About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volumeย 2011ย (2011), Article IDย 238138, 39 pages
http://dx.doi.org/10.1155/2011/238138
Research Article

Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation

Graduate School of Humanities and Sciences, Nara Women's University, Nara 630-8506, Japan

Received 6 October 2010; Revised 23 December 2010; Accepted 26 January 2011

Academic Editor: M.ย Lakshmanan

Copyright ยฉ 2011 Kyoko Tomoeda. 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

We consider the initial value problem for the reduced fifth-order KdV-type equation: ๐œ•๐‘ก๐‘ขโˆ’๐œ•5๐‘ฅ๐‘ขโˆ’10๐œ•๐‘ฅ(๐‘ข3)+10๐œ•๐‘ฅ(๐œ•๐‘ฅ๐‘ข)2=0, ๐‘ก,๐‘ฅโˆˆโ„, ๐‘ข(0,๐‘ฅ)=๐œ™(๐‘ฅ), ๐‘ฅโˆˆโ„. This equation is obtained by removing the nonlinear term 10๐‘ข๐œ•3๐‘ฅ๐‘ข from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data ๐œ™โˆˆ๐ป๐‘ (โ„)(๐‘ >1/8) satisfies the condition โˆ‘โˆž๐‘˜=0(๐ด๐‘˜0/๐‘˜!)โ€–(๐‘ฅ๐œ•๐‘ฅ)๐‘˜๐œ™โ€–๐ป๐‘ <โˆž, for some constant ๐ด0(0<๐ด0<1). Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of ๐œ•๐‘ฅ(๐œ•๐‘ฅ๐‘ข)2 on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).

1. Introduction

The KdV hierarchy is well known as the series of the Lax pair formulation [1, 2], which are presented as 1st-orderKdV๐œ•๐‘ก๐‘ขโˆ’๐œ•๐‘ฅ๐‘ข=0,(1.1)03rd-orderKdV๐œ•๐‘ก๐‘ข+๐œ•3๐‘ฅ๐‘ขโˆ’6๐‘ข๐œ•๐‘ฅ๐‘ข=0,(1.1)15th-orderKdV๐œ•๐‘ก๐‘ขโˆ’๐œ•5๐‘ฅ๐‘ขโˆ’10๐œ•๐‘ฅ๎€ท๐‘ข3๎€ธ+10๐œ•๐‘ฅ๎€ท๐œ•๐‘ฅ๐‘ข๎€ธ2+10๐‘ข๐œ•3๐‘ฅโ‹ฎ๐‘ข=0.(1.1)2 These equations describe mathematical models of some water waves [3, 4]. We are interested in the existence theory of the analytic solution and the smoothing effect of the KdV hierarchy. There are some results concerning the analyticity for the third-order KdV equation (1.1)1. To. Kato and Masuda [5] considered the initial value problem of the following equation: ๐œ•๐‘ก๐‘ข+๐œ•3๐‘ฅ๐‘ข+๐‘Ž(๐‘ข)๐œ•๐‘ฅ๐‘ข=0,๐‘ก,๐‘ฅโˆˆโ„,(1.2) where ๐‘Ž(๐œ†) is the real analytic and the no growth rate function in ๐œ†โˆˆโ„. They showed that if the initial data is real analytic, then, the global solution of (1.2) is real analytic in the space variable. Hayashi [6] also considered (1.2) in which ๐‘Ž(๐œ†) is the polynomial. He showed that if the initial data is analytic and has an analytic continuation to a strip containing the real axis, then, the local solution also has the same property. When ๐‘Ž(๐‘ข)=โˆ’6๐‘ข, (1.2) becomes the third-order KdV equation (1.1)1. K. Kato and Ogawa [7] proved that (1.1)1 has not only the real analytic solution in both time and space variables but also the smoothing effect.

Recently, it is shown that the nonlinear dispersive equations including the KdV hierarchy have the local analytic solution in the space variable (see [8]). However, neither the existence of the real analytic solution in both time and space variables nor the smoothing effect is obtained for (1.1)๐‘— with ๐‘—โ‰ฅ2, because the bilinear estimate of ๐‘ข๐œ•๐‘ฅ2๐‘—โˆ’1๐‘ข with ๐‘—โ‰ฅ2 cannot be obtained by their method used in [9].

On the other hand, we may expect that the method used in [7] can work for the reduced equation given by removing ๐‘ข๐œ•๐‘ฅ2๐‘—โˆ’1๐‘ข from the higher-order KdV equations (1.1)0 with ๐‘—โ‰ฅ2. In this paper, as a starting point for this attempt, we consider the following initial value problem of the reduced fifth-order KdV-type equation: ๐œ•๐‘ก๐‘ขโˆ’๐œ•5๐‘ฅ๐‘ข=๐œ•๐‘ฅ๎€ท๐‘ข3๎€ธ+๐œ•๐‘ฅ๎€ท๐œ•๐‘ฅ๐‘ข๎€ธ2,๐‘ก,๐‘ฅโˆˆโ„,๐‘ข(0,๐‘ฅ)=๐œ™(๐‘ฅ),๐‘ฅโˆˆโ„,(1.3) where we may take all coefficients of the nonlinear terms to be equal to 1 without loss of generality. This equation is obtained by removing the nonlinear term 10๐‘ข๐œ•3๐‘ฅ๐‘ข from the original fifth-order KdV equation (1.1)2. Our main purpose is to prove not only the existence of a local real analytic solution of (1.3) in both time and space variables but also the smoothing effect.

Before stating the main result precisely, we introduce the function space introduced by Bourgain (see [10]): for ๐‘ ,๐‘โˆˆโ„, define that ๐‘‹๐‘ ๐‘=๎‚†๐‘“โˆˆ๐’ฎ๎…ž๎€ทโ„2๎€ธ;โ€–๐‘“โ€–๐‘‹๐‘ ๐‘๎‚‡<โˆž,(1.4) where โ€–๐‘“โ€–2๐‘‹๐‘ ๐‘=๎€โ„2๎€ท||1+๐œโˆ’๐œ‰5||๎€ธ2๐‘๎€ท||๐œ‰||๎€ธ1+2๐‘ ||โ„ฑ๐‘ก,๐‘ฅ||๐‘“(๐œ,๐œ‰)2๐‘‘๐œ๐‘‘๐œ‰,(1.5) and โ„ฑ๐‘ก,๐‘ฅ๐‘“ is the Fourier transform of ๐‘“ in both ๐‘ฅ and ๐‘ก variables; that is, โ„ฑ๐‘ก,๐‘ฅ๐‘“(๐œ,๐œ‰)=(2๐œ‹)โˆ’1๎€โ„2๐‘“(๐‘ก,๐‘ฅ)๐‘’โˆ’๐‘–๐‘ก๐œโˆ’๐‘–๐‘ฅ๐œ‰๐‘‘๐‘ก๐‘‘๐‘ฅ.(1.6)

Our main result is the following theorem.

Theorem 1.1. Let ๐‘ >1/8 and let ๐‘โˆˆ(1/2,23/40). Then, for any ๐œ™(๐‘ฅ)โˆˆ๐ป๐‘ (โ„) such that ๎€ท๐‘ฅ๐œ•๐‘ฅ๎€ธ๐‘˜๐œ™(๐‘ฅ)โˆˆ๐ป๐‘ (โ„)(๐‘˜=0,1,2,โ€ฆ),โˆž๎“๐‘˜=0๐ด๐‘˜0โ€–โ€–๎€ท๐‘˜!๐‘ฅ๐œ•๐‘ฅ๎€ธ๐‘˜๐œ™โ€–โ€–๐ป๐‘ <โˆž,forsome0<๐ด0<1,(1.7) there exist a constant ๐‘‡=๐‘‡(๐œ™)>0 and a unique solution ๐‘ขโˆˆ๐ถ((โˆ’๐‘‡,๐‘‡),๐ป๐‘ )โˆฉ๐‘‹๐‘ ๐‘ of (1.3) satisfying ๐‘ƒ๐‘˜๐‘ขโˆˆ๐ถ((โˆ’๐‘‡,๐‘‡),๐ป๐‘ )โˆฉ๐‘‹๐‘ ๐‘,โˆž๎“๐‘˜=0๐ด๐‘˜0โ€–โ€–๐‘ƒ๐‘˜!๐‘˜๐‘ขโ€–โ€–๐‘‹๐‘ ๐‘<โˆž,(1.8) where ๐‘ƒ=5๐‘ก๐œ•๐‘ก+๐‘ฅ๐œ•๐‘ฅ is the generator of dilation for the linear part of the equation of (1.3).
Moreover this solution becomes real analytic in both time and space variables; that is, there exist the positive constants ๐ถ and ๐ด1 such that ||๐œ•๐‘š๐‘ก๐œ•๐‘™๐‘ฅ๐‘ข||(๐‘ก,๐‘ฅ)โ‰ค๐ถ๐ด1๐‘š+๐‘™(๐‘š+๐‘™)!(1.9) holds for all (๐‘ก,๐‘ฅ)โˆˆ(โˆ’๐‘‡,0)โˆช(0,๐‘‡)ร—โ„ and ๐‘™,๐‘š=0,1,2,โ€ฆ.

