Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 825240 | https://doi.org/10.1155/2012/825240

Lei Qiao, "Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone", Journal of Function Spaces, vol. 2012, Article ID 825240, 9 pages, 2012. https://doi.org/10.1155/2012/825240

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

Academic Editor: Dachun Yang
Received17 May 2012
Accepted08 Jul 2012
Published01 Aug 2012

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 ๐‘ and ๐‘+ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by ๐‘๐‘›(๐‘›โ‰ฅ2) the ๐‘›-dimensional Euclidean space. A point in ๐‘๐‘› is denoted by ๐‘ƒ=(๐‘‹,๐‘ฅ๐‘›), where ๐‘‹=(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆ’1). The Euclidean distance of two points ๐‘ƒ and ๐‘„ in ๐‘๐‘› is denoted by |๐‘ƒโˆ’๐‘„|. Also |๐‘ƒโˆ’๐‘‚| with the origin ๐‘‚ of ๐‘๐‘› is simply denoted by |๐‘ƒ|. The boundary, the closure, and the complement of a set ๐’ in ๐‘๐‘› are denoted by ๐œ•๐’, ๐’, and ๐’๐‘, respectively.

We introduce a system of spherical coordinates (๐‘Ÿ,ฮ˜), โ€‰ฮ˜=(๐œƒ1,๐œƒ2,โ€ฆ,๐œƒ๐‘›โˆ’1), in ๐‘๐‘› which are related to cartesian coordinates (๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆ’1,๐‘ฅ๐‘›) by ๐‘ฅ๐‘›=๐‘Ÿcos๐œƒ1.

For positive functions โ„Ž1 and โ„Ž2, we say that โ„Ž1โ‰ฒโ„Ž2 if โ„Ž1โ‰ค๐‘€โ„Ž2 for some constant ๐‘€>0. If โ„Ž1โ‰ฒโ„Ž2 and โ„Ž2โ‰ฒโ„Ž1, we say that โ„Ž1โ‰ˆโ„Ž2.

For ๐‘ƒโˆˆ๐‘๐‘› and ๐‘…>0, let ๐ต(๐‘ƒ,๐‘…) denote the open ball with center at ๐‘ƒ and radius ๐‘… in ๐‘๐‘›. The unit sphere and the upper half unit sphere are denoted by ๐’๐‘›โˆ’1 and ๐’+๐‘›โˆ’1, respectively. For simplicity, a point (1,ฮ˜) on ๐’๐‘›โˆ’1 and the set {ฮ˜;(1,ฮ˜)โˆˆฮฉ} for a set ฮฉ, ฮฉโŠ‚๐’๐‘›โˆ’1 are often identified with ฮ˜ and ฮฉ, respectively. For two sets ฮžโŠ‚๐‘+ and ฮฉโŠ‚๐’๐‘›โˆ’1, the set {(๐‘Ÿ,ฮ˜)โˆˆ๐‘๐‘›;๐‘Ÿโˆˆฮž,(1,ฮ˜)โˆˆฮฉ} in ๐‘๐‘› is simply denoted by ฮžร—ฮฉ. In particular, the half space ๐‘+ร—๐’+๐‘›โˆ’1={(๐‘‹,๐‘ฅ๐‘›)โˆˆ๐‘๐‘›;๐‘ฅ๐‘›>0} will be denoted by ๐“๐‘›.

By ๐ถ๐‘›(ฮฉ), we denote the set ๐‘+ร—ฮฉ in ๐‘๐‘› with the domain ฮฉ on ๐’๐‘›โˆ’1. We call it a cone. Then ๐‘‡๐‘› is a special cone obtained by putting ฮฉ=๐’+๐‘›โˆ’1. We denote the sets ๐ผร—ฮฉ and ๐ผร—๐œ•ฮฉ with an interval on ๐‘ by ๐ถ๐‘›(ฮฉ;๐ผ) and ๐‘†๐‘›(ฮฉ;๐ผ). By ๐‘†๐‘›(ฮฉ) we denote ๐‘†๐‘›(ฮฉ;(0,+โˆž)) which is ๐œ•๐ถ๐‘›(ฮฉ)โˆ’{๐‘‚}.

