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(𝑛≥2) the 𝑛-dimensional Euclidean space. A point in 𝐑𝑛 is denoted by 𝑃=(𝑋,𝑥𝑛), where 𝑋=(𝑥1,𝑥2,…,𝑥𝑛−1). The Euclidean distance of two points 𝑃 and 𝑄 in 𝐑𝑛 is denoted by |𝑃−𝑄|. Also |𝑃−𝑂| 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 𝜕𝐒, 𝐒, and 𝐒𝑐, respectively.

We introduce a system of spherical coordinates (𝑟,Θ),  Θ=(𝜃1,𝜃2,…,𝜃𝑛−1), in 𝐑𝑛 which are related to cartesian coordinates (𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑛) by 𝑥𝑛=𝑟cos𝜃1.

For positive functions ℎ1 and ℎ2, we say that ℎ1≲ℎ2 if ℎ1â‰¤ğ‘€â„Ž2 for some constant 𝑀>0. If ℎ1≲ℎ2 and ℎ2≲ℎ1, we say that ℎ1≈ℎ2.

For 𝑃∈𝐑𝑛 and 𝑅>0, let 𝐵(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 𝐒𝑛−1 and 𝐒+𝑛−1, respectively. For simplicity, a point (1,Θ) on 𝐒𝑛−1 and the set {Θ;(1,Θ)∈Ω} for a set Ω, Ω⊂𝐒𝑛−1 are often identified with Θ and Ω, respectively. For two sets ÎžâŠ‚ğ‘+ and Ω⊂𝐒𝑛−1, the set {(𝑟,Θ)∈𝐑𝑛;ğ‘ŸâˆˆÎž,(1,Θ)∈Ω} in 𝐑𝑛 is simply denoted by Ξ×Ω. In particular, the half space 𝐑+×𝐒+𝑛−1={(𝑋,𝑥𝑛)∈𝐑𝑛;𝑥𝑛>0} will be denoted by 𝐓𝑛.

By 𝐶𝑛(Ω), we denote the set 𝐑+×Ω in 𝐑𝑛 with the domain Ω on 𝐒𝑛−1. We call it a cone. Then 𝑇𝑛 is a special cone obtained by putting Ω=𝐒+𝑛−1. We denote the sets 𝐼×Ω and 𝐼×𝜕Ω with an interval on 𝐑 by 𝐶𝑛(Ω;𝐼) and 𝑆𝑛(Ω;𝐼). By 𝑆𝑛(Ω) we denote 𝑆𝑛(Ω;(0,+∞)) which is 𝜕𝐶𝑛(Ω)−{𝑂}.

Let Ω be a domain on 𝐒𝑛−1 with smooth boundary. Consider the Dirichlet problem: Λ𝑛+𝜆𝜑=0onΩ,𝜑=0on𝜕Ω,(1.1) where Λ𝑛 is the spherical part of the Laplace operator Δ𝑛∶Δ𝑛=𝑛−1𝑟𝜕+𝜕𝜕𝑟2𝜕𝑟2+Λ𝑛𝑟2.(1.2) We denote the least positive eigenvalue of this boundary value problem by 𝜆Ω and the normalized positive eigenfunction corresponding to 𝜆Ω by 𝜑Ω(Θ), Ω𝜑2Ω(Θ)ğ‘‘ğœŽÎ˜=1,(1.3) where ğ‘‘ğœŽÎ˜ is the surface area on 𝑆𝑛−1. We denote the solutions of the equation 𝑡2+(𝑛−2)𝑡−𝜆Ω=0 by 𝛼Ω,−𝛽Ω (𝛼Ω,𝛽Ω>0). If Ω=𝐒+𝑛−1, then 𝛼Ω=1,𝛽Ω=𝑛−1, and 𝜑1(Θ)=(2𝑛𝑠𝑛−1)1/2cos𝜃1, where 𝑠𝑛 is the surface area 2𝜋𝑛/2(Γ(𝑛/2))−1 of 𝐒1.