Remark 1.2. The initial data ๐œ™(๐‘ฅ) has to be analytic except for ๐‘ฅ=0 but is allowed to have ๐ป๐‘ -singularity at ๐‘ฅ=0. It follows from (1.9) that the singularity of ๐œ™(๐‘ฅ) disappears after time passes and the regularity of the local solution of (1.3) reaches real analyticity in both time and space variables; that is, the fifth-order KdV-type equation has the smoothing effect.

Remark 1.3. A typical example of the initial data satisfying the condition (1.7) is given by |๐‘ฅ|๐›พ๐‘’โˆ’๐‘ฅ23with๐›พ>โˆ’8.(1.10)

The existence results of the higher-order KdV equation are studied by many authors. Saut [11] and Schwarz [12] proved that each equation of the KdV hierarchy has a unique global solution in the spatially periodic Sobolev space. Kenig, Ponce, and Vega studied the initial value problem of the higher-order dispersive equation๐œ•๐‘ก๐‘ข+๐œ•๐‘ฅ2๐‘—+1๎‚€๐‘ข+๐‘ƒ๐‘ข,๐œ•๐‘ฅ๐‘ข,โ€ฆ,๐œ•๐‘ฅ2๐‘—โˆ’1๐‘ข๎‚=0,(1.11) where ๐‘—โ‰ฅ1 and ๐‘ƒ(โ‹…) is a polynomial having no constant or linear part. They proved the local well-posedness in the weighted Sobolev space [13, 14]. Recently, Kwon [15] studied the simplified fifth-order KdV-type equation ๐œ•๐‘ก๐‘ข+๐œ•5๐‘ฅ๐‘ข+๐œ•๐‘ฅ๐‘ข๐œ•2๐‘ฅ๐‘ข+๐‘ข๐œ•3๐‘ฅ๐‘ข=0,(1.12) which is obtained by removing the term 10๐œ•๐‘ฅ(๐‘ข3) from (1.1)2. He showed the local well-posedness for the IVP of this equation in ๐ป๐‘ (โ„) with ๐‘ >5/2. On the other hand, Ta. Kato [16] proved the following result for (1.1)2.

Well-Posedness Theorem (Ta. Kato)
(1)Let 1๐‘ >โˆ’43,๐‘ โ‰ฅโˆ’2๐‘Žโˆ’2,whereโˆ’21<๐‘Žโ‰คโˆ’4.(1.13) Then, the local well-posedness for the IVP of (1.1)2 holds in ๐ป๐‘ ,๐‘Ž(โ„), where ๐ป๐‘ ,๐‘Ž๎‚ป(โ„)โ‰ก๐‘“โˆˆ๐’ฎ๎…ž(โ„);โ€–๐‘“โ€–๐ป๐‘ ,๐‘Žโ‰กโ€–โ€–โŸจ๐œ‰โŸฉ๐‘ โˆ’๐‘Ž||๐œ‰||๐‘Ž๎๐‘“โ€–โ€–๐ฟ2๐œ‰๎‚ผ<โˆž.(1.14)(2) Let 1๐‘ โ‰ฅ1,โˆ’21โ‰ค๐‘Žโ‰คโˆ’4.(1.15) Then, the global well-posedness for the IVP of (1.1)2 holds in ๐ป๐‘ ,๐‘Ž(โ„).

The plan of this paper is as follows. In Section 2, we give the existence and uniqueness of the local solution of (1.3), which is shown by the contraction argument consisting of Lemmas 2.3โ€“2.5. In Section 3, we prove Lemma 2.4 which gives the bilinear estimate for ๐œ•๐‘ฅ(๐œ•๐‘ฅ๐‘ข)2 in the Bourgain space. Kenig et al. [9] showed the bilinear estimate for ๐‘ข๐œ•๐‘ฅ๐‘ข of (1.1)1 by estimating the potential which appears in an expression of the Bourgain norm of this term via duality. They divided the domain of integration of the potential into 17 subregions. However, their method of the domain decomposition is consistent with (1.1)2, but not with the fifth-order KdV-type equation. We divide this domain into 30 subregions to derive the bilinear estimate for ๐œ•๐‘ฅ(๐œ•๐‘ฅ๐‘ข)2. In Section 4, we show the analyticity of the solution stated in Theorem 1.1 by the bootstrap argument. The result of this paper is announced in Proceedings of the Japan Academy [17].

Notation. Let โ„ฑx be the Fourier transform in the x variable, and let โ„ฑ๐œ‰โˆ’1 and โ„ฑโˆ’1๐œ,๐œ‰ be the Fourier inverse transform in the ๐œ‰ and (๐œ,๐œ‰) variables, respectively. The Riesz operator Dx and its fractional derivative โŸจDxโŸฉs are defined by ๐ท๐‘ฅ=โ„ฑ๐œ‰โˆ’1||๐œ‰||โ„ฑ๐‘ฅ,โŸจ๐ท๐‘ฅโŸฉ๐‘ =โ„ฑ๐œ‰โˆ’1โŸจ๐œ‰โŸฉ๐‘ โ„ฑ๐‘ฅ,(1.16) respectively, where โŸจโ‹…โŸฉ=(1+|โ‹…|). Similarly, โŸจ๐ท๐‘ก,๐‘ฅโŸฉ๐‘  is defined by โŸจ๐ท๐‘ก,๐‘ฅโŸฉ๐‘ =โ„ฑโˆ’1๐œ,๐œ‰๎ซ||๐œ‰||๎ฌ|๐œ|+๐‘ โ„ฑ๐‘ก,๐‘ฅ.(1.17)[๐ด,๐ต] denotes the commutator relation of two operators given by ๐ด๐ตโˆ’๐ต๐ด. ๐ฟ๐‘๐‘ก๐ฟ๐‘ž๐‘ฅ denotes the space ๐ฟ๐‘(โ„๐‘ก;๐ฟ๐‘ž(โ„๐‘ฅ)) for 1โ‰ค๐‘,๐‘žโ‰คโˆž with the norm โ€–๐‘“โ€–๐ฟ๐‘๐‘ก๐ฟ๐‘ž๐‘ฅ=๎ƒฉ๎€œโˆžโˆ’โˆž๎‚ต๎€œโˆžโˆ’โˆž||||๐‘“(๐‘ก,๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ก๐‘ž/๐‘๎ƒช๐‘‘๐‘ฅ1/๐‘ž.(1.18) We use Sobolev spaces with both time and space variables ๐ป๐‘ ๐‘ก,๐‘ฅ๎€ทโ„2๎€ธ=๎€ฝ๐‘ขโˆˆ๐’ฎ๎…ž๎€ทโ„2๎€ธโˆถ๎ซ๐ท๐‘ก,๐‘ฅ๎ฌ๐‘ ๐‘ขโˆˆ๐ฟ2๐‘ก๐ฟ2๐‘ฅ๎€พ,(1.19) with the norm โ€–โ‹…โ€–๐ป๐‘ ๐‘ก,๐‘ฅ(โ„2)=โ€–โŸจ๐ท๐‘ก,๐‘ฅโŸฉ๐‘ โ‹…โ€–๐ฟ2๐‘ก๐ฟ2๐‘ฅ. Moreover, ๐ฟ2๐‘ก(โ„;๐ป๐‘ ๐‘ฅ) denotes the space ๐ฟ2(โ„๐‘ก;๐ป๐‘ (โ„๐‘ฅ)) with the norm โ€–โ‹…โ€–๐ฟ2๐‘ก(โ„;๐ป๐‘ ๐‘ฅ)=โ€–โŸจ๐ท๐‘ฅโŸฉ๐‘ โ‹…โ€–๐ฟ2๐‘ก๐ฟ2๐‘ฅ. The dual coupling is expressed as โŸจ๐‘“,๐‘”โŸฉ. The convolution of ๐‘“ and ๐‘” with both space and time variables is denoted by ๐‘“โˆ—๐‘”. For the constant ๐ด0 appearing in Theorem 1.1, we put ๐’œ๐ด0๎€ท๐‘‹๐‘ ๐‘๎€ธ=๎‚†๎€ท๐‘“๐Ÿ=0,๐‘“1๎€ธ,โ€ฆ;๐‘“๐‘˜โˆˆ๐‘‹๐‘ ๐‘||๐Ÿ||(๐‘˜=0,1,โ€ฆ),๐’œ๐ด0(๐‘‹๐‘ ๐‘)๎‚‡,<โˆž(1.20) where ||||โ€–๐Ÿโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰กโˆž๎“๐‘˜=0๐ด๐‘˜0โ€–โ€–๐‘“๐‘˜!๐‘˜โ€–โ€–๐‘‹๐‘ ๐‘.(1.21) For simplicity we make use of three notations: ๎“๐ค=๎“๐‘˜=๐‘˜1+๐‘˜2+๐‘˜3+๐‘˜4,๎“๐ฅ=๎“๐‘™=๐‘™1+๐‘™2+๐‘™3,๎“๐ฆ=๎“๐‘š=๐‘š1+๐‘š2+๐‘š3.(1.22)