Let ฮฉ be a domain on ๐’๐‘›โˆ’1 with smooth boundary. Consider the Dirichlet problem: ๎€ทฮ›๐‘›๎€ธ+๐œ†๐œ‘=0onฮฉ,๐œ‘=0on๐œ•ฮฉ,(1.1) where ฮ›๐‘› is the spherical part of the Laplace operator ฮ”๐‘›โˆถฮ”๐‘›=๐‘›โˆ’1๐‘Ÿ๐œ•+๐œ•๐œ•๐‘Ÿ2๐œ•๐‘Ÿ2+ฮ›๐‘›๐‘Ÿ2.(1.2) We denote the least positive eigenvalue of this boundary value problem by ๐œ†ฮฉ and the normalized positive eigenfunction corresponding to ๐œ†ฮฉ by ๐œ‘ฮฉ(ฮ˜), ๎€œฮฉ๐œ‘2ฮฉ(ฮ˜)๐‘‘๐œŽฮ˜=1,(1.3) where ๐‘‘๐œŽฮ˜ is the surface area on ๐‘†๐‘›โˆ’1. We denote the solutions of the equation ๐‘ก2+(๐‘›โˆ’2)๐‘กโˆ’๐œ†ฮฉ=0 by ๐›ผฮฉ,โˆ’๐›ฝฮฉ (๐›ผฮฉ,๐›ฝฮฉ>0). If ฮฉ=๐’+๐‘›โˆ’1, then ๐›ผฮฉ=1,๐›ฝฮฉ=๐‘›โˆ’1, and ๐œ‘1(ฮ˜)=(2๐‘›๐‘ ๐‘›โˆ’1)1/2cos๐œƒ1, where ๐‘ ๐‘› is the surface area 2๐œ‹๐‘›/2(ฮ“(๐‘›/2))โˆ’1 of ๐’1.

To simplify our consideration in the following, we will assume that if ๐‘›โ‰ฅ3, then ฮฉ is a ๐ถ2,๐›ผ-domain (0<๐›ผ<1) on ๐’๐‘›โˆ’1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see ([1], pages 88-89) for the definition of ๐ถ2,๐›ผ-domain). Then by modifying Mirandaโ€™s method ([2], pages 7-8), we can prove the following inequality: ๐œ‘ฮฉ(ฮ˜)โ‰ˆdist(ฮ˜,๐œ•ฮฉ)(ฮ˜โˆˆฮฉ).(1.4)

For any (1,ฮ˜)โˆˆฮฉ, we have (see [3]) ๐œ‘ฮฉ๎€ท(ฮ˜)โ‰ˆdist(1,ฮ˜),๐œ•๐ถ๐‘›๎€ธ(ฮฉ),(1.5) which yields that ๐›ฟ(๐‘ƒ)โ‰ˆ๐‘Ÿ๐œ‘ฮฉ(ฮ˜),(1.6) where ๐›ฟ(๐‘ƒ)=dist(๐‘ƒ,๐œ•๐ถ๐‘›(ฮฉ)) and ๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆ๐ถ๐‘›(ฮฉ).

Let ๐บฮฉ(๐‘ƒ,๐‘„)(๐‘ƒ=(๐‘Ÿ,ฮ˜),๐‘„=(๐‘ก,ฮฆ)โˆˆ๐ถ๐‘›(ฮฉ)) be the Green function of ๐ถ๐‘›(ฮฉ). We define the Poisson kernel ๐พฮฉ(๐‘ƒ,๐‘„) by ๐พฮฉ1(๐‘ƒ,๐‘„)=๐‘๐‘›๐œ•๐œ•๐‘›๐‘„๐บฮฉ(๐‘ƒ,๐‘„),(1.7) where ๐‘๐‘›=๎‚ป(2๐œ‹๐‘›=2,๐‘›โˆ’2)๐‘ ๐‘›๐‘›โ‰ฅ3,(1.8)๐‘„โˆˆ๐‘†๐‘›(ฮฉ) and ๐œ•/๐œ•๐‘›๐‘„ denotes the differentiation at ๐‘„ along the inward normal into ๐ถ๐‘›(ฮฉ).

