€œπΈ2+ξ€œπΈ3=π‘ˆ1(𝑃)+π‘ˆ2(𝑃)+π‘ˆ3(𝑃),(2.4) where 𝐸1=𝑆𝑛4Ξ©;0,5π‘Ÿξ‚„ξ‚,𝐸2=𝑆𝑛5Ξ©;4π‘Ÿ,βˆžξ‚ξ‚,𝐸3=𝑆𝑛4Ξ©;55π‘Ÿ,4π‘Ÿ.(2.5)
By (2.1), we have the following estimates π‘ˆ1(𝑃)β‰ˆπ‘Ÿβˆ’π›½Ξ©πœ‘Ξ©(ξ€œΞ˜)𝐸1π‘‘π›ΌΞ©βˆ’1π‘‘πœŽπ‘„β‰ˆπ‘ π‘›π›½Ξ©ξ‚€45ξ‚π›½Ξ©πœ‘Ξ©π‘ˆ(Θ),(2.6)2𝑠(𝑃)β‰ˆπ‘›π›ΌΞ©ξ‚€45ξ‚π›ΌΞ©πœ‘Ξ©(Θ).(2.7)
Next, we will estimate π‘ˆ3(𝑃). Take a sufficiently small positive number π‘˜ such that 𝐸3βŠ‚ξšπ‘ƒ=(π‘Ÿ,Θ)βˆˆΞ›(π‘˜)𝐡1𝑃,2π‘Ÿξ‚,(2.8) where ξ‚»Ξ›(π‘˜)=𝑃=(π‘Ÿ,Θ)βˆˆπΆπ‘›(Ξ©);infπ‘§βˆˆπœ•Ξ©||||ξ‚Ό(1,Θ)βˆ’(1,𝑧)<π‘˜,0<π‘Ÿ<∞.(2.9)
Since π‘ƒβ†’πœβˆˆπΊ, we only consider the case π‘ƒβˆˆΞ›(π‘˜). Now put 𝐻𝑖(𝑃)=π‘„βˆˆπΈ3;2π‘–βˆ’1𝛿||||(𝑃)β‰€π‘ƒβˆ’π‘„<2𝑖𝛿.(𝑃)(2.10)
Since 𝑆𝑛(Ξ©)∩{π‘„βˆˆπ‘π‘›βˆΆ|π‘ƒβˆ’π‘„|<𝛿(𝑃)}=Ø, we have π‘ˆ3(𝑃)β‰ˆπ‘–(𝑃)𝑖=0ξ€œπ»π‘–(𝑃)π‘Ÿπœ‘Ξ©(Θ)||||π‘ƒβˆ’π‘„π‘›π‘‘πœŽπ‘„,(2.11) where 𝑖(𝑃) is a positive integer satisfying 2𝑖(𝑃)βˆ’1𝛿(𝑃)β‰€π‘Ÿ/2<2𝑖(𝑃)𝛿(𝑃).
By (1.6) we have ξ€œπ»π‘–(𝑃)π‘Ÿπœ‘Ξ©(Θ)||||π‘ƒβˆ’π‘„π‘›π‘‘πœŽπ‘„β‰ˆπ‘Ÿπœ‘Ξ©ξ€œ(Θ)𝐻𝑖(𝑃)1𝛿(𝑃)π‘‘πœŽπ‘„=π‘Ÿπœ‘Ξ©(Θ)𝑠𝛿(𝑃)𝑛2𝑖(𝑃)β‰ˆπ‘ π‘›2𝑖(𝑃)(2.12) for 𝑖=0,1,2,…,𝑖(𝑃).
So π‘ˆ3(𝑃)β‰ˆπ‘‚(1).(2.13)
Combining (2.6)–(2.13), Lemma 2.2 is proved.