2. Existence and Uniqueness of the Solution

In this section, we give the proof of the existence and uniqueness of the solution of (1.3). Let ๐‘ข๐‘˜=๐‘ƒ๐‘˜๐‘ข and ๐œ™๐‘˜(๐‘ฅ)=(๐‘ฅ๐œ•๐‘ฅ)๐‘˜๐œ™(๐‘ฅ), and we derive the equation which ๐‘ข๐‘˜ and ๐œ™๐‘˜(๐‘ฅ) satisfy. Since [๐‘ฅ๐œ•๐‘ฅ,๐œ•๐‘ฅ]=โˆ’๐œ•๐‘ฅ, it follows that(๐‘ƒ+๐‘™)๐‘˜๐œ•๐‘ฅ=๐œ•๐‘ฅ(๐‘ƒ+(๐‘™โˆ’1))๐‘˜,๐‘˜,๐‘™=0,1,2,โ€ฆ.(2.1) Using (2.1) and the following relations ๎€บ๐œ•๐‘กโˆ’๐œ•5๐‘ฅ๎€ป๎€ท๐œ•,๐‘ƒ=5๐‘กโˆ’๐œ•5๐‘ฅ๎€ธ,๎€ท๐œ•๐‘กโˆ’๐œ•5๐‘ฅ๎€ธ๐‘ƒ๐‘˜=(๐‘ƒ+5)๐‘˜๎€ท๐œ•๐‘กโˆ’๐œ•5๐‘ฅ๎€ธ,(2.2) we have from (1.3)๐œ•๐‘ก๐‘ข๐‘˜โˆ’๐œ•5๐‘ฅ๐‘ข๐‘˜=๐’ฉ๐‘˜๐‘ข(๐‘ข),๐‘ก,๐‘ฅโˆˆโ„,๐‘˜(0,๐‘ฅ)=๐œ™๐‘˜(๐‘ฅ),๐‘ฅโˆˆโ„,๐‘˜=0,1,2,โ€ฆ,(2.3) where๐’ฉ๐‘˜(๐‘ข)=๐œ•๐‘ฅ(๐‘ƒ+4)๐‘˜๎€ท๐‘ข3๎€ธ+๐œ•๐‘ฅ(๐‘ƒ+4)๐‘˜๎€ท๐œ•๐‘ฅ๐‘ข๎€ธ2.(2.4) Using the Leibniz rule and (2.1), we can see that ๐’ฉ๐‘˜(๐‘ข)=๐œ•๐‘ฅ๐‘˜๎“๐‘™=0๎‚ต๐‘˜๐‘™๎‚ถ4๐‘˜โˆ’๐‘™๐‘ƒ๐‘™๎€ท๐‘ข3๎€ธ+๐œ•๐‘ฅ๐‘˜๎“๐‘™=0๎‚ต๐‘˜๐‘™๎‚ถ3๐‘˜โˆ’๐‘™(๐‘ƒ+1)๐‘™๎€ท๐œ•๐‘ฅ๐‘ข๎€ธ2=๎“๐ค๐‘˜!๐‘˜1!๐‘˜2!๐‘˜3!๐‘˜4!4๐‘˜4๐œ•๐‘ฅ๎€ท๐‘ข๐‘˜1๐‘ข๐‘˜2๐‘ข๐‘˜3๎€ธ+๎“๐ค๐‘˜!๐‘˜1!๐‘˜2!๐‘˜3!๐‘˜4!3๐‘˜4(โˆ’1)๐‘˜3๐œ•๐‘ฅ๐œ•๎€ท๎€ท๐‘ฅ๐‘ข๐‘˜1๐œ•๎€ธ๎€ท๐‘ฅ๐‘ข๐‘˜2.๎€ธ๎€ธ(2.5) We will show the existence and uniqueness of the solution of (2.3).

Proposition 2.1. Let 1๐‘ >โˆ’4๎‚€1,๐‘โˆˆ2,12๎‚,+๐œŽ(2.6) where ๐œŽ=min{๐‘ /5+1/20,3/16}. Then, for any ๐œ™โ‰ก(๐œ™0,๐œ™1,โ€ฆ) such that ๐œ™๐‘˜โˆˆ๐ป๐‘ (โ„)(๐‘˜=0,1,โ€ฆ) and ||โ€–||๐œ™โ€–๐’œ๐ด0(๐ป๐‘ )<โˆž,(2.7) there exist a constant ๐‘‡=๐‘‡(๐œ™)>0 and a unique solution ๐‘ข๐‘˜โˆˆ๐ถ((โˆ’๐‘‡,๐‘‡),๐ป๐‘ )โˆฉ๐‘‹๐‘ ๐‘ of (2.3) satisfying ||||โ€–๐ฎโ€–๐’œ๐ด0(๐‘‹s๐‘)๎€ท๐‘ข<โˆž,๐ฎโ‰ก0,๐‘ข1๎€ธ.,โ€ฆ(2.8)

Remark 2.2. The uniqueness of the solution of (2.3) yields ๐‘ข๐‘˜=๐‘ƒ๐‘˜๐‘ข for ๐‘˜=0,1,2,โ€ฆ. Moreover, ๐‘ข0 becomes a solution of (1.3), the uniqueness of which also follows.

To prove this proposition we prepare three lemmas (Lemmas 2.3, 2.4 and 2.5), which play an important role in applying the contraction principle to the following system of the integral equations:๐œ“(๐‘ก)๐‘ข๐‘˜=๐œ“(๐‘ก)๐‘’๐‘ก๐œ•5๐‘ฅ๐œ™๐‘˜๎€œ+๐œ“(๐‘ก)๐‘ก0๐‘’(๐‘กโˆ’๐‘กโ€ฒ)๐œ•5๐‘ฅ๐œ“๐‘‡๎€ท๐‘ก๎…ž๎€ธ๐’ฉ๐‘˜(๎€ท๐‘ก๐‘ข)๎…ž๎€ธ๐‘‘๐‘ก๎…ž,(2.9) where ๐‘’๐‘ก๐œ•5๐‘ฅ๐‘“โ‰กโ„ฑ๐œ‰โˆ’1๎‚€๐‘’๐‘–๐œ‰5๐‘ก๎๎‚๐‘“(๐œ‰),(2.10)๐œ“(๐‘ก) denotes a cut-off function in ๐ถโˆž0(โ„) satisfying ๎ƒฏ๐œ“(๐‘ก)=1,if|๐‘ก|โ‰ค1,0,if|๐‘ก|>2,(2.11) and ๐œ“๐‘‡(๐‘ก)=๐œ“(๐‘ก/๐‘‡).

Lemma 2.3. Let 0<๐‘‡<1 and let ๎‚€1๐‘ โˆˆโ„,๐‘โˆˆ2๎‚๎‚€1,1,๐‘Žโ€ฒ,๐‘Žโˆˆ0,2๎‚๎€ท๐‘Ž๎…ž๎€ธ.<๐‘Ž(2.12) Then โ€–โ€–๐œ“(๐‘ก)๐‘’๐‘ก๐œ•5๐‘ฅโ€–โ€–๐œ™(๐‘ฅ)๐‘‹๐‘ ๐‘โ‰ค๐ถ0,๐‘ ,๐‘โ€–๐œ™โ€–๐ป๐‘ ,โ€–โ€–โ€–๎€œ(2.13)๐œ“(๐‘ก)๐‘ก0๐‘’(๐‘กโˆ’๐‘กโ€ฒ)๐œ•5๐‘ฅโ„Ž๎€ท๐‘ก๎…ž๎€ธ๐‘‘๐‘ก๎…žโ€–โ€–โ€–๐‘‹๐‘ ๐‘โ‰ค๐ถ1,๐‘ ,๐‘โ€–โ„Žโ€–๐‘‹๐‘ ๐‘โˆ’1โ€–โ€–๐œ“,(2.14)๐‘‡โ„Žโ€–โ€–๐‘‹๐‘ โˆ’๐‘Žโ‰ค๐ถ2,๐‘ ,โˆ’๐‘Ž,โˆ’๐‘Ž๎…ž๐‘‡(๐‘Žโˆ’aโ€ฒ)/4(1โˆ’๐‘Žโ€ฒ)โ€–โ„Žโ€–๐‘‹๐‘ โˆ’aโ€ฒ,(2.15) where ๐ถ0,๐‘ ,๐‘, ๐ถ1,๐‘ ,๐‘, and ๐ถ2,๐‘ ,โˆ’๐‘Ž,โˆ’๐‘Ž๎…ž are constants depending on ๐‘ , ๐‘, ๐‘Ž, and ๐‘Ž๎…ž.

Proof. Equations (2.13) and (2.14) are obtained by replacing the generator ๐‘’โˆ’๐‘ก๐œ•3๐‘ฅ by ๐‘’๐‘ก๐œ•5๐‘ฅ in the argument given by Kenig et al. [18] (see [19]). For the proof of (2.15), we refer to Lemmaโ€‰โ€‰2.5 in the study by Ginibre-Tsutsumi-Velo [20].

Lemma 2.4. Let 1๐‘ >โˆ’4๎‚€1,๐‘,๐‘โ€ฒโˆˆ2,12๎‚๎€ท+๐œŽ๐‘โ‰ค๐‘๎…ž๎€ธ,(2.16) where ๐œŽ=min{๐‘ /5+1/20,3/16}. Then โ€–โ€–๐œ•๐‘ฅ๐œ•๎€ท๎€ท๐‘ฅ๐‘ข๐œ•๎€ธ๎€ท๐‘ฅ๐‘ฃโ€–โ€–๎€ธ๎€ธ๐‘‹๐‘ ๐‘โ€ฒโˆ’1โ‰ค๐ถ3,๐‘ ,๐‘,๐‘โ€ฒโ€–๐‘ขโ€–๐‘‹๐‘ ๐‘โ€–๐‘ฃโ€–๐‘‹๐‘ ๐‘,(2.17) where ๐ถ3,๐‘ ,๐‘,๐‘๎…ž is a constant depending on ๐‘ , ๐‘, and ๐‘โ€ฒ.