In this paper, we consider functions ๐‘“โˆˆ๐ฟ๐‘(๐œ•๐ถ๐‘›(ฮฉ)), where 1โ‰ค๐‘<โˆž. Then the Poisson integral ๐‘Šฮฉ๐‘“(๐‘ƒ)(๐‘ƒโˆˆ๐ถ๐‘›(ฮฉ)) is defined by ๐‘Šฮฉ๎€œ๐‘“(๐‘ƒ)=๐‘†๐‘›(ฮฉ)๐พฮฉ(๐‘ƒ,๐‘„)๐‘“(๐‘„)๐‘‘๐œŽ๐‘„,(1.9) where ๐‘‘๐œŽ๐‘„ is the surface area element on ๐‘†๐‘›(ฮฉ).

Remark 1.1. Let ฮฉ=๐’+๐‘›โˆ’1. Then ๐บ๐’+๐‘›โˆ’1๎‚ป||(๐‘ƒ,๐‘„)=log๐‘ƒโˆ’๐‘„โˆ—||||||||||โˆ’log๐‘ƒโˆ’๐‘„๐‘›=2,๐‘ƒโˆ’๐‘„2โˆ’๐‘›โˆ’||๐‘ƒโˆ’๐‘„โˆ—||2โˆ’๐‘›๐‘›โ‰ฅ3,(1.10) where ๐‘„โˆ—=(๐‘Œ,โˆ’๐‘ฆ๐‘›), that is, ๐‘„โˆ— is the mirror image of ๐‘„=(๐‘Œ,๐‘ฆ๐‘›) with respect to ๐œ•๐‘‡๐‘›. Hence, for the two points ๐‘ƒ=(๐‘‹,๐‘ฅ๐‘›)โˆˆ๐‘‡๐‘› and ๐‘„=(๐‘Œ,๐‘ฆ๐‘›)โˆˆ๐œ•๐‘‡๐‘›, we have ๐‘๐‘›๐พ๐’+๐‘›โˆ’1๐œ•(๐‘ƒ,๐‘„)=๐œ•๐‘›๐‘„๐บ๐’+๐‘›โˆ’1๎‚ป2||||(๐‘ƒ,๐‘„)=๐‘ƒโˆ’๐‘„โˆ’2๐‘ฅ๐‘›||||๐‘›=2,2(๐‘›โˆ’2)๐‘ƒโˆ’๐‘„โˆ’๐‘›๐‘ฅ๐‘›๐‘›โ‰ฅ3.(1.11)

We fix an open, nonempty, and bounded set ๐บ(ฮฉ)โŠ‚๐œ•๐ถ๐‘›(ฮฉ). In ๐ถ๐‘›(ฮฉ), we normalise the extension, with respect to ๐บ(ฮฉ), by ๐’ซโ„ฮฉ๐‘Š๐‘“(๐‘ƒ)=ฮฉ๐‘“(๐‘ƒ)๐‘Šฮฉ๐œ’๐บ(ฮฉ)(,๐‘ƒ)(1.12) where ๐œ’๐บ(ฮฉ) denotes the characteristic function of ๐บ(ฮฉ).

Let ฮ“๎€ฝ(ฮฉ,๐œ)=๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆ๐ถ๐‘›||||๎€พ(ฮฉ)โˆถ(๐‘Ÿ,ฮ˜)โˆ’๐œโ‰ฒ๐›ฟ(๐‘ƒ)(1.13) be a nontangential cone in ๐ถ๐‘›(ฮฉ) with vertex ๐œโˆˆ๐œ•๐ถ๐‘›(ฮฉ).