To simplify our consideration in the following, we will assume that if 𝑛≥3, then Ω is a 𝐶2,𝛼-domain (0<𝛼<1) on 𝐒𝑛−1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see ([1], pages 88-89) for the definition of 𝐶2,𝛼-domain). Then by modifying Miranda’s method ([2], pages 7-8), we can prove the following inequality: 𝜑Ω(Θ)≈dist(Θ,𝜕Ω)(Θ∈Ω).(1.4)

For any (1,Θ)∈Ω, we have (see [3]) 𝜑Ω(Θ)≈dist(1,Θ),𝜕𝐶𝑛(Ω),(1.5) which yields that 𝛿(𝑃)≈𝑟𝜑Ω(Θ),(1.6) where 𝛿(𝑃)=dist(𝑃,𝜕𝐶𝑛(Ω)) and 𝑃=(𝑟,Θ)∈𝐶𝑛(Ω).

Let 𝐺Ω(𝑃,𝑄)(𝑃=(𝑟,Θ),𝑄=(𝑡,Φ)∈𝐶𝑛(Ω)) be the Green function of 𝐶𝑛(Ω). We define the Poisson kernel 𝐾Ω(𝑃,𝑄) by 𝐾Ω1(𝑃,𝑄)=𝑐𝑛𝜕𝜕𝑛𝑄𝐺Ω(𝑃,𝑄),(1.7) where 𝑐𝑛=(2𝜋𝑛=2,𝑛−2)𝑠𝑛𝑛≥3,(1.8)𝑄∈𝑆𝑛(Ω) and 𝜕/𝜕𝑛𝑄 denotes the differentiation at 𝑄 along the inward normal into 𝐶𝑛(Ω).

In this paper, we consider functions 𝑓∈𝐿𝑝(𝜕𝐶𝑛(Ω)), where 1≤𝑝<∞. Then the Poisson integral 𝑊Ω𝑓(𝑃)(𝑃∈𝐶𝑛(Ω)) is defined by 𝑊Ω𝑓(𝑃)=𝑆𝑛(Ω)𝐾Ω(𝑃,𝑄)𝑓(𝑄)ğ‘‘ğœŽğ‘„,(1.9) where ğ‘‘ğœŽğ‘„ is the surface area element on 𝑆𝑛(Ω).

Remark 1.1. Let Ω=𝐒+𝑛−1. Then 𝐺𝐒+𝑛−1||(𝑃,𝑄)=log𝑃−𝑄∗||||||||||−log𝑃−𝑄𝑛=2,𝑃−𝑄2−𝑛−||𝑃−𝑄∗||2−𝑛𝑛≥3,(1.10) where 𝑄∗=(𝑌,−𝑦𝑛), that is, 𝑄∗ is the mirror image of 𝑄=(𝑌,𝑦𝑛) with respect to 𝜕𝑇𝑛. Hence, for the two points 𝑃=(𝑋,𝑥𝑛)∈𝑇𝑛 and 𝑄=(𝑌,𝑦𝑛)∈𝜕𝑇𝑛, we have 𝑐𝑛𝐾𝐒+𝑛−1𝜕(𝑃,𝑄)=𝜕𝑛𝑄𝐺𝐒+𝑛−12||||(𝑃,𝑄)=𝑃−𝑄−2𝑥𝑛||||𝑛=2,2(𝑛−2)𝑃−𝑄−𝑛𝑥𝑛𝑛≥3.(1.11)

We fix an open, nonempty, and bounded set 𝐺(Ω)⊂𝜕𝐶𝑛(Ω). In 𝐶𝑛(Ω), we normalise the extension, with respect to 𝐺(Ω), by 𝒫ℐΩ𝑊𝑓(𝑃)=Ω𝑓(𝑃)𝑊Ω𝜒𝐺(Ω)(,𝑃)(1.12) where 𝜒𝐺(Ω) denotes the characteristic function of 𝐺(Ω).

Let Γ(Ω,𝜁)=𝑃=(𝑟,Θ)∈𝐶𝑛||||(Ω)∶(𝑟,Θ)−𝜁≲𝛿(𝑃)(1.13) be a nontangential cone in 𝐶𝑛(Ω) with vertex 𝜁∈𝜕𝐶𝑛(Ω).