We give for the proof of this lemma, in Section 3.

Lemma 2.5. Let 1๐‘ >โˆ’4,๐‘,๐‘๎…žโˆˆ๎‚€12,34๎‚(๐‘โ‰ค๐‘โ€ฒ).(2.18) Then โ€–โ€–๐œ•๐‘ฅโ€–โ€–(๐‘ข๐‘ฃ๐‘ค)๐‘‹๐‘ ๐‘โ€ฒโˆ’1โ‰ค๐ถ4,๐‘ ,๐‘,๐‘โ€ฒโ€–๐‘ขโ€–๐‘‹๐‘ ๐‘โ€–๐‘ฃโ€–๐‘‹๐‘ ๐‘โ€–๐‘คโ€–๐‘‹๐‘ ๐‘,(2.19) where ๐ถ4,๐‘ ,๐‘,๐‘โ€ฒ is a constant depending on ๐‘ , ๐‘, and ๐‘๎…ž.

Proof. This lemma is proved by improving Chen et al.โ€™s argument used in the case where ๐‘=๐‘๎…žโˆˆ(1/2,3/4) [21].

Proof of Proposition 2.1. We define ๐‘‹๐‘€0=๎‚†๐Ÿโˆˆ๐’œ๐ด0๎€ท๐‘‹๐‘ ๐‘๎€ธ;||โ€–||๐Ÿโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰ค2๐ถ0๐‘€0๎‚‡,(2.20) where ๐ถ0=๐ถ0,๐‘ ,๐‘,๐‘€0=||โ€–||๐œ™โ€–๐’œ๐ด0(๐ป๐‘ ).(2.21) We define a map ฮฆโˆถ๐‘‹๐‘€0โ†’๐‘‹๐‘€0 by ฮฆ(๐‘ข)=(ฮฆ0(๐‘ข),ฮฆ1(๐‘ข),โ€ฆ) and ฮฆ๐‘˜(๐‘ข)=๐œ“(๐‘ก)๐‘’๐‘ก๐œ•5๐‘ฅ๐œ™๐‘˜๎€œ+๐œ“(๐‘ก)๐‘ก0๐‘’(๐‘กโˆ’๐‘กโ€ฒ)๐œ•5๐‘ฅ๐œ“๐‘‡๎€ท๐‘ก๎…ž๎€ธ๐’ฉ๐‘˜(๎€ท๐‘ก๐‘ข)๎…ž๎€ธ๐‘‘๐‘ก๎…ž.(2.22) Let ๐‘๎…ž and ๐‘‡ be positive constants satisfying ๐‘<๐‘๎…ž<1/2+๐œŽ and ๎‚†๎€ท๐‘‡<๐‘š๐‘–๐‘›1,24๐ถ20๐ถ5๐‘’4๐ด0๐‘€20+8๐ถ0๐ถ6๐‘’4๐ด0๐‘€0๎€ธโˆ’1/๐œ‡๎‚‡,(2.23) respectively, where ๐ถ5=๐ถ1,๐‘ ,๐‘๐ถ2,๐‘ ,๐‘โˆ’1,๐‘โ€ฒโˆ’1๐ถ4,๐‘ ,๐‘,๐‘โ€ฒ๐ถ6=๐ถ1,๐‘ ,๐‘๐ถ2,๐‘ ,๐‘โˆ’1,๐‘โ€ฒโˆ’1๐ถ3,๐‘ ,๐‘,๐‘โ€ฒ.(2.24) We now show that ฮฆ is a contraction mapping from ๐‘‹๐‘€0 to itself. According to Lemmas 2.3, 2.4, and 2.5, we have for ๐‘ขโˆˆ๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ€–โ€–ฮฆ๐‘˜โ€–โ€–(๐‘ข)๐‘‹๐‘ ๐‘โ‰ค๐ถ0โ€–โ€–ฮฆ๐‘˜โ€–โ€–๐ป๐‘ +๐ถ5๐‘‡๐œ‡๎“๐ค๐‘˜!๐‘˜1!๐‘˜2!๐‘˜3!๐‘˜4!4๐‘˜4โ€–โ€–๐‘ข๐‘˜1โ€–โ€–๐‘‹๐‘ ๐‘โ€–โ€–๐‘ข๐‘˜2โ€–โ€–๐‘‹๐‘ ๐‘โ€–โ€–๐‘ข๐‘˜3โ€–โ€–๐‘‹๐‘ ๐‘+๐ถ6๐‘‡๐œ‡๎“๐ค๐‘˜!๐‘˜1!๐‘˜2!๐‘˜3!๐‘˜4!3๐‘˜4โ€–โ€–๐‘ข๐‘˜1โ€–โ€–๐‘‹๐‘ ๐‘โ€–โ€–๐‘ข๐‘˜2โ€–โ€–๐‘‹๐‘ ๐‘,(2.25) for any ๐‘˜โ‰ฅ0. Here ๐œ‡=(๐‘๎…žโˆ’๐‘)/4๐‘๎…ž>0. By taking a sum over ๐‘˜, we have ||(||โ€–ฮฆ๐‘ข)โ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰กโˆž๎“๐‘˜=0๐ด0๐‘˜โ€–โ€–ฮฆ๐‘˜!๐‘˜โ€–โ€–(๐‘ข)๐‘‹๐‘ ๐‘โ‰ค๐ถ0||||โ€–๐œ™โ€–๐’œ๐ด0(๐ป๐‘ )+๐ถ5๐‘‡๐œ‡โˆž๎“๐‘˜4=0๎€ท4๐ด0๎€ธ๐‘˜4๐‘˜4!โˆž๎“๐‘˜1=0๐ด๐‘˜10๐‘˜1!โ€–โ€–๐‘ข๐‘˜1โ€–โ€–๐‘‹๐‘ ๐‘โˆž๎“๐‘˜2=0๐ด๐‘˜20๐‘˜2!โ€–โ€–๐‘ข๐‘˜2โ€–โ€–๐‘‹๐‘ ๐‘โˆž๎“๐‘˜3=0๐ด๐‘˜30๐‘˜3!โ€–โ€–๐‘ข๐‘˜3โ€–โ€–๐‘‹๐‘ ๐‘+๐ถ6๐‘‡๐œ‡โˆž๎“๐‘˜4=0๎€ท3๐ด0๎€ธ๐‘˜4๐‘˜4!โˆž๎“๐‘˜3=0๐ด๐‘˜30๐‘˜3!โˆž๎“๐‘˜1=0๐ด๐‘˜10๐‘˜1!โ€–โ€–๐‘ข๐‘˜1โ€–โ€–๐‘‹๐‘ ๐‘โˆž๎“๐‘˜2=0๐ด๐‘˜20๐‘˜2!โ€–โ€–๐‘ข๐‘˜2โ€–โ€–๐‘‹๐‘ ๐‘.(2.26) Since ๐ฎโˆˆ๐‘‹๐‘€0, we have from (2.23) ||โ€–||โ€–ฮฆ(๐‘ข)๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰ค๐ถ0||||โ€–๐œ™โ€–๐’œ๐ด0(๐ป๐‘ )+๐ถ5๐‘’4๐ด0๐‘‡๐œ‡||||โ€–๐ฎโ€–3๐’œ๐ด0(๐‘‹๐‘ ๐‘)+๐ถ6๐‘’4๐ด0๐‘‡๐œ‡||||โ€–๐ฎโ€–2๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰ค๐ถ0๐‘€0+8๐ถ30๐ถ5๐‘’4๐ด0๐‘‡๐œ‡๐‘€30+4๐ถ20๐ถ6๐‘’4๐ด0๐‘‡๐œ‡๐‘€20โ‰ค32๐ถ0๐‘€0,(2.27) which implies ฮฆ(๐‘ข)โˆˆ๐‘‹๐‘€0. Similarly, we have for ๐‘ข and ฬƒ๐‘ขโˆˆ๐’œ๐ด0(๐‘‹๐‘ ๐‘)||โ€–||ฮฆ(๐‘ข)โˆ’ฮฆ(ฬƒ๐‘ข)โ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰ค๐ถ5๐‘’4๐ด0๐‘‡๐œ‡ร—๎‚€||||โ€–๐ฎโ€–2๐’œ๐ด0๎€ท๐‘‹๐‘ ๐‘๎€ธ+||||โ€–๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)||โ€–ฬƒ||๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)+||โ€–ฬƒ||๐ฎโ€–2๐’œ๐ด0๎€ท๐‘‹๐‘ ๐‘๎€ธ๎‚||ฬƒ||โ€–๐ฎโˆ’๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)+๐ถ6๐‘’4๐ด0๐‘‡๐œ‡๎‚€||||โ€–๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)+||โ€–ฬƒ||๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)๎‚||ฬƒ||โ€–๐ฎโˆ’๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰ค๎€ท12๐ถ20๐ถ5๐‘’4๐ด0๐‘€20+4๐ถ0๐ถ6๐‘’4๐ด0๐‘€0๎€ธ๐‘‡๐œ‡||ฬƒ||โ€–๐ฎโˆ’๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘)โ‰ค12||ฬƒ||โ€–๐ฎโˆ’๐ฎโ€–๐’œ๐ด0(๐‘‹๐‘ ๐‘).(2.28) Thus, the mapping ฮฆ is contraction from ๐‘‹๐‘€0 to itself. We obtain a unique fixed point ๐‘ข๐‘˜โˆˆ๐‘‹๐‘ ๐‘ satisfying ๐‘ข๐‘˜(๐‘ก)=๐œ“(๐‘ก)๐‘’โˆ’๐‘ก๐œ•5๐‘ฅ๐œ™๐‘˜๎€œ+๐œ“(๐‘ก)๐‘ก0๐‘’โˆ’(๐‘กโˆ’๐‘กโ€ฒ)๐œ•5๐‘ฅ๐œ“๐‘‡๎€ท๐‘ก๎…ž๎€ธ๐’ฉ๐‘˜(๎€ท๐‘ก๐‘ข)๎…ž๎€ธ๐‘‘๐‘ก๎…ž(2.29) on the time interval [โˆ’๐‘‡,๐‘‡] and ๐‘˜=0,1,2,โ€ฆ. Uniqueness of the solution is also shown by using Bekiranov et al.โ€™s argument in [22]. This completes the proof.

