Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 361807 | 19 pages | https://doi.org/10.1155/2012/361807

Duality of Variable Exponent Triebel-Lizorkin and Besov Spaces

Academic Editor: V. M. Kokilashvili
Received12 Dec 2010
Accepted25 May 2011
Published12 Jan 2012

Abstract

We will prove the duality and reflexivity of variable exponent Triebel-Lizorkin and Besov spaces. It was shown by many authors that variable exponent Triebel-Lizorkin spaces coincide with variable exponent Bessel potential spaces, Sobolev spaces, and Lebesgue spaces when appropriate indices are chosen. In consequence of the results, these variable exponent function spaces are shown to be reflexive.

1. Introduction

Recently, variable exponent function spaces have been studied by many authors [1โ€“13], and in particular, the papers about the variable exponent Triebel-Lizorkin and Besov spaces have been published in [14โ€“18]. Diening et al. [15] and Almeida and Hรคstรถ [14] studied the spaces ๐น๐›ผ(โ‹…)๐‘(โ‹…),๐‘ž(โ‹…) and ๐ต๐›ผ(โ‹…)๐‘(โ‹…),๐‘ž(โ‹…) and Xu studied [16โ€“18] the spaces ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›).

In this paper, we will show the duality of variable exponent Triebel-Lizorkin spaces ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and Besov spaces ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) under suitable conditions by using the same arguments in Triebel [19]. The duality follows from the fact that ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) are reflexive under same conditions.

Xu [16] showed that variable exponent Bessel potential space ๐ฟ๐‘ ,๐‘(โ‹…)(โ„๐‘›) coincides with ๐น๐‘ ๐‘(โ‹…),2(โ„๐‘›) if ๐‘ โ‰ฅ0 and ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). Diening et al. [15] showed that the variable exponent Lebesgue space ๐ฟ๐‘(โ‹…)(โ„๐‘›) coincides with ๐น0๐‘(โ‹…),2(โ„๐‘›) under suitable assumptions on ๐‘(โ‹…). Gurka et al. [7] showed that ๐ฟ๐‘˜,๐‘(โ‹…)(โ„๐‘›) coincides with variable exponent Sobolev spaces ๐‘Š๐‘˜,๐‘(โ‹…)(โ„๐‘›) if ๐‘˜โˆˆโ„• and ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). In consequences of these results, we have the duality and reflexivity of ๐ฟ๐‘ ,๐‘(โ‹…)(โ„๐‘›)(๐‘ โ‰ฅ0), although the duality and reflexivity of ๐ฟ๐‘(โ‹…)(โ„๐‘›) have been obtained in Kovรกฤik and Rรกkosnรญk [9] under the assumptions on ๐‘(โ‹…) which are weaker than ours.

2. Definitions of Variable Exponent Function Spaces

We first introduce variable exponent Lebesgue spaces. Let ๐‘(โ‹…) be a measurable function on โ„๐‘› with range in (1,โˆž). Let ๐ฟ๐‘(โ‹…)(โ„๐‘›) denote the set of all complex-valued functions ๐‘“ on โ„๐‘› such that, for some ๐œ†>0, ๎€œโ„๐‘›๎‚ต|๐‘“(๐‘ฅ)|๐œ†๎‚ถ๐‘(๐‘ฅ)d๐‘ฅ<โˆž.(2.1) The set becomes a Banach function space when it is equipped with the Luxemburg-Nakano norm: โ€–๐‘“โ€–๐ฟ๐‘(โ‹…)๎ƒฏ๎€œ=inf๐œ†>0โˆถโ„๐‘›๎‚ต|๐‘“(๐‘ฅ)|๐œ†๎‚ถ๐‘(๐‘ฅ)๎ƒฐd๐‘ฅโ‰ค1.(2.2) If ๐‘(๐‘ฅ)โ‰ก๐‘ is a constant function, then the above norm coincides with the usual ๐ฟ๐‘-norm and so the notation is not confusional. In Kovรกฤik and Rรกkosnรญk [9], variable exponent Lebesgue spaces are defined on arbitrary measurable subset of โ„๐‘›. Denote by ๐’ซ(โ„๐‘›) the set of measurable functions ๐‘(โ‹…) on โ„๐‘› with range in (1,โˆž) such that 1<๐‘โˆ’=essinf๐‘ฅโˆˆโ„