We define โ„ต๐‘๎‚ต1(๐‘“,๐‘™,๐‘ƒ)=๐‘™๐‘›โˆ’1๎€œ๐ต(๐‘ƒ,๐‘™)||||๐‘“(๐‘„)๐‘๐‘‘๐œŽ๐‘„๎‚ถ1/๐‘,๐”ผ๐‘๐‘“๎€ฝ(๐บ(ฮฉ))=๐‘ƒโˆˆ๐บ(ฮฉ)โˆถโ„ต๐‘๎€พ.(๐‘“โˆ’๐‘“(๐‘ƒ),๐‘™,๐‘ƒ)โŸถ0as๐‘™โŸถ0(1.14)

Note that, if ๐‘“โˆˆ๐ฟ๐‘(๐œ•๐ถ๐‘›(ฮฉ)), then |๐บ(ฮฉ)โงต๐”ผ๐‘๐‘“(๐บ(ฮฉ))|=0 (a.e. point is a Lebesgue point).

In ๐‘‡๐‘›, the following conclusion was proved by Brundin (see ([4], pages 11โ€“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 Sjรถgren (see [6, 7]), Rรถnning (see [8]), and Brundin (see [9]).

Theorem A. For a.e. ๐œโˆˆ๐บ(๐’+๐‘›โˆ’1), ๐’ซโ„๐’+๐‘›โˆ’1๐‘“(๐‘ƒ)โ†’๐‘“(๐œ) (see Remark 1.1 for the definition of ๐’ซโ„๐’+๐‘›โˆ’1๐‘“(๐‘ƒ)) as ๐‘ƒโ†’๐œ along ฮ“(๐’+๐‘›โˆ’1,๐œ).

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

Theorem 1.2. For any ๐œโˆˆ๐”ผ๐‘๐‘“(๐บ(ฮฉ)) (in particular, for a.e. ๐œโˆˆ๐บ(ฮฉ)) one has that ๐’ซโ„ฮฉ๐‘“(๐‘ƒ)โ†’๐‘“(๐œ) as ๐‘ƒโ†’๐œ along ฮ“(ฮฉ,๐œ).

2. Some Lemmas

Lemma 2.1. One has ๐พฮฉ(๐‘ƒ,๐‘„)โ‰ˆ๐‘Ÿโˆ’๐›ฝฮฉ๐‘ก๐›ผฮฉโˆ’1๐œ‘ฮฉ๎€ท(ฮ˜),resp.๐พฮฉ(๐‘ƒ,๐‘„)โ‰ˆ๐‘Ÿ๐›ผฮฉ๐‘กโˆ’๐›ฝฮฉโˆ’1๐œ‘ฮฉ๎€ธ,(ฮ˜)(2.1) for any ๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆ๐ถ๐‘›(ฮฉ) and any ๐‘„=(๐‘ก,ฮฆ)โˆˆ๐‘†๐‘›(ฮฉ) satisfying 0<๐‘ก/๐‘Ÿโ‰ค4/5(resp.0<๐‘Ÿ/๐‘กโ‰ค4/5); ๐พฮฉ(๐‘ƒ,๐‘„)โ‰ˆ๐‘Ÿ๐œ‘ฮฉ(ฮ˜)||||๐‘ƒโˆ’๐‘„๐‘›,(2.2) for any ๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆ๐ถ๐‘›(ฮฉ) and any ๐‘„=(๐‘ก,ฮฆ)โˆˆ๐‘†๐‘›(ฮฉ;(4๐‘Ÿ/5,5๐‘Ÿ/4)).