3. Proof of Lemma 2.4

In this section we prove Lemma 2.4. To prove Lemma 2.4 we prepare the following lemma.

Lemma 3.1 (see [22]). (1) Let ๐›ผ,๐›ฝ>0 and let ๐œ…=min{๐›ผ,๐›ฝ}. If ๐›ผ+๐›ฝ>1+๐œ…,(3.1) then ๎ƒฉ๎€œโˆžโˆ’โˆž๐‘‘๐‘ฅ๎€ท||||๎€ธ1+๐‘ฅโˆ’๐œ๐›ผ๎€ท||||๎€ธ1+๐‘ฅโˆ’๐œ‚๐›ฝ๎ƒช1/2โ‰ค๐ถ7,๐›ผ,๐›ฝ๎ƒฉ1๎€ท||||๎€ธ1+๐œโˆ’๐œ‚๐œ…๎ƒช1/2,forany๐œ,๐œ‚โˆˆโ„,(3.2) where ๐ถ7,๐›ผ,๐›ฝ is a constant depending on ๐›ผ and ๐›ฝ.
(2) If ๐›พ>1, then ๎ƒฉ๎€œโˆžโˆ’โˆž๐‘‘๐‘ฅ๎€ท||||๎€ธ1+๐‘ฅ+๐œ‚๐›พ๎ƒช1/2โ‰ค๐ถ8,๐›พ,forany๐œ‚โˆˆโ„,(3.3) where ๐ถ8,๐›พ is a constant depending on ๐›พ.