Lemma 2.3. One has π‘ŠΞ©πœ’πΊ(Ξ©)(𝑃)=π‘ŠΞ©1(𝑃)+𝑂(1)asπ‘ƒβŸΆπœβˆˆπΊ(Ξ©).(2.14)

Proof. In fact, we only need to prove π‘ˆ4(ξ€œπ‘ƒ)=𝑆𝑛(Ξ©)βˆ’πΊ(Ξ©)𝐾Ω(𝑃,𝑄)π‘‘πœŽπ‘„β‰²π‘‚(1).(2.15)
Write π‘ˆ4(ξ€œπ‘ƒ)=(𝑆𝑛(Ξ©)βˆ’πΊ(Ξ©))∩𝐸1+ξ€œ(𝑆𝑛(Ξ©)βˆ’πΊ(Ξ©))∩𝐸2+ξ€œ(𝑆𝑛(Ξ©)βˆ’πΊ(Ξ©))∩𝐸3=π‘ˆ5(𝑃)+π‘ˆ6(𝑃)+π‘ˆ7(𝑃),(2.16) where 𝐸1, 𝐸2, and 𝐸3 are sets on 𝑆𝑛(Ξ©) used in Lemma 2.2.
Obviously, π‘ˆ5(𝑃)β‰²π‘ˆ1π‘ˆ(𝑃)β‰ˆπ‘‚(1),(2.17)6(𝑃)β‰²π‘ˆ2(𝑃)β‰ˆπ‘‚(1).(2.18)
Further, we have by (2.2) π‘ˆ7(𝑃)β‰ˆπ‘Ÿπœ‘Ξ©(ξ€œΞ˜)(𝑆𝑛(Ξ©)βˆ’πΊ(Ξ©))∩𝐸31||||π‘ƒβˆ’π‘„π‘›π‘‘πœŽπ‘„β‰²π‘ π‘›π‘‘||𝜁||πœ‘Ξ©(Θ)(π‘ƒβŸΆπœβˆˆπΊ(Ξ©)),(2.19) where 𝑑=infπ‘„βˆˆπœ•πΆπ‘›(Ξ©)βˆ’πΊ(Ξ©)||||.π‘„βˆ’πœ(2.20)
Combining (2.17)–(2.19), (2.15) holds which gives the conclusion.

3. Proof of the Theorem 1.2

As π‘ƒβ†’πœβˆˆπΊ(Ξ©), π‘ŠΞ©πœ’πΊ(Ξ©)(𝑃)=𝑂(1) from Lemmas 2.2 and 2.3.

Now, let π‘“βˆˆπΏπ‘(πœ•πΆπ‘›(Ξ©)) and πœβˆˆπ”Όπ‘π‘“(𝐺(Ξ©)) be given. We may, without loss of generality, assume that 𝑓(𝜁)=0. Furthermore, we assume that 𝑃=(π‘Ÿ,Θ)βˆˆΞ“(Ξ©,𝜁). For short, let 𝑠=|(π‘Ÿ,Θ)βˆ’πœ|. We write π‘ŠΞ©ξ€œπ‘“(𝑃)=𝐸1+ξ€œπΈ2+ξ€œπΈ3∩𝐡(𝜁,2𝑠)+ξ€œπΈ3βˆ©π΅π‘(𝜁,2𝑠)=𝑉1𝑓(𝑃)+𝑉2𝑓(𝑃)+𝑉3𝑓(𝑃)+𝑉4𝑓(𝑃),(3.1) where 𝐸1, 𝐸2, and 𝐸3 are sets on 𝑆𝑛(Ξ©) used in Lemma 2.2.

By using HΓΆlder’s inequality, (2.1), we have the following estimates ||𝑉1||𝑓(𝑃)β‰²π‘Ÿβˆ’π›½Ξ©πœ‘Ξ©(ξ€œΞ˜)𝐸1π‘‘π›ΌΞ©βˆ’1𝑓(𝑄)π‘‘πœŽπ‘„β‰²π‘Ÿ(1βˆ’π‘›)/𝑝‖𝑓‖𝑝,||𝑉2||𝑓(𝑃)β‰²π‘Ÿ(1βˆ’π‘›)/𝑝‖𝑓‖𝑝.(3.2)

Similar to the estimate of π‘ˆ3(𝑃) in Lemma 2.2, we only consider the following inequality by (1.6) ξ€œπ»π‘–(𝑃)π‘Ÿπœ‘Ξ©(Θ)||||π‘ƒβˆ’π‘„π‘›π‘‘πœŽπ‘„β‰ˆπ‘Ÿπœ‘Ξ©ξ€œ(Θ)𝐻𝑖(𝑃)1ξ€½2π‘–βˆ’1𝛿(𝑃)π‘›π‘‘πœŽπ‘„β‰²π‘Ÿπ›ΌΞ©πœ‘Ξ©ξ€œ(Θ)𝐸2π‘‘βˆ’π›½Ξ©βˆ’1||𝑓||(𝑄)π‘‘πœŽπ‘„β‰²π‘Ÿ(1βˆ’π‘›)/𝑝‖𝑓‖𝑝(3.3) for 𝑖=0,1,2,…,𝑖(𝑃), which is similar to the estimate of 𝑉2𝑓(𝑃).