We define ℵ𝑝1(𝑓,𝑙,𝑃)=𝑙𝑛−1𝐵(𝑃,𝑙)||||𝑓(𝑄)ğ‘ğ‘‘ğœŽğ‘„î‚¶1/𝑝,𝔼𝑝𝑓(𝐺(Ω))=𝑃∈𝐺(Ω)∶ℵ𝑝.(𝑓−𝑓(𝑃),𝑙,𝑃)⟶0as𝑙⟶0(1.14)

Note that, if 𝑓∈𝐿𝑝(𝜕𝐶𝑛(Ω)), then |𝐺(Ω)⧵𝔼𝑝𝑓(𝐺(Ω))|=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 Sjögren (see [6, 7]), Rönning (see [8]), and Brundin (see [9]).

Theorem A. For a.e. 𝜁∈𝐺(𝐒+𝑛−1), 𝒫ℐ𝐒+𝑛−1𝑓(𝑃)→𝑓(𝜁) (see Remark 1.1 for the definition of 𝒫ℐ𝐒+𝑛−1𝑓(𝑃)) as 𝑃→𝜁 along Γ(𝐒+𝑛−1,𝜁).

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

Theorem 1.2. For any 𝜁∈𝔼𝑝𝑓(𝐺(Ω)) (in particular, for a.e. 𝜁∈𝐺(Ω)) one has that 𝒫ℐΩ𝑓(𝑃)→𝑓(𝜁) as 𝑃→𝜁 along Γ(Ω,𝜁).

2. Some Lemmas

Lemma 2.1. One has 𝐾Ω(𝑃,𝑄)≈𝑟−𝛽Ω𝑡𝛼Ω−1𝜑Ω(Θ),resp.𝐾Ω(𝑃,𝑄)≈𝑟𝛼Ω𝑡−𝛽Ω−1𝜑Ω,(Θ)(2.1) for any 𝑃=(𝑟,Θ)∈𝐶𝑛(Ω) and any 𝑄=(𝑡,Φ)∈𝑆𝑛(Ω) satisfying 0<𝑡/𝑟≤4/5(resp.0<𝑟/𝑡≤4/5); 𝐾Ω(𝑃,𝑄)≈𝑟𝜑Ω(Θ)||||𝑃−𝑄𝑛,(2.2) for any 𝑃=(𝑟,Θ)∈𝐶𝑛(Ω) and any 𝑄=(𝑡,Φ)∈𝑆𝑛(Ω;(4𝑟/5,5𝑟/4)).

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

Lemma 2.2. One has 𝑊Ω1(𝑃)=𝑂(1)as𝑃⟶𝜁∈𝐺.(2.3)

Proof. Write 𝑊Ω1(𝑃)=𝐸1+𝐸2+𝐸3=𝑈1(𝑃)+𝑈2(𝑃)+𝑈3(𝑃),(2.4) where 𝐸1=𝑆𝑛4Ω;0,5𝑟,𝐸2=𝑆𝑛5Ω;4𝑟,∞,𝐸3=𝑆𝑛4Ω;55𝑟,4𝑟.(2.5)
By (2.1), we have the following estimates 𝑈1(𝑃)≈𝑟−𝛽Ω𝜑Ω(Θ)𝐸1𝑡𝛼Ω−1ğ‘‘ğœŽğ‘„â‰ˆğ‘ ğ‘›ğ›½Î©î‚€45𝛽Ω𝜑Ω𝑈(Θ),(2.6)2𝑠(𝑃)≈𝑛𝛼Ω45𝛼Ω𝜑Ω(Θ).(2.7)
Next, we will estimate 𝑈3(𝑃). Take a sufficiently small positive number 𝑘 such that 𝐸3⊂𝑃=(𝑟,Θ)∈Λ(𝑘)𝐵1𝑃,2𝑟,(2.8) where Λ(𝑘)=𝑃=(𝑟,Θ)∈𝐶𝑛(Ω);inf𝑧∈𝜕Ω||||(1,Θ)−(1,𝑧)<𝑘,0<𝑟<∞.(2.9)
Since 𝑃→𝜁∈𝐺, we only consider the case 𝑃∈Λ(𝑘). Now put 𝐻𝑖(𝑃)=𝑄∈𝐸3;2𝑖−1𝛿||||(𝑃)≤𝑃−𝑄<2𝑖𝛿.(𝑃)(2.10)
Since 𝑆𝑛(Ω)∩{𝑄∈𝐑𝑛∶|𝑃−𝑄|<𝛿(𝑃)}=Ø, we have 𝑈3(𝑃)≈𝑖(𝑃)𝑖=0𝐻𝑖(𝑃)𝑟𝜑Ω(Θ)||||ğ‘ƒâˆ’ğ‘„ğ‘›ğ‘‘ğœŽğ‘„,(2.11) where 𝑖(𝑃) is a positive integer satisfying 2𝑖(𝑃)−1𝛿(𝑃)≤𝑟/2<2𝑖(𝑃)𝛿(𝑃).
By (1.6) we have 𝐻𝑖(𝑃)𝑟𝜑Ω(Θ)||||ğ‘ƒâˆ’ğ‘„ğ‘›ğ‘‘ğœŽğ‘„â‰ˆğ‘Ÿğœ‘Î©î€œ(Θ)𝐻𝑖(𝑃)1𝛿(𝑃)ğ‘‘ğœŽğ‘„=𝑟𝜑Ω(Θ)𝑠𝛿(𝑃)𝑛2𝑖(𝑃)≈𝑠𝑛2𝑖(𝑃)(2.12) for 𝑖=0,1,2,…,𝑖(𝑃).
So 𝑈3(𝑃)≈𝑂(1).(2.13)
Combining (2.6)–(2.13), Lemma 2.2 is proved.