Proof of Lemma 2.4. By duality, we have โ€–โ€–๐œ•๐‘ฅ๐œ•๎€ท๎€ท๐‘ฅ๐‘ข๐œ•๎€ธ๎€ท๐‘ฅ๐‘ฃโ€–โ€–๎€ธ๎€ธ๐‘‹๐‘ ๐‘โ€ฒโˆ’1=โ€–โ€–โ€–๎ซ๐œโˆ’๐œ‰5๎ฌ(๐‘โ€ฒโˆ’1)โŸจ๐œ‰โŸฉ๐‘ ๐œ‰๎€ทโ„ฑ๐‘ก,๐‘ฅ๐œ•๐‘ฅ๐‘ข๎€ธโˆ—๎€ทโ„ฑ๐‘ก,๐‘ฅ๐œ•๐‘ฅ๐‘ข๎€ธโ€–โ€–โ€–๐ฟ2๐œ๐ฟ2๐œ‰=supโ„Žโˆˆ๐ฟ2๐œ๐ฟ2๐œ‰,โ€–โ„Žโ€–๐ฟ2๐œ๐ฟ2๐œ‰โ‰ค1||||||โŽ›โŽœโŽœโŽโŸจ๐œ‰โŸฉ๐‘ ๐œ‰๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒ๎€ทโ„ฑ๐‘ก,๐‘ฅ๐œ•๐‘ฅ๐‘ข๎€ธโˆ—๎€ทโ„ฑ๐‘ก,๐‘ฅ๐œ•๐‘ฅ๐‘ฃ๎€ธโŽžโŽŸโŽŸโŽ ,โ„Ž๐ฟ2๐œ๐ฟ2๐œ‰||||||,(3.4) where (โ‹…,โ‹…)๐ฟ2๐œ๐ฟ2๐œ‰ is the inner product in ๐ฟ2(โ„๐œร—โ„๐œ‰). Setting ๐‘“๎ซ(๐œ,๐œ‰)=๐œโˆ’๐œ‰5๎ฌ๐‘โŸจ๐œ‰โŸฉ๐‘ โ„ฑ๐‘ก,๐‘ฅ๐‘ข๎ซ(๐œ,๐œ‰),๐‘”(๐œ,๐œ‰)=๐œโˆ’๐œ‰5๎ฌ๐‘โŸจ๐œ‰โŸฉ๐‘ โ„ฑ๐‘ก,๐‘ฅ๐‘ฃ(๐œ,๐œ‰),(3.5) we have โŽ›โŽœโŽœโŽโŸจ๐œ‰โŸฉ๐‘ ๐œ‰๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒ๎€ทโ„ฑ๐‘ก,๐‘ฅ๐œ•๐‘ฅ๐‘ข๎€ธโˆ—๎€ทโ„ฑ๐‘ก,๐‘ฅ๐œ•๐‘ฅ๐‘ฃ๎€ธโŽžโŽŸโŽŸโŽ ,โ„Ž๐ฟ2๐œ๐ฟ2๐œ‰=๎€โ„2โŸจ๐œ‰โŸฉ๐‘ ๐œ‰๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒร—โŽ›โŽœโŽœโŽœโŽ๎€โ„2๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๎€ธ๐‘“๎€ท๐œ1,๐œ‰1๎€ธ๐‘”๎€ท๐œโˆ’๐œ1,๐œ‰โˆ’๐œ‰1๎€ธโŸจ๐œ‰1โŸฉ๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ๐‘ ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ๐‘๐‘‘๐œ1๐‘‘๐œ‰1โŽžโŽŸโŽŸโŽŸโŽ โ„Ž(๐œ,๐œ‰)๐‘‘๐œ๐‘‘๐œ‰=๐ผฮฉ0.0.0+๐ผฮฉ๐‘0.0.0,(3.6) where ๐ผฮฉ0.0.0=๎€œ๎€œ๎€œ๎€œฮฉ0.0.0โŸจ๐œ‰โŸฉ๐‘ ๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๎€ธโ„Ž๎€ท๐œ(๐œ,๐œ‰)๐‘“1,๐œ‰1๎€ธ๐‘”๎€ท๐œโˆ’๐œ1,๐œ‰โˆ’๐œ‰1๎€ธ๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ๐‘ ๐‘‘๐œ1๐‘‘๐œ‰1๐ผ๐‘‘๐œ๐‘‘๐œ‰,ฮฉ๐‘0.0.0=๎€œ๎€œ๎€œ๎€œฮฉ๐‘0.0.0โŸจ๐œ‰โŸฉ๐‘ ๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๎€ธ๎€ท๐œโ„Ž(๐œ,๐œ‰)๐‘“1,๐œ‰1๎€ธ๐‘”๎€ท๐œโˆ’๐œ1,๐œ‰โˆ’๐œ‰1๎€ธ๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ๐‘ ๐‘‘๐œ1๐‘‘๐œ‰1ฮฉ๐‘‘๐œ๐‘‘๐œ‰,(3.7)0.0.0=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆโ„4โˆถ||๐œ‰||,||๐œ‰1||,||๐œ‰โˆ’๐œ‰1||๎€พโ‰ค5,ฮฉ๐‘0.0.0=โ„4โงตฮฉ0.0.0.(3.8)
We split ฮฉ๐‘0.0.0 into three regions, ฮฉ1,ฮฉ2,andฮฉ3: ฮฉ1=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘0.0.0โˆถ||๐œ‰||โ‰ค14||๐œ‰1||๎‚‡,ฮฉ2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘0.0.0โˆถ14||๐œ‰1||โ‰ค||๐œ‰||||๐œ‰โ‰ค41||๎‚‡,ฮฉ3=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘0.0.0||๐œ‰โˆถ41||โ‰ค||๐œ‰||๎€พ,(3.9) and then, we split ฮฉ๐‘–(๐‘–=1,2,3) into three regions: ฮฉ๐‘–.1=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘–โˆถ||๐œ1โˆ’๐œ‰51||,|||๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5|||โ‰ค||๐œโˆ’๐œ‰5||๎‚‡,ฮฉ๐‘–.2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘–โˆถ||๐œโˆ’๐œ‰5||,|||๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5|||โ‰ค||๐œ1โˆ’๐œ‰51||๎‚‡,ฮฉ๐‘–.3=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘–โˆถ||๐œโˆ’๐œ‰5||,||๐œ1โˆ’๐œ‰51||โ‰ค|||๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5|||๎‚‡.(3.10) We further split ฮฉ1.๐‘—, ฮฉ2.๐‘—, and ฮฉ3.๐‘—(๐‘—=1,2,3) into the following regions: ฮฉ1.๐‘—.1=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ1.๐‘—โˆถ||๐œ‰||๎€พ,ฮฉโ‰ฅ11.๐‘—.2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ1.๐‘—โˆถ||๐œ‰||||๐œ‰||||๐œ‰โ‰ค1,1||4๎‚‡,ฮฉโ‰ฅ11.๐‘—.3=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ1.๐‘—โˆถ||๐œ‰||||๐œ‰1||4๎‚‡,ฮฉโ‰ค12.1.1=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.1โˆถ||๐œ‰โˆ’๐œ‰1||๎€พ,ฮฉโ‰ค12.1.2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.1โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,๐œ‰โˆ’2๐œ‰1||โ‰ฅ||๐œ‰||โˆ’3/2๎‚‡,ฮฉ2.1.3=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.1โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,๐œ‰โˆ’2๐œ‰1||โ‰ค||๐œ‰||โˆ’3/2๎‚‡,ฮฉ2.2.1=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.2โˆถ||๐œ‰โˆ’๐œ‰1||๎€พ,ฮฉโ‰ค12.2.2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.2โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,2๐œ‰โˆ’๐œ‰1||โ‰ฅ||๐œ‰1||โˆ’3/2๎‚‡,ฮฉ2.2.3=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.2โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,2๐œ‰โˆ’๐œ‰1||โ‰ค||๐œ‰1||โˆ’3/2๎‚‡,ฮฉ2.3.1=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.3โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ค1,๐œ‰โˆ’๐œ‰1||||๐œ‰1||4๎‚‡,ฮฉโ‰ฅ12.3.2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.3โˆถ||๐œ‰โˆ’๐œ‰1||||๐œ‰1||4๎‚‡,ฮฉโ‰ค12.3.3=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.3โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,๐œ‰+๐œ‰1||๎€พ,ฮฉโ‰ฅ12.3.4=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.3โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,๐œ‰โˆ’๐œ‰1||โˆ’3/2โ‰ค||๐œ‰+๐œ‰1||๎‚‡,ฮฉโ‰ค12.3.5=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ2.3โˆถ||๐œ‰โˆ’๐œ‰1||||โ‰ฅ1,๐œ‰+๐œ‰1||โ‰ค||๐œ‰โˆ’๐œ‰1||โˆ’3/2๎‚‡,ฮฉ3.๐‘—.1=๎€ฝ๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ3.๐‘—โˆถ||๐œ‰1||๎€พ,ฮฉโ‰ฅ13.๐‘—.2=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ3.๐‘—โˆถ||๐œ‰1||||๐œ‰โ‰ค1,1||||๐œ‰||4๎‚‡,ฮฉโ‰ฅ13.๐‘—.3=๎‚†๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ3.๐‘—โˆถ||๐œ‰1||||๐œ‰||4๎‚‡,โ‰ค1(3.11) so that, we have ||๐ผฮฉ0.0.0+๐ผฮฉ๐‘0.0.0||โ‰ค||๐ผฮฉ0.0.0||+3๎“๐‘–,๐‘—,๐‘˜=1|||๐ผฮฉ๐‘–.๐‘—.๐‘˜|||+||๐ผฮฉ2.3.4||+||๐ผฮฉ2.3.5||.(3.12) Now we will estimate |๐ผฮฉ0.0.0|, |๐ผฮฉ๐‘–.๐‘—.๐‘˜|(๐‘–,๐‘—,๐‘˜=1,2,3), and |๐ผฮฉ2.3.๐‘™|(๐‘™=4,5). To estimate these terms, we prepare some estimates. By (3.8), (3.9) we obtain 34||๐œ‰1||โ‰ค||๐œ‰โˆ’๐œ‰1||โ‰ค54||๐œ‰1||,inฮฉ1,34||๐œ‰||โ‰ค||๐œ‰โˆ’๐œ‰1||โ‰ค54||๐œ‰||,inฮฉ3,||๐œ‰(3.13)1||โ‰ฅ4,inฮฉ1โˆชฮฉ2.2.3,๎€ฝ||๐œ‰min1||,||๐œ‰||๎€พโ‰ฅ1,inฮฉ2โงต๎€ฝฮฉ4โˆชฮฉ2.1.3โˆชฮฉ2.2.3๎€พ,๎€ฝ||๐œ‰min1||,||๐œ‰||๎€พโ‰ฅ3,inฮฉ4,||๐œ‰||โ‰ฅ4,inฮฉ3โˆชฮฉ2.1.3,(3.14) where ฮฉ4=ฮฉ2.1.1โˆชฮฉ2.2.1โˆชฮฉ2.3.1โˆชฮฉ2.3.2. Since (3.13) and (3.14) yield ||๐œ‰โˆ’๐œ‰1||โ‰ฅ3,inฮฉ1,ฮฉ3,(3.15) we have by (3.9) and (3.13)โ€“(3.15) ||๐œ‰1||2||๐œ‰โˆ’๐œ‰1||2โŸจ๐œ‰1โŸฉ2๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โ‰ค๐ถ29,๐‘ ||๐œ‰1||4โˆ’4๐‘ ,inฮฉ1,||๐œ‰||2||๐œ‰1||2โŸจ๐œ‰โŸฉ2๐‘ โŸจ๐œ‰1โŸฉ2๐‘ โ‰ค๐ถ29,๐‘ ||๐œ‰1||4,inฮฉ2,||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2โŸจ๐œ‰โŸฉ2๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โ‰ค๐ถ29,๐‘ ||๐œ‰||4,inฮฉ3,(3.16) where ๐ถ9,๐‘ =4|๐‘ |+1. Using (3.11), (3.14), and (3.15), we have โŸจ๐œ‰โŸฉ2๐‘ โ‰ค22|๐‘ |๎€ฝ||๐œ‰||๎€พmax1,2๐‘ inฮฉ1,โŸจ๐œ‰โˆ’๐œ‰1โŸฉโˆ’2๐‘ โ‰ค22|๐‘ |๎€ฝ||max1,๐œ‰โˆ’๐œ‰1||๎€พโˆ’2๐‘ inฮฉ2,โŸจ๐œ‰1โŸฉโˆ’2๐‘ โ‰ค22|๐‘ |๎€ฝ||๐œ‰max1,1||๎€พโˆ’2๐‘ inฮฉ3.(3.17)