So ||𝑉3||𝑓(𝑃)β‰²π‘Ÿ(1βˆ’π‘›)/𝑝‖𝑓‖𝑝.(3.4)

Notice that |π‘ƒβˆ’π‘„|>(1/2)|πœβˆ’π‘„| in the case π‘„βˆˆπΈ3βˆ©π΅π‘(𝜁,2𝑠). By (1.6) and (2.2), we have ||𝑉4||ξ€œπ‘“(𝑃)≲𝛿(𝑃)𝐸3βˆ©π΅π‘(𝜁,2𝑠)||||𝑓(𝑄)||||π‘ƒβˆ’π‘„π‘›π‘‘πœŽπ‘„β‰²π›Ώ(𝑃)βˆžξ“π‘–=1ξ€œπΈ3∩(𝐡(𝜁,2𝑖+1𝑠)⧡𝐡(𝜁,2𝑖𝑠))||𝑓||(𝑄)||||πœβˆ’π‘„π‘›π‘‘πœŽπ‘„β‰²π›Ώ(𝑃)βˆžξ“π‘–=1ξ‚΅12π‘–π‘ ξ‚Άπ‘›ξ€œπΈ3∩𝐡(𝜁,2𝑖+1𝑠)||||𝑓(𝑄)π‘‘πœŽπ‘„β‰²π›Ώ(𝑃)βˆžξ“π‘–=1β„΅1𝑓,2𝑖+1𝑠,πœβ‰²π›Ώ(𝑃)βˆžξ“π‘–=1ξ€œ2𝑖+2𝑠2𝑖+1𝑠℡1(𝑓,𝑙,𝜁)π‘™ξ€œπ‘‘π‘™β‰²π›Ώ(𝑃)βˆžπ‘ β„΅1(𝑓,𝑙,𝜁)π‘™ξ€œπ‘‘π‘™β‰²π›Ώ(𝑃)βˆžπ›Ώ(𝑃)β„΅1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.5)

Thus, it follows that ||𝒫ℐΩ||≲1𝑓(𝑃)𝑂||𝑉(1)1||+||𝑉𝑓(𝑃)2||+||𝑉𝑓(𝑃)3||+||𝑉𝑓(𝑃)4||𝑓(𝑃)β‰²π‘Ÿ(1βˆ’π‘›)/π‘β€–π‘“β€–π‘ξ€œ+𝛿(𝑃)βˆžπ›Ώ(𝑃)β„΅1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.6)

Using the fact that 𝑠≲𝛿(𝑃)β‰²π‘Ÿπœ‘Ξ©(Θ), we get ||𝒫ℐΩ||𝑓(𝑃)≲℡1ξ€œ(𝑓,2𝑠,𝜁)+𝛿(𝑃)βˆžπ›Ώ(𝑃)β„΅1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.7)

It is clear that ξ€œβˆžπ›Ώ(𝑃)β„΅1(𝑓,𝑙,𝜁)𝑙𝑑𝑙(3.8) is a convergent integral, since β„΅1(𝑓,l,𝜁)π‘™β‰²π‘ βˆ’1βˆ’π‘›π‘ π‘›/π‘žβ€–π‘“β€–π‘β‰²π‘ βˆ’1βˆ’(𝑛/𝑝)‖𝑓‖𝑝(3.9) from the HΓΆlder’s inequality.

Now, as 𝛿(𝑃)β†’0, we also have 𝑠→0. Since 𝑓(𝜁)=0 and since we have assumed that πœβˆˆπ”Όπ‘π‘“(𝐺(Ξ©)) (and thus that πœβˆˆπ”Ό1𝑓(𝐺(Ξ©))), it follows that 𝒫ℐΩ𝑓(𝑃)β†’0=𝑓(𝜁) as 𝑃=(π‘Ÿ,Θ)β†’πœ along Ξ“(Ξ©,𝜁). This concludes the proof.

Acknowledgments

This paper is supported by SRFDP (no. 20100003110004) and NSF of China (no. 11071020).

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