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

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 𝐑^𝑛(𝑛≥2) the 𝑛-dimensional Euclidean space. A point in 𝐑^𝑛 is denoted by 𝑃=(𝑋,𝑥_𝑛), where 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛−1). The Euclidean distance of two points 𝑃 and 𝑄 in 𝐑^𝑛 is denoted by |𝑃−𝑄|. Also |𝑃−𝑂| with the origin 𝑂 of 𝐑^𝑛 is simply denoted by |𝑃|. The boundary, the closure, and the complement of a set 𝐒 in 𝐑^𝑛 are denoted by 𝜕𝐒, 𝐒, and 𝐒^𝑐, respectively.

We introduce a system of spherical coordinates (𝑟,Θ),  Θ=(𝜃₁,𝜃₂,…,𝜃_𝑛−1), in 𝐑^𝑛 which are related to cartesian coordinates (𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑛) by 𝑥_𝑛=𝑟cos𝜃₁.

For positive functions ℎ₁ and ℎ₂, we say that ℎ₁≲ℎ₂ if ℎ₁≤𝑀ℎ₂ for some constant 𝑀>0. If ℎ₁≲ℎ₂ and ℎ₂≲ℎ₁, we say that ℎ₁≈ℎ₂.

For 𝑃∈𝐑^𝑛 and 𝑅>0, let 𝐵(𝑃,𝑅) denote the open ball with center at 𝑃 and radius 𝑅 in 𝐑^𝑛. The unit sphere and the upper half unit sphere are denoted by 𝐒^𝑛−1 and 𝐒^+𝑛−1, respectively. For simplicity, a point (1,Θ) on 𝐒^𝑛−1 and the set {Θ;(1,Θ)∈Ω} for a set Ω, Ω⊂𝐒^𝑛−1 are often identified with Θ and Ω, respectively. For two sets Ξ⊂𝐑₊ and Ω⊂𝐒^𝑛−1, the set {(𝑟,Θ)∈𝐑^𝑛;𝑟∈Ξ,(1,Θ)∈Ω} in 𝐑^𝑛 is simply denoted by Ξ×Ω. In particular, the half space 𝐑₊×𝐒^+𝑛−1={(𝑋,𝑥_𝑛)∈𝐑^𝑛;𝑥_𝑛>0} will be denoted by 𝐓_𝑛.

By 𝐶_𝑛(Ω), we denote the set 𝐑₊×Ω in 𝐑^𝑛 with the domain Ω on 𝐒^𝑛−1. We call it a cone. Then 𝑇_𝑛 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 𝐶_𝑛(Ω;𝐼) and 𝑆_𝑛(Ω;𝐼). By 𝑆_𝑛(Ω) we denote 𝑆_𝑛(Ω;(0,+∞)) which is 𝜕𝐶_𝑛(Ω)−{𝑂}.

Let Ω be a domain on 𝐒^𝑛−1 with smooth boundary. Consider the Dirichlet problem: Λ_𝑛𝜑+=𝜆0𝜑o=n0𝜕Ωon,Ω,(1.1) where Λ_𝑛 is the spherical part of the Laplace operator Δ_𝑛∶Δ_𝑛=𝑛−1𝑟𝜕𝜕𝑟+𝜕𝜕𝑟²²+Λ^𝑛𝑟².(1.2) We denote the least positive eigenvalue of this boundary value problem by 𝜆_Ω and the normalized positive eigenfunction corresponding to 𝜆_Ω by 𝜑_Ω(Θ), _Ω𝜑^2Ω(Θ)𝑑𝜎_Θ=1,(1.3) where 𝑑𝜎_Θ is the surface area on 𝑆^𝑛−1. We denote the solutions of the equation 𝑡²+(𝑛−2)𝑡−𝜆_Ω=0 by 𝛼_Ω,−𝛽_Ω (𝛼_Ω,𝛽_Ω>0). If Ω=𝐒^+𝑛−1, then 𝛼_Ω=1,𝛽_Ω=𝑛−1, and 𝜑₁(Θ)=(2𝑛𝑠^𝑛−1)^1/2cos𝜃₁, where 𝑠_𝑛 is the surface area 2𝜋^𝑛/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 𝐶^2,𝛼-domain (0<𝛼<1) on 𝐒^𝑛−1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see ([1], pages 88-89) for the definition of 𝐶^2,𝛼-domain). Then by modifying Miranda’s method ([2], pages 7-8), we can prove the following inequality: 𝜑_Ω(Θ)≈dist(Θ,𝜕Ω)(Θ∈Ω).(1.4)

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

Let 𝐺_Ω(𝑃,𝑄)(𝑃=(𝑟,Θ),𝑄=(𝑡,Φ)∈𝐶_𝑛(Ω)) be the Green function of 𝐶_𝑛(Ω). We define the Poisson kernel 𝐾_Ω(𝑃,𝑄) by 𝐾_Ω(𝑃,𝑄)=1𝑐_𝑛𝜕𝜕𝑛_𝑄𝐺_Ω(𝑃,𝑄),(1.7) where 𝑐_𝑛=(2𝑛𝜋−2)𝑠_𝑛𝑛=2,𝑛≥3,(1.8)𝑄∈𝑆_𝑛(Ω) and 𝜕/𝜕𝑛_𝑄 denotes the differentiation at 𝑄 along the inward normal into 𝐶_𝑛(Ω).

In this paper, we consider functions 𝑓∈𝐿^𝑝(𝜕𝐶_𝑛(Ω)), where 1≤𝑝<∞. Then the Poisson integral 𝑊_Ω𝑓(𝑃)(𝑃∈𝐶_𝑛(Ω)) is defined by 𝑊_Ω𝑓(𝑃)=_𝑆𝑛(Ω)𝐾_Ω(𝑃,𝑄)𝑓(𝑄)𝑑𝜎_𝑄,(1.9) where 𝑑𝜎_𝑄 is the surface area element on 𝑆_𝑛(Ω).

Remark 1.1. Let Ω=𝐒^+𝑛−1. Then 𝐺_𝐒^+𝑛−1(𝑃,𝑄)=log||𝑃−𝑄^∗||𝑃−𝑄||_2−𝑛||−log||𝑃−𝑄||𝑛=2,−||𝑃−𝑄^∗||^2−𝑛𝑛≥3,(1.10) where 𝑄^∗=(𝑌,−𝑦_𝑛), that is, 𝑄^∗ is the mirror image of 𝑄=(𝑌,𝑦_𝑛) with respect to 𝜕𝑇_𝑛. Hence, for the two points 𝑃=(𝑋,𝑥_𝑛)∈𝑇_𝑛 and 𝑄=(𝑌,𝑦_𝑛)∈𝜕𝑇_𝑛, we have 𝑐_𝑛𝐾_𝐒^+𝑛−1(𝑃,𝑄)=𝜕𝜕𝑛_𝑄𝐺_𝐒^+𝑛−1(𝑃,𝑄)=2||𝑃−𝑄||⁻²2(𝑛−2)||𝑃𝑥−_𝑛𝑄||_−𝑛𝑛=2,𝑥_𝑛𝑛≥3.(1.11)

We fix an open, nonempty, and bounded set 𝐺(Ω)⊂𝜕𝐶_𝑛(Ω). In 𝐶_𝑛(Ω), we normalise the extension, with respect to 𝐺(Ω), by 𝒫ℐ_Ω𝑓(𝑃)=𝑊^Ω𝑓(𝑃)𝑊_Ω𝜒_𝐺(Ω)(𝑃),(1.12) where 𝜒_𝐺(Ω) denotes the characteristic function of 𝐺(Ω).

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

We define ℵ_𝑝(𝑓,𝑙,𝑃)=1𝑙^𝑛−1_{𝐵(𝑃,𝑙)}||𝑓(𝑄)||^𝑝𝑑𝜎_𝑄^1/𝑝𝔼_𝑝𝑓,(𝐺(Ω))=𝑃∈𝐺(Ω)∶ℵ_𝑝(𝑓−𝑓(𝑃),𝑙,𝑃)⟶0as𝑙⟶0.(1.14)

Note that, if 𝑓∈𝐿^𝑝(𝜕𝐶_𝑛(Ω)), then |𝐺(Ω)⧵𝔼^𝑝𝑓(𝐺(Ω))|=0 (a.e. point is a Lebesgue point).

In 𝑇_𝑛, 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), 𝒫ℐ_𝐒^+𝑛−1𝑓(𝑃)→𝑓(𝜁) (see Remark 1.1 for the definition of 𝒫ℐ_𝐒^+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∈𝔼^𝑝𝑓(𝐺(Ω)) (in particular, for a.e. 𝜁∈𝐺(Ω)) one has that 𝒫ℐ_Ω𝑓(𝑃)→𝑓(𝜁) as 𝑃→𝜁 along Γ(Ω,𝜁).

2. Some Lemmas

Lemma 2.1. One has res𝐾p._Ω𝐾(𝑃_Ω,𝑄)≈𝑟^−𝛽Ω𝑡^𝛼Ω−1𝜑_Ω(Θ),(𝑃,𝑄)≈𝑟^𝛼Ω𝑡^{−𝛽Ω−1}𝜑_Ω(Θ),(2.1) for any 𝑃=(𝑟,Θ)∈𝐶_𝑛(Ω) and any 𝑄=(𝑡,Φ)∈𝑆_𝑛(Ω) satisfying 0<𝑡/𝑟≤4/5(resp.0<𝑟/𝑡≤4/5); 𝐾_Ω(𝑃,𝑄)≈𝑟𝜑^Ω(Θ)||𝑃−𝑄||^𝑛,(2.2) for any 𝑃=(𝑟,Θ)∈𝐶_𝑛(Ω) and any 𝑄=(𝑡,Φ)∈𝑆_𝑛(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 𝑊_Ω1(𝑃)=𝑂(1)as𝑃⟶𝜁∈𝐺.(2.3)

Proof. Write 𝑊_Ω1(𝑃)=_𝐸1+_𝐸2+_𝐸3=𝑈₁(𝑃)+𝑈₂(𝑃)+𝑈₃(𝑃),(2.4) where 𝐸₁=𝑆_𝑛Ω;0,45𝑟,𝐸²=𝑆_𝑛Ω;54𝑟,∞,𝐸³=𝑆_𝑛Ω;45𝑟,45𝑟.(2.5)
By (2.1), we have the following estimates 𝑈₁(𝑃)≈𝑟^−𝛽Ω𝜑_Ω(Θ)_𝐸1𝑡^𝛼Ω−1𝑑𝜎_𝑄≈𝑠^𝑛𝛽_Ω54^𝛽Ω𝑈𝜑_Ω(Θ),(2.6)₂(𝑃)≈𝑠^𝑛𝛼_Ω54^𝛼Ω𝜑_Ω(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𝑃,12𝑟,(2.8) where Λ(𝑘)=𝑃=(𝑟,Θ)∈𝐶_𝑛(Ω);inf_{𝑧∈𝜕Ω}||(1,Θ)−(1,𝑧)||<𝑘,0<𝑟<∞.(2.9)
Since 𝑃→𝜁∈𝐺, we only consider the case 𝑃∈Λ(𝑘). Now put 𝐻_𝑖(𝑃)=𝑄∈𝐸₃;2^𝑖−1𝛿(𝑃)≤||𝑃−𝑄||<2^𝑖𝛿(𝑃).(2.10)
Since 𝑆_𝑛(Ω)∩{𝑄∈𝐑^𝑛∶|𝑃−𝑄|<𝛿(𝑃)}=Ø, we have 𝑈₃(𝑃)≈^𝑖(𝑃)_𝑖=0_{𝐻𝑖(𝑃)}𝑟𝜑_Ω(Θ)||𝑃−𝑄||^𝑛𝑑𝜎_𝑄,(2.11) where 𝑖(𝑃) is a positive integer satisfying 2^{𝑖(𝑃)−1}𝛿(𝑃)≤𝑟/2<2^𝑖(𝑃)𝛿(𝑃).
By (1.6) we have _{𝐻𝑖(𝑃)}𝑟𝜑_Ω(Θ)||𝑃−𝑄||^𝑛𝑑𝜎_𝑄≈𝑟𝜑_Ω(Θ)_{𝐻𝑖(𝑃)}1𝛿(𝑃)𝑑𝜎^𝑄=𝑟𝜑^Ω(Θ)𝛿(𝑃)2𝑠^{𝑛𝑖(𝑃)}≈𝑠^𝑛2^𝑖(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 𝑊_Ω𝜒_𝐺(Ω)(𝑃)=𝑊_Ω1(𝑃)+𝑂(1)as𝑃⟶𝜁∈𝐺(Ω).(2.14)

Proof. In fact, we only need to prove 𝑈₄(𝑃)=_{𝑆𝑛(Ω)−𝐺(Ω)}𝐾_Ω(𝑃,𝑄)𝑑𝜎_𝑄≲𝑂(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 𝑆_𝑛(Ω) used in Lemma 2.2.
Obviously, 𝑈𝑈₅(𝑃)≲𝑈₁(𝑃)≈𝑂(1),(2.17)₆(𝑃)≲𝑈₂(𝑃)≈𝑂(1).(2.18)
Further, we have by (2.2) 𝑈₇(𝑃)≈𝑟𝜑_Ω(Θ)_{(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸3}1||𝑃−𝑄||^𝑛𝑑𝜎_𝑄≲𝑠^𝑛𝑑||𝜁||𝜑^Ω(Θ)(𝑃⟶𝜁∈𝐺(Ω)),(2.19) where 𝑑=inf_{𝑄∈𝜕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), 𝑊_Ω𝜒_𝐺(Ω)(𝑃)=𝑂(1) from Lemmas 2.2 and 2.3.