In ฮฉ๐‘–1.๐‘—1.๐‘˜1 ((๐‘–1.๐‘—1.๐‘˜1)=(0.0.0),(1.1.3),(3.2.1),(3.2.2),(๐‘–.2.๐‘˜),(๐‘–=1,2,๐‘˜=1,2,3)), we integrate with respect to ๐œ and ๐œ‰ first, then, we use Schwarzโ€™s inequality, Fubiniโ€™s theorem, and note that โ€–โ„Žโ€–๐ฟ2๐œ๐ฟ2๐œ‰โ‰ค1 to have |||๐ผฮฉ๐‘–111.๐‘—.๐‘˜|||โ‰คโ€–๐‘“โ€–๐ฟ2๐œ1๐ฟ2๐œ‰1ร—โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2๎€ทโ„Ž(๐œ,๐œ‰)๐‘”๐œโˆ’๐œ1,๐œ‰โˆ’๐œ‰1๎€ธโŸจ๐œ‰โŸฉ๐‘ ||๐œ‰||||๐œ‰โˆ’๐œ‰1||๐œ’ฮฉ๐‘–111.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰๐‘‘๐œ๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒ๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ๐‘โŸจ๐œ‰โˆ’๐œ‰1โŸฉ๐‘ โŽžโŽŸโŽŸโŽŸโŽ โ€–โ€–โ€–โ€–โ€–๐ฟ2๐œ1๐ฟ2๐œ‰1โ‰คโ€–๐‘“โ€–๐ฟ2๐œ๐ฟ2๐œ‰โ€–โ€–โ€–โ€–โ€–โ€–๎‚ต๎€โ„2||||โ„Ž(๐œ,๐œ‰)2||๐‘”๎€ท๐œโˆ’๐œ1,๐œ‰โˆ’๐œ‰1๎€ธ||2๎‚ถ๐‘‘๐œ‰๐‘‘๐œ1/2ร—||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰โŸฉ2๐‘ ||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ๐‘–111.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰๐‘‘๐œ๎ซ๐œโˆ’๐œ‰5๎ฌ2(1โˆ’๐‘โ€ฒ)๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2๐‘โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟ2๐œ1๐ฟ2๐œ‰1โ‰คโ€–๐‘“โ€–๐ฟ2๐œ๐ฟ2๐œ‰โ€–๐‘”โ€–๐ฟ2๐œ๐ฟ2๐œ‰ร—โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰โŸฉ2๐‘ ||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ๐‘–111.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰๐‘‘๐œ๎ซ๐œโˆ’๐œ‰5๎ฌ2(1โˆ’๐‘โ€ฒ)๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2๐‘โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1,(3.18) where ๐œ’ฮฉ๐‘–111.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโ‰ก๎ƒฏ๎€ท1,if๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆˆฮฉ๐‘–1.๐‘—1.๐‘˜1,๎€ท0,if๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธโˆ‰ฮฉ๐‘–1.๐‘—1.๐‘˜1.(3.19) In ฮฉ๐‘–2.๐‘—2.๐‘˜2 ((๐‘–2.๐‘—2.๐‘˜2)=(๐‘–.1.๐‘˜),(๐‘–=1,2,3,๐‘˜=1,2),(2.1.3),(3.1.3),(3.2.3)), we integrate with respect to ๐œ and ๐œ‰ first, then, we use the same way as in (3.18) to have |||๐ผฮฉ๐‘–222.๐‘—.๐‘˜|||โ‰คโ€–๐‘“โ€–๐ฟ2๐œ๐ฟ2๐œ‰โ€–๐‘”โ€–๐ฟ2๐œ๐ฟ2๐œ‰ร—โ€–โ€–โ€–โ€–โ€–โ€–โŸจ๐œ‰โŸฉ๐‘ ||๐œ‰||๎ซ๐œโˆ’๐œ‰5๎ฌ(1โˆ’๐‘โ€ฒ)โŽ›โŽœโŽœโŽœโŽ๎€โ„2||๐œ‰1||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ๐‘–222.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰1๐‘‘๐œ1๎ซ๐œ1โˆ’๐œ‰51๎ฌ2๐‘๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2๐‘โŸจ๐œ‰1โŸฉ2๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ๐ฟโˆž๐œ‰.(3.20)
In ฮฉ2.3.2 we use the change of variables ๐œ2=๐œ1โˆ’๐œ,๐œ‰2=๐œ‰1โˆ’๐œ‰(3.21) to obtain ๎€œ๎€œ๎€œ๎€œฮฉ2.3.2โŸจ๐œ‰โŸฉ๐‘ ๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๎€ธโ„Ž๎€ท๐œ(๐œ,๐œ‰)๐‘“1,๐œ‰1๎€ธ๐‘”๎€ท๐œโˆ’๐œ1,๐œ‰โˆ’๐œ‰1๎€ธ๎ซ๐œโˆ’๐œ‰5๎ฌ1โˆ’๐‘โ€ฒ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ๐‘ ๐‘‘๐œ1๐‘‘๐œ‰1=๎‚ฮฉ๐‘‘๐œ๐‘‘๐œ‰๎€œ๎€œ๎€œ๎€œ2.3.2โŸจ๐œ‰1โˆ’๐œ‰2โŸฉ๐‘ ๐œ‰1๎€ท๐œ‰1โˆ’๐œ‰2๎€ธ๎€ทโˆ’๐œ‰2๎€ธโ„Ž๎€ท๐œ1โˆ’๐œ2,๐œ‰1โˆ’๐œ‰2๎€ธ๐‘“๎€ท๐œ1,๐œ‰1๎€ธ๐‘”๎€ทโˆ’๐œ2,โˆ’๐œ‰2๎€ธ๎‚ฌ๐œ1โˆ’๐œ2โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ1โˆ’๐‘โ€ฒ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘๎ซ๐œ2โˆ’๐œ‰52๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŸจ๐œ‰1โŸฉ๐‘ โŸจ๐œ‰2โŸฉ๐‘ ๐‘‘๐œ1๐‘‘๐œ‰1๐‘‘๐œ2๐‘‘๐œ‰2๎‚ฮฉโ‰ก๐ฝ2.3.2,(3.22) where ๎‚ฮฉ2.3.2=๎‚†๎€ท๐œ1,๐œ2,๐œ‰1,๐œ‰2๎€ธโˆˆฮฉ๐‘0.0.0โˆถ14||๐œ‰1โˆ’๐œ‰2||โ‰ค||๐œ‰1||||๐œ‰โ‰ค41โˆ’๐œ‰2||,|||๐œ1โˆ’๐œ2โˆ’๎€ท๐œ‰1โˆ’๐œ‰2๎€ธ5|||,||๐œ1โˆ’๐œ‰51||โ‰ค||๐œ2โˆ’๐œ‰52||,||๐œ‰2||||๐œ‰1||4๎‚‡.โ‰ค1(3.23) We integrate with respect to ๐œ2 and ๐œ‰2 first, then, we use the same way as in (3.18) to have |||๐ฝ๎‚ฮฉ2.3.2|||โ‰คโ€–๐‘“โ€–๐ฟ2๐œ๐ฟ2๐œ‰โ€–๐‘”โ€–๐ฟ2๐œ๐ฟ2๐œ‰ร—โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰1โˆ’๐œ‰2โŸฉ2๐‘ ||๐œ‰2||2||๐œ‰1โˆ’๐œ‰2||2๐œ’๎‚ฮฉ2.3.2๎€ท๐œ1,๐œ2,๐œ‰1,๐œ‰2๎€ธ๐‘‘๐œ‰2๐‘‘๐œ2๎‚ฌ๐œ1โˆ’๐œ2โˆ’๎€ท๐œ‰1โˆ’๐œ‰2๎€ธ5๎‚ญ2(1โˆ’๐‘โ€ฒ)๎ซ๐œ2โˆ’๐œ‰52๎ฌ2๐‘โŸจ๐œ‰2โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1.(3.24) In ฮฉ๐‘–3.3.๐‘˜3 ((๐‘–3.3.๐‘˜3)=(๐‘–.3.๐‘˜),(๐‘–=1,3,๐‘˜=1,2,3),(2.3.1),(2.3.3),(2.3.4),(3.3.5)), we have by a similar argument to (3.22) ๐ผฮฉ๐‘–33.3.๐‘˜๎‚ฮฉ=๐ฝ๐‘–33.3.๐‘˜,(3.25) where ๎‚ฮฉ๐‘–3.3.๐‘˜3 is the region which is obtained from ฮฉ๐‘–3.3.๐‘˜3 by the change of variables ๐œ2=๐œ1โˆ’๐œ and ๐œ‰2=๐œ‰1โˆ’๐œ‰. We integrate with respect to ๐œ1 and ๐œ‰1 first, then, we use the same way as in (3.18) to have ||||๐ฝ๎‚ฮฉ๐‘–33.3.๐‘˜||||โ‰คโ€–๐‘“โ€–๐ฟ2๐œ๐ฟ2๐œ‰โ€–๐‘”โ€–๐ฟ2๐œ๐ฟ2๐œ‰ร—โ€–โ€–โ€–โ€–โ€–โ€–|๐œ‰2|๎ซ๐œ2โˆ’๐œ‰52๎ฌ๐‘โŸจ๐œ‰2โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰1โˆ’๐œ‰2โŸฉ2๐‘ ||๐œ‰1||2||๐œ‰1โˆ’๐œ‰2||2๐œ’๎‚ฮฉ๐‘–33.3.๐‘˜๎€ท๐œ1,๐œ2,๐œ‰1,๐œ‰2๎€ธ๐‘‘๐œ‰1๐‘‘๐œ1๎‚ฌ๐œ1โˆ’๐œ2โˆ’๎€ท๐œ‰1โˆ’๐œ‰2๎€ธ5๎‚ญ2(1โˆ’๐‘โ€ฒ)๎ซ๐œ1โˆ’๐œ‰51๎ฌ2๐‘โŸจ๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ2๐ฟโˆž๐œ‰2.(3.26)