Lemma 2.3. One has 𝑊Ω𝜒𝐺(Ω)(𝑃)=𝑊Ω1(𝑃)+𝑂(1)as𝑃⟶𝜁∈𝐺(Ω).(2.14)

Proof. In fact, we only need to prove 𝑈4(𝑃)=𝑆𝑛(Ω)−𝐺(Ω)𝐾Ω(𝑃,𝑄)ğ‘‘ğœŽğ‘„â‰²ğ‘‚(1).(2.15)
Write 𝑈4(𝑃)=(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸1+(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸2+(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸3=𝑈5(𝑃)+𝑈6(𝑃)+𝑈7(𝑃),(2.16) where 𝐸1, 𝐸2, and 𝐸3 are sets on 𝑆𝑛(Ω) used in Lemma 2.2.
Obviously, 𝑈5(𝑃)≲𝑈1𝑈(𝑃)≈𝑂(1),(2.17)6(𝑃)≲𝑈2(𝑃)≈𝑂(1).(2.18)
Further, we have by (2.2) 𝑈7(𝑃)≈𝑟𝜑Ω(Θ)(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸31||||ğ‘ƒâˆ’ğ‘„ğ‘›ğ‘‘ğœŽğ‘„â‰²ğ‘ ğ‘›ğ‘‘||𝜁||𝜑Ω(Θ)(𝑃⟶𝜁∈𝐺(Ω)),(2.19) where 𝑑=inf𝑄∈𝜕𝐶𝑛(Ω)−𝐺(Ω)||||.𝑄−𝜁(2.20)
Combining (2.17)–(2.19), (2.15) holds which gives the conclusion.

3. Proof of the Theorem 1.2

As 𝑃→𝜁∈𝐺(Ω), 𝑊Ω𝜒𝐺(Ω)(𝑃)=𝑂(1) from Lemmas 2.2 and 2.3.

Now, let 𝑓∈𝐿𝑝(𝜕𝐶𝑛(Ω)) and 𝜁∈𝔼𝑝𝑓(𝐺(Ω)) be given. We may, without loss of generality, assume that 𝑓(𝜁)=0. Furthermore, we assume that 𝑃=(𝑟,Θ)∈Γ(Ω,𝜁). For short, let 𝑠=|(𝑟,Θ)−𝜁|. We write 𝑊Ω𝑓(𝑃)=𝐸1+𝐸2+𝐸3∩𝐵(𝜁,2𝑠)+𝐸3∩𝐵𝑐(𝜁,2𝑠)=𝑉1𝑓(𝑃)+𝑉2𝑓(𝑃)+𝑉3𝑓(𝑃)+𝑉4𝑓(𝑃),(3.1) where 𝐸1, 𝐸2, and 𝐸3 are sets on 𝑆𝑛(Ω) used in Lemma 2.2.