Now, let 𝑓∈𝐿^𝑝(𝜕𝐶_𝑛(Ω)) and 𝜁∈𝔼^𝑝𝑓(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 𝑊_Ω𝑓(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 𝑆_𝑛(Ω) used in Lemma 2.2.

By using Hölder’s inequality, (2.1), we have the following estimates ||𝑉₁𝑓(𝑃)||≲𝑟^−𝛽Ω𝜑_Ω(Θ)_𝐸1𝑡^𝛼Ω−1𝑓(𝑄)𝑑𝜎_𝑄||𝑉₂≲𝑟^{(1−𝑛)/𝑝}‖𝑓‖_𝑝,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)}𝑟𝜑_Ω(Θ)||𝑃−𝑄||^𝑛𝑑𝜎_𝑄≈𝑟𝜑_Ω(Θ)_{𝐻𝑖(𝑃)}12^𝑖−1𝛿(𝑃)^𝑛𝑑𝜎_𝑄≲𝑟^𝛼Ω𝜑_Ω(Θ)_𝐸2𝑡^{−𝛽Ω−1}||𝑓(𝑄)||𝑑𝜎_𝑄≲𝑟^{(1−𝑛)/𝑝}‖𝑓‖_𝑝(3.3) for 𝑖=0,1,2,…,𝑖(𝑃), which is similar to the estimate of 𝑉₂𝑓(𝑃).

So ||𝑉₃𝑓(𝑃)||≲𝑟^{(1−𝑛)/𝑝}‖𝑓‖_𝑝.(3.4)

Notice that |𝑃−𝑄|>(1/2)|𝜁−𝑄| in the case 𝑄∈𝐸₃∩𝐵^𝑐(𝜁,2𝑠). By (1.6) and (2.2), we have ||𝑉₄𝑓(𝑃)||≲𝛿(𝑃)_{𝐸3∩𝐵𝑐(𝜁,2𝑠)}||𝑓(𝑄)||||𝑃−𝑄||^𝑛𝑑𝜎_𝑄≲𝛿(𝑃)^∞_𝑖=1_{𝐸3∩(𝐵(𝜁,2𝑖+1𝑠)⧵𝐵(𝜁,2𝑖𝑠))}||𝑓(𝑄)||||𝜁−𝑄||^𝑛𝑑𝜎_𝑄≲𝛿(𝑃)^∞_𝑖=121^𝑖𝑠^𝑛_{𝐸3∩𝐵(𝜁,2𝑖+1𝑠)}||𝑓(𝑄)||𝑑𝜎_𝑄≲𝛿(𝑃)^∞_𝑖=1≲𝛿(𝑃)_∞ℵ₁𝑓,2^𝑖+1𝑠,𝜁_𝑖=1^2𝑖+2𝑠2_𝑖+1𝑠ℵ₁(𝑓,𝑙,𝜁)≲𝛿(𝑃)_∞𝑠𝑙𝑑𝑙ℵ₁(𝑓,𝑙,𝜁)≲𝛿(𝑃)_{∞𝛿(𝑃)}𝑙𝑑𝑙ℵ₁(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.5)

Thus, it follows that ||𝒫ℐ_Ω𝑓(𝑃)||≲1𝑂(1)||𝑉¹≲𝑟_{(1−𝑛)/𝑝}𝑓(𝑃)||+||𝑉₂𝑓(𝑃)||+||𝑉₃𝑓(𝑃)||+||𝑉₄𝑓(𝑃)||‖𝑓‖_𝑝+𝛿(𝑃)^{∞𝛿(𝑃)}ℵ₁(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.6)

Using the fact that 𝑠≲𝛿(𝑃)≲𝑟𝜑_Ω(Θ), we get ||𝒫ℐ_Ω𝑓(𝑃)||≲ℵ₁(𝑓,2𝑠,𝜁)+𝛿(𝑃)^{∞𝛿(𝑃)}ℵ₁(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.7)

It is clear that ^{∞𝛿(𝑃)}ℵ₁(𝑓,𝑙,𝜁)𝑙𝑑𝑙(3.8) is a convergent integral, since ℵ₁(𝑓,l,𝜁)𝑙≲𝑠^−1−𝑛𝑠^𝑛/𝑞‖𝑓‖_𝑝≲𝑠^{−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∈𝔼^𝑝𝑓(𝐺(Ω)) (and thus that 𝜁∈𝔼^1𝑓(𝐺(Ω))), it follows that 𝒫ℐ_Ω𝑓(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).