Now we will get bounds for โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰โŸฉ2๐‘ ||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ๐‘–111.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰๐‘‘๐œ๎ซ๐œโˆ’๐œ‰5๎ฌ2(1โˆ’๐‘โ€ฒ)๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2๐‘โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1,โ€–โ€–โ€–โ€–โ€–โ€–โŸจ๐œ‰โŸฉ๐‘ ||๐œ‰||๎ซ๐œโˆ’๐œ‰5๎ฌ(1โˆ’๐‘โ€ฒ)โŽ›โŽœโŽœโŽœโŽ๎€โ„2||๐œ‰1||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ๐‘–222.๐‘—.๐‘˜๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰1๐‘‘๐œ1๎ซ๐œ1โˆ’๐œ‰51๎ฌ2๐‘๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2๐‘โŸจ๐œ‰1โŸฉ2๐‘ โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ๐ฟโˆž๐œ‰,โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽ๎€โ„2โŸจ๐œ‰1โˆ’๐œ‰2โŸฉ2๐‘ ||๐œ‰2||2||๐œ‰1โˆ’๐œ‰2||2๐œ’๎‚ฮฉ2.3.2๐‘‘๐œ‰2๐‘‘๐œ2๎ซ๐œ1โˆ’๐œ2โˆ’(๐œ‰1โˆ’๐œ‰2)5๎ฌ2(1โˆ’๐‘โ€ฒ)๎ซ๐œ2โˆ’๐œ‰52๎ฌ2๐‘โŸจ๐œ‰2โŸฉ2๐‘ โŽžโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1,โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰2||๎ซ๐œ2โˆ’๐œ‰52๎ฌ๐‘โŸจ๐œ‰2โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰1โˆ’๐œ‰2โŸฉ2๐‘ ||๐œ‰2||2||๐œ‰1โˆ’๐œ‰2||2๐œ’๎‚ฮฉ๐‘–33.3.๐‘˜๐‘‘๐œ‰1๐‘‘๐œ1๎ซ๐œ1โˆ’๐œ2โˆ’(๐œ‰1โˆ’๐œ‰2)5๎ฌ2(1โˆ’๐‘โ€ฒ)๎ซ๐œ1โˆ’๐œ‰51๎ฌ2๐‘โŸจ๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ2๐ฟโˆž๐œ‰2.(3.27) By using the following methods, we estimate (3.27).
The Case of ฮฉ0.0.0
Since ๎‚€๐œ1+๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚โˆ’๐œ‰5=๐œ1โˆ’๐œ‰51โˆ’5๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๐œ‰๎€ธ๎€ท2โˆ’๐œ‰๐œ‰1+๐œ‰21๎€ธ,(3.28) it follows from (3.2) in Lemma 3.1 with ๐›ผ=2๐‘, ๐›ฝ=๐œ…=2(1โˆ’๐‘๎…ž) that โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„2โŸจ๐œ‰โŸฉ2๐‘ ||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ0.0.0๎€ท๐œ,๐œ1,๐œ‰,๐œ‰1๎€ธ๐‘‘๐œ‰๐‘‘๐œ๎ซ๐œโˆ’๐œ‰5๎ฌ2(1โˆ’๐‘โ€ฒ)๎‚ฌ๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2๐‘โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1โ‰ค๐ถ7,2(1โˆ’๐‘โ€ฒ),2๐‘ร—โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๐œ’ฮฉ๐ต๎€ท๐œ‰1๎€ธ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽ๎€œโ„โŸจ๐œ‰โŸฉ2๐‘ ||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ1๐ด;๐œ‰(๐œ‰)๎ซ๐œ1โˆ’๐œ‰51โˆ’5๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๐œ‰๎€ธ๎€ท2โˆ’๐œ‰๐œ‰1+๐œ‰21๎€ธ๎ฌ2(1โˆ’๐‘โ€ฒ)โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽ ๐‘‘๐œ‰1/2โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1,(3.29) where ฮฉ๐ด;๐œ‰1={๐œ‰โˆถ|๐œ‰|,|๐œ‰โˆ’๐œ‰1|โ‰ค5} and ฮฉ๐ต={๐œ‰1โˆถ|๐œ‰1|โ‰ค5}. Since โŸจ๐œ‰โŸฉ๐‘ โ‰คmax{1,6๐‘ }, we have ๐ถ7,2(1โˆ’๐‘โ€ฒ),2๐‘ร—โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๐œ’ฮฉ๐ต๎€ท๐œ‰1๎€ธ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽ๎€œโ„โŸจ๐œ‰โŸฉ2๐‘ ||๐œ‰||2||๐œ‰โˆ’๐œ‰1||2๐œ’ฮฉ1๐ด;๐œ‰(๐œ‰)๎ซ๐œ1โˆ’๐œ‰51โˆ’5๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๐œ‰๎€ธ๎€ท2โˆ’๐œ‰๐œ‰1+๐œ‰21๎€ธ๎ฌ2(1โˆ’๐‘โ€ฒ)โŸจ๐œ‰โˆ’๐œ‰1โŸฉ2๐‘ โŽžโŽŸโŽŸโŽ ๐‘‘๐œ‰1/2โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1โ‰ค๐ถ7,2(1โˆ’๐‘โ€ฒ),2๐‘๎€ฝmax1,6๐‘ ๎€พ52๎ƒฉ๎€œ||๐œ‰||โ‰ค5||๐œ‰||2๎ƒช๐‘‘๐œ‰1/2โ‰ค๐‘€1,๐‘ ,๐‘,๐‘โ€ฒ,(3.30) where ๐‘€1,๐‘ ,๐‘,๐‘โ€ฒ is some constant.
The Case of ฮฉ1.๐‘—.3,ฮฉ3.๐‘—.3๎‚ฮฉ(๐‘—=1,2),๐‘–.3.3(๐‘–=1,3) and ๎‚ฮฉ2.3.2
We consider ๎‚ฮฉ2.3.2. By (3.2), we have โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŸจ๐œ‰1โŸฉ๐‘ โŽ›โŽœโŽœโŽœโŽ๎€โ„โŸจ๐œ‰1โˆ’๐œ‰2โŸฉ2๐‘ ||๐œ‰2||2||๐œ‰โˆ’๐œ‰1||2๐œ’๎‚ฮฉ2.3.2๐‘‘๐œ‰2๐‘‘๐œ2๎‚ฌ๐œ1โˆ’๐œ2โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚ญ2(1โˆ’๐‘โ€ฒ)๎ซ๐œ2โˆ’๐œ‰52๎ฌ2๐‘โŸจ๐œ‰2โŸฉ2๐‘ โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1โ‰ค๐ถ7,2(1โˆ’๐‘โ€ฒ),2๐‘๐ถ9,๐‘ 2|๐‘ |ร—โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||2๐œ’๎‚ฮฉ๐ถ๎€ท๐œ‰1๎€ธ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŽ›โŽœโŽœโŽœโŽ๎€œโ„||๐œ‰2||2๐œ’๎‚ฮฉ1๐ท;๐œ‰๎€ท๐œ‰2๎€ธ๎ซ๐œ1โˆ’๐œ‰51+5๐œ‰1๐œ‰2๎€ท๐œ‰1โˆ’๐œ‰2๐œ‰๎€ธ๎€ท21โˆ’๐œ‰1๐œ‰2+๐œ‰22๎€ธ๎ฌ2(1โˆ’๐‘โ€ฒ)๐‘‘๐œ‰2โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1,(3.31) where ๎‚ฮฉ๐ถ={๐œ‰1โˆถ|๐œ‰1|โ‰ฅ4} and ๎‚ฮฉ๐ท;๐œ‰1={๐œ‰2โˆถ|๐œ‰2|โ‰ค|๐œ‰1|โˆ’4}. Here we have used (3.16) and (3.17) with the change of variables ๐œ‰2=๐œ‰1โˆ’๐œ‰. Noting ๎€ท2๐‘>0,21โˆ’๐‘๎…ž๎€ธ>0,(3.32) we have โ€–โ€–โ€–โ€–โ€–โ€–||๐œ‰1||2๐œ’๎‚ฮฉ๐ถ๎€ท๐œ‰1๎€ธ๎ซ๐œ1โˆ’๐œ‰51๎ฌ๐‘โŽ›โŽœโŽœโŽœโŽ๎€œโ„||๐œ‰2||2๐œ’๎‚ฮฉ1๐ท;๐œ‰๎€ท๐œ‰2๎€ธ๎ซ๐œ1โˆ’๐œ‰51+5๐œ‰1๐œ‰2๎€ท๐œ‰1โˆ’๐œ‰2๐œ‰๎€ธ๎€ท21โˆ’๐œ‰1๐œ‰2+๐œ‰22๎€ธ๎ฌ2(1โˆ’๐‘โ€ฒ)๐‘‘๐œ‰2โŽžโŽŸโŽŸโŽŸโŽ 1/2โ€–โ€–โ€–โ€–โ€–โ€–๐ฟโˆž๐œ1๐ฟโˆž๐œ‰1โ‰คโ€–โ€–โ€–โ€–๎‚ป||๐œ‰1||4๐œ’๎‚ฮฉ๐ถ๎€ท๐œ‰1๎€ธ๎‚ต๎€œ|๐œ‰2|โ‰ค|๐œ‰1|โˆ’4||๐œ‰2||2๐‘‘๐œ‰2๎‚ถ๎‚ผ1/2โ€–โ€–โ€–โ€–๐ฟโˆž๐œ‰1โ‰ค21/2โ€–โ€–โ€–โ€–๎‚ต||๐œ‰1||โˆ’8๐œ’๎‚ฮฉ๐ถ๎€ท๐œ‰1๎€ธ๎‚ถ1/2โ€–โ€–โ€–โ€–๐ฟโˆž๐œ‰1โ‰ค21/2.(3.33) Thus, (3.27) is bounded by ๐‘€2,๐‘ ,๐‘,๐‘โ€ฒ=๐ถ9,๐‘ 2|๐‘ |+1/2๎€ฝ๐ถmax7,2(1โˆ’๐‘โ€ฒ),2๐‘,๐ถ7,2๐‘,2๐‘๎€พ(3.34) in ๎‚ฮฉ2.3.2. In the same manner as (3.31)-(3.33), (3.27) are bounded by ๐‘€2,๐‘ ,๐‘,๐‘โ€ฒ in ฮฉ1.๐‘—.3, ฮฉ3.๐‘—.3(๐‘—=1,2), and ๎‚ฮฉ๐‘–.3.3(๐‘–=1,3).
The Case of ฮฉ2.1.1, ฮฉ2.2.1
We consider ฮฉ2.2.1. Since ||๐œโˆ’๐œ‰5||+||๐œ1โˆ’๐œ‰51||+|||๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5|||โ‰ฅ|||๐œโˆ’๐œ‰5โˆ’๎€ท๐œ1โˆ’๐œ‰51๎€ธโˆ’๎‚€๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5๎‚|||||๐œ‰||||๐œ‰=51||||๐œ‰โˆ’๐œ‰1||||๐œ‰2โˆ’๐œ‰๐œ‰1+๐œ‰21||,(3.35) we obtain ๎‚†||max๐œโˆ’๐œ‰5||,||๐œ1โˆ’๐œ‰51||,|||๐œโˆ’๐œ1โˆ’๎€ท๐œ‰โˆ’๐œ‰1๎€ธ5|||๎‚‡โ‰ฅ53||๐œ‰||||๐œ‰1||||๐œ‰โˆ’๐œ‰1||||๐œ‰2โˆ’๐œ‰๐œ‰1+๐œ‰21||.(3.36) Noting that โˆ’2๐‘<0, we have ๎ซ๐œ1โˆ’๐œ‰51๎ฌโˆ’2๐‘โ‰ค32๐‘๎ซ5๐œ‰๐œ‰1๎€ท๐œ‰โˆ’๐œ‰1๐œ‰๎€ธ๎€ท2โˆ’๐œ‰๐œ‰1+๐œ‰21๎€ธ๎ฌโˆ’2๐‘inฮฉ2.2.1.(3.37) Since (3.14) and |๐œ‰โˆ’๐œ‰1|โ‰ค1 yield |