By using Hölder’s inequality, (2.1), we have the following estimates ||𝑉1||𝑓(𝑃)≲𝑟−𝛽Ω𝜑Ω(Θ)𝐸1𝑡𝛼Ω−1𝑓(𝑄)ğ‘‘ğœŽğ‘„â‰²ğ‘Ÿ(1−𝑛)/𝑝‖𝑓‖𝑝,||𝑉2||𝑓(𝑃)≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝.(3.2)

Similar to the estimate of 𝑈3(𝑃) in Lemma 2.2, we only consider the following inequality by (1.6) 𝐻𝑖(𝑃)𝑟𝜑Ω(Θ)||||ğ‘ƒâˆ’ğ‘„ğ‘›ğ‘‘ğœŽğ‘„â‰ˆğ‘Ÿğœ‘Î©î€œ(Θ)𝐻𝑖(𝑃)12𝑖−1𝛿(𝑃)ğ‘›ğ‘‘ğœŽğ‘„â‰²ğ‘Ÿğ›¼Î©ğœ‘Î©î€œ(Θ)𝐸2𝑡−𝛽Ω−1||𝑓||(𝑄)ğ‘‘ğœŽğ‘„â‰²ğ‘Ÿ(1−𝑛)/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)≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝.(3.4)

Notice that |𝑃−𝑄|>(1/2)|𝜁−𝑄| in the case 𝑄∈𝐸3∩𝐵𝑐(𝜁,2𝑠). By (1.6) and (2.2), we have ||𝑉4||𝑓(𝑃)≲𝛿(𝑃)𝐸3∩𝐵𝑐(𝜁,2𝑠)||||𝑓(𝑄)||||ğ‘ƒâˆ’ğ‘„ğ‘›ğ‘‘ğœŽğ‘„â‰²ğ›¿(𝑃)âˆžî“ğ‘–=1𝐸3∩(𝐵(𝜁,2𝑖+1𝑠)⧵𝐵(𝜁,2𝑖𝑠))||𝑓||(𝑄)||||ğœâˆ’ğ‘„ğ‘›ğ‘‘ğœŽğ‘„â‰²ğ›¿(𝑃)âˆžî“ğ‘–=112𝑖𝑠𝑛𝐸3∩𝐵(𝜁,2𝑖+1𝑠)||||𝑓(𝑄)ğ‘‘ğœŽğ‘„â‰²ğ›¿(𝑃)âˆžî“ğ‘–=1ℵ1𝑓,2𝑖+1𝑠,𝜁≲𝛿(𝑃)âˆžî“ğ‘–=12𝑖+2𝑠2𝑖+1𝑠ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙≲𝛿(𝑃)âˆžğ‘ â„µ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙≲𝛿(𝑃)âˆžğ›¿(𝑃)ℵ1(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)≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝+𝛿(𝑃)âˆžğ›¿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.6)

Using the fact that 𝑠≲𝛿(𝑃)≲𝑟𝜑Ω(Θ), we get ||𝒫ℐΩ||𝑓(𝑃)≲ℵ1(𝑓,2𝑠,𝜁)+𝛿(𝑃)âˆžğ›¿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.7)

It is clear that î€œâˆžğ›¿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙(3.8) is a convergent integral, since ℵ1(𝑓,l,𝜁)𝑙≲𝑠−1−𝑛𝑠𝑛/ğ‘žâ€–ğ‘“â€–ğ‘â‰²ğ‘ âˆ’1−(𝑛/𝑝)‖𝑓‖𝑝(3.9) from the Hölder’s inequality.

Now, as 𝛿(𝑃)→0, we also have 𝑠→0. Since 𝑓(𝜁)=0 and since we have assumed that 𝜁∈𝔼𝑝𝑓(𝐺(Ω)) (and thus that 𝜁∈𝔼1𝑓(𝐺(Ω))), it follows that 𝒫ℐΩ𝑓(𝑃)→0=𝑓(𝜁) as 𝑃=(𝑟,Θ)→𝜁 along Γ(Ω,𝜁). This concludes the proof.

Acknowledgments

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