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

Lemma 2.2. One has ๐‘Šฮฉ1(๐‘ƒ)=๐‘‚(1)as๐‘ƒโŸถ๐œโˆˆ๐บ.(2.3)

Proof. Write ๐‘Šฮฉ๎€œ1(๐‘ƒ)=๐ธ1+๎€œ๐ธ2+๎€œ๐ธ3=๐‘ˆ1(๐‘ƒ)+๐‘ˆ2(๐‘ƒ)+๐‘ˆ3(๐‘ƒ),(2.4) where ๐ธ1=๐‘†๐‘›๎‚€๎‚€4ฮฉ;0,5๐‘Ÿ๎‚„๎‚,๐ธ2=๐‘†๐‘›๎‚€๎‚ƒ5ฮฉ;4๐‘Ÿ,โˆž๎‚๎‚,๐ธ3=๐‘†๐‘›๎‚€๎‚€4ฮฉ;55๐‘Ÿ,4๐‘Ÿ.๎‚๎‚(2.5)
By (2.1), we have the following estimates ๐‘ˆ1(๐‘ƒ)โ‰ˆ๐‘Ÿโˆ’๐›ฝฮฉ๐œ‘ฮฉ(๎€œฮ˜)๐ธ1๐‘ก๐›ผฮฉโˆ’1๐‘‘๐œŽ๐‘„โ‰ˆ๐‘ ๐‘›๐›ฝฮฉ๎‚€45๎‚๐›ฝฮฉ๐œ‘ฮฉ๐‘ˆ(ฮ˜),(2.6)2๐‘ (๐‘ƒ)โ‰ˆ๐‘›๐›ผฮฉ๎‚€45๎‚๐›ผฮฉ๐œ‘ฮฉ(ฮ˜).(2.7)
Next, we will estimate ๐‘ˆ3(๐‘ƒ). Take a sufficiently small positive number ๐‘˜ such that ๐ธ3โŠ‚๎š๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆฮ›(๐‘˜)๐ต๎‚€1๐‘ƒ,2๐‘Ÿ๎‚,(2.8) where ๎‚ปฮ›(๐‘˜)=๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆ๐ถ๐‘›(ฮฉ);inf๐‘งโˆˆ๐œ•ฮฉ||||๎‚ผ(1,ฮ˜)โˆ’(1,๐‘ง)<๐‘˜,0<๐‘Ÿ<โˆž.(2.9)
Since ๐‘ƒโ†’๐œโˆˆ๐บ, we only consider the case ๐‘ƒโˆˆฮ›(๐‘˜). Now put ๐ป๐‘–๎€ฝ(๐‘ƒ)=๐‘„โˆˆ๐ธ3;2๐‘–โˆ’1๐›ฟ||||(๐‘ƒ)โ‰ค๐‘ƒโˆ’๐‘„<2๐‘–๐›ฟ๎€พ.(๐‘ƒ)(2.10)
Since ๐‘†๐‘›(ฮฉ)โˆฉ{๐‘„โˆˆ๐‘๐‘›โˆถ|๐‘ƒโˆ’๐‘„|<๐›ฟ(๐‘ƒ)}=ร˜, we have ๐‘ˆ3(๐‘ƒ)โ‰ˆ๐‘–(๐‘ƒ)๎“๐‘–=0๎€œ๐ป๐‘–(๐‘ƒ)๐‘Ÿ๐œ‘ฮฉ(ฮ˜)||||๐‘ƒโˆ’๐‘„๐‘›๐‘‘๐œŽ๐‘„,(2.11) where ๐‘–(๐‘ƒ) is a positive integer satisfying 2๐‘–(๐‘ƒ)โˆ’1๐›ฟ(๐‘ƒ)โ‰ค๐‘Ÿ/2<2๐‘–(๐‘ƒ)๐›ฟ(๐‘ƒ).
By (1.6) we have ๎€œ๐ป๐‘–(๐‘ƒ)๐‘Ÿ๐œ‘ฮฉ(ฮ˜)||||๐‘ƒโˆ’๐‘„๐‘›๐‘‘๐œŽ๐‘„โ‰ˆ๐‘Ÿ๐œ‘ฮฉ๎€œ(ฮ˜)๐ป๐‘–(๐‘ƒ)1๐›ฟ(๐‘ƒ)๐‘‘๐œŽ๐‘„=๐‘Ÿ๐œ‘ฮฉ(ฮ˜)๐‘ ๐›ฟ(๐‘ƒ)๐‘›2๐‘–(๐‘ƒ)โ‰ˆ๐‘ ๐‘›2๐‘–(๐‘ƒ)(2.12) for ๐‘–=0,1,2,โ€ฆ,๐‘–(๐‘ƒ).
So ๐‘ˆ3(๐‘ƒ)โ‰ˆ๐‘‚(1).(2.13)
Combining (2.6)โ€“(2.13), Lemma 2.2 is proved.

Lemma 2.3. One has ๐‘Šฮฉ๐œ’๐บ(ฮฉ)(๐‘ƒ)=๐‘Šฮฉ1(๐‘ƒ)+๐‘‚(1)as๐‘ƒโŸถ๐œโˆˆ๐บ(ฮฉ).(2.14)

Proof. In fact, we only need to prove ๐‘ˆ4(๎€œ๐‘ƒ)=๐‘†๐‘›(ฮฉ)โˆ’๐บ(ฮฉ)๐พฮฉ(๐‘ƒ,๐‘„)๐‘‘๐œŽ๐‘„โ‰ฒ๐‘‚(1).(2.15)
Write ๐‘ˆ4(๎€œ๐‘ƒ)=(๐‘†๐‘›(ฮฉ)โˆ’๐บ(ฮฉ))โˆฉ๐ธ1+๎€œ(๐‘†๐‘›(ฮฉ)โˆ’๐บ(ฮฉ))โˆฉ๐ธ2+๎€œ(๐‘†๐‘›(ฮฉ)โˆ’๐บ(ฮฉ))โˆฉ๐ธ3=๐‘ˆ5(๐‘ƒ)+๐‘ˆ6(๐‘ƒ)+๐‘ˆ7(๐‘ƒ),(2.16) where ๐ธ1, ๐ธ2, and ๐ธ3 are sets on ๐‘†๐‘›(ฮฉ) used in Lemma 2.2.
Obviously, ๐‘ˆ5(๐‘ƒ)โ‰ฒ๐‘ˆ1๐‘ˆ(๐‘ƒ)โ‰ˆ๐‘‚(1),(2.17)6(๐‘ƒ)โ‰ฒ๐‘ˆ2(๐‘ƒ)โ‰ˆ๐‘‚(1).(2.18)
Further, we have by (2.2) ๐‘ˆ7(๐‘ƒ)โ‰ˆ๐‘Ÿ๐œ‘ฮฉ(๎€œฮ˜)(๐‘†๐‘›(ฮฉ)โˆ’๐บ(ฮฉ))โˆฉ๐ธ31||||๐‘ƒโˆ’๐‘„๐‘›๐‘‘๐œŽ๐‘„โ‰ฒ๐‘ ๐‘›๐‘‘||๐œ||๐œ‘ฮฉ(ฮ˜)(๐‘ƒโŸถ๐œโˆˆ๐บ(ฮฉ)),(2.19) where ๐‘‘=inf๐‘„โˆˆ๐œ•๐ถ๐‘›(ฮฉ)โˆ’๐บ(ฮฉ)||||.๐‘„โˆ’๐œ(2.20)
Combining (2.17)โ€“(2.19), (2.15) holds which gives the conclusion.

3. Proof of the Theorem 1.2

As ๐‘ƒโ†’๐œโˆˆ๐บ(ฮฉ), ๐‘Šฮฉ๐œ’๐บ(ฮฉ)(๐‘ƒ)=๐‘‚(1) from Lemmas 2.2 and 2.3.

Now, let ๐‘“โˆˆ๐ฟ๐‘(๐œ•๐ถ๐‘›(ฮฉ)) and ๐œโˆˆ๐”ผ๐‘๐‘“(๐บ(ฮฉ)) be given. We may, without loss of generality, assume that ๐‘“(๐œ)=0. Furthermore, we assume that ๐‘ƒ=(๐‘Ÿ,ฮ˜)โˆˆฮ“(ฮฉ,๐œ). For short, let ๐‘ =|(๐‘Ÿ,ฮ˜)โˆ’๐œ|. We write ๐‘Šฮฉ๎€œ๐‘“(๐‘ƒ)=๐ธ1+๎€œ๐ธ2+๎€œ๐ธ3โˆฉ๐ต(๐œ,2๐‘ )+๎€œ๐ธ3โˆฉ๐ต๐‘(๐œ,2๐‘ )=๐‘‰1๐‘“(๐‘ƒ)+๐‘‰2๐‘“(๐‘ƒ)+๐‘‰3๐‘“(๐‘ƒ)+๐‘‰4๐‘“(๐‘ƒ),(3.1) where ๐ธ1, ๐ธ2, and ๐ธ3 are sets on ๐‘†๐‘›(ฮฉ) used in Lemma 2.2.

By using Hรถlderโ€™s inequality, (2.1), we have the following estimates ||๐‘‰1||๐‘“(๐‘ƒ)โ‰ฒ๐‘Ÿโˆ’๐›ฝฮฉ๐œ‘ฮฉ(๎€œฮ˜)๐ธ1๐‘ก๐›ผฮฉโˆ’1๐‘“(๐‘„)๐‘‘๐œŽ๐‘„โ‰ฒ๐‘Ÿ(1โˆ’๐‘›)/๐‘โ€–๐‘“โ€–๐‘,||๐‘‰2||๐‘“(๐‘ƒ)โ‰ฒ๐‘Ÿ(1โˆ’๐‘›)/๐‘โ€–๐‘“โ€–๐‘.(3.2)

Similar to the estimate of ๐‘ˆ3(๐‘ƒ) in Lemma 2.2, we only consider the following inequality by (1.6) ๎€œ๐ป๐‘–(๐‘ƒ)๐‘Ÿ๐œ‘ฮฉ(ฮ˜)||||๐‘ƒโˆ’๐‘„๐‘›๐‘‘๐œŽ๐‘„โ‰ˆ๐‘Ÿ๐œ‘ฮฉ๎€œ(ฮ˜)๐ป๐‘–(๐‘ƒ)1๎€ฝ2๐‘–โˆ’1๎€พ๐›ฟ(๐‘ƒ)๐‘›๐‘‘๐œŽ๐‘„โ‰ฒ๐‘Ÿ๐›ผฮฉ๐œ‘ฮฉ๎€œ(ฮ˜)๐ธ2๐‘กโˆ’๐›ฝฮฉโˆ’1||๐‘“||(๐‘„)๐‘‘๐œŽ๐‘„โ‰ฒ๐‘Ÿ(1โˆ’๐‘›)/๐‘โ€–๐‘“โ€–๐‘(3.3) for ๐‘–=0,1,2,โ€ฆ,๐‘–(๐‘ƒ), which is similar to the estimate of ๐‘‰2๐‘“(๐‘ƒ).

So ||๐‘‰3||๐‘“(๐‘ƒ)โ‰ฒ๐‘Ÿ(1โˆ’๐‘›)/๐‘โ€–๐‘“โ€–๐‘.(3.4)

Notice that |๐‘ƒโˆ’๐‘„|>(1/2)|๐œโˆ’๐‘„| in the case ๐‘„โˆˆ๐ธ3โˆฉ๐ต๐‘(๐œ,2๐‘ ). By (1.6) and (2.2), we have ||๐‘‰4||๎€œ๐‘“(๐‘ƒ)โ‰ฒ๐›ฟ(๐‘ƒ)๐ธ3โˆฉ๐ต๐‘(๐œ,2๐‘ )||||๐‘“(๐‘„)||||๐‘ƒโˆ’๐‘„๐‘›๐‘‘๐œŽ๐‘„โ‰ฒ๐›ฟ(๐‘ƒ)โˆž๎“๐‘–=1๎€œ๐ธ3โˆฉ(๐ต(๐œ,2๐‘–+1๐‘ )โงต๐ต(๐œ,2๐‘–๐‘ ))||๐‘“||(๐‘„)||||๐œโˆ’๐‘„๐‘›๐‘‘๐œŽ๐‘„โ‰ฒ๐›ฟ(๐‘ƒ)โˆž๎“๐‘–=1๎‚ต12๐‘–๐‘ ๎‚ถ๐‘›๎€œ๐ธ3โˆฉ๐ต(๐œ,2๐‘–+1๐‘ )||||๐‘“(๐‘„)๐‘‘๐œŽ๐‘„โ‰ฒ๐›ฟ(๐‘ƒ)โˆž๎“๐‘–=1โ„ต1๎€ท๐‘“,2๐‘–+1๎€ธ๐‘ ,๐œโ‰ฒ๐›ฟ(๐‘ƒ)โˆž๎“๐‘–=1๎€œ2๐‘–+2๐‘ 2๐‘–+1๐‘ โ„ต1(๐‘“,๐‘™,๐œ)๐‘™๎€œ๐‘‘๐‘™โ‰ฒ๐›ฟ(๐‘ƒ)โˆž๐‘ โ„ต1(๐‘“,๐‘™,๐œ)๐‘™๎€œ๐‘‘๐‘™โ‰ฒ๐›ฟ(๐‘ƒ)โˆž๐›ฟ(๐‘ƒ)โ„ต1(๐‘“,๐‘™,๐œ)๐‘™๐‘‘๐‘™.(3.5)

Thus, it follows that ||๐’ซโ„ฮฉ||โ‰ฒ1๐‘“(๐‘ƒ)๐‘‚๎€บ||๐‘‰(1)1||+||๐‘‰๐‘“(๐‘ƒ)2||+||๐‘‰๐‘“(๐‘ƒ)3||+||๐‘‰๐‘“(๐‘ƒ)4||๎€ป๐‘“(๐‘ƒ)โ‰ฒ๐‘Ÿ(1โˆ’๐‘›)/๐‘โ€–๐‘“โ€–๐‘๎€œ+๐›ฟ(๐‘ƒ)โˆž๐›ฟ(๐‘ƒ)โ„ต1(๐‘“,๐‘™,๐œ)๐‘™๐‘‘๐‘™.(3.6)

Using the fact that ๐‘ โ‰ฒ๐›ฟ(๐‘ƒ)โ‰ฒ๐‘Ÿ๐œ‘ฮฉ(ฮ˜), we get ||๐’ซโ„ฮฉ||๐‘“(๐‘ƒ)โ‰ฒโ„ต1๎€œ(๐‘“,2๐‘ ,๐œ)+๐›ฟ(๐‘ƒ)โˆž๐›ฟ(๐‘ƒ)โ„ต1(๐‘“,๐‘™,๐œ)๐‘™๐‘‘๐‘™.(3.7)

It is clear that ๎€œโˆž๐›ฟ(๐‘ƒ)โ„ต1(๐‘“,๐‘™,๐œ)๐‘™๐‘‘๐‘™(3.8) is a convergent integral, since โ„ต1(๐‘“,l,๐œ)๐‘™โ‰ฒ๐‘ โˆ’1โˆ’๐‘›๐‘ ๐‘›/๐‘žโ€–๐‘“โ€–๐‘โ‰ฒ๐‘ โˆ’1โˆ’(๐‘›/๐‘)โ€–๐‘“โ€–๐‘(3.9) from the Hรถlderโ€™s inequality.

Now, as ๐›ฟ(๐‘ƒ)โ†’0, we also have ๐‘ โ†’0. Since ๐‘“(๐œ)=0 and since we have assumed that ๐œโˆˆ๐”ผ๐‘๐‘“(๐บ(ฮฉ)) (and thus that ๐œโˆˆ๐”ผ1๐‘“(๐บ(ฮฉ))), it follows that ๐’ซโ„ฮฉ๐‘“(๐‘ƒ)โ†’0=๐‘“(๐œ) as ๐‘ƒ=(๐‘Ÿ,ฮ˜)โ†’๐œ along ฮ“(ฮฉ,๐œ). This concludes the proof.

Acknowledgments

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

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Copyright © 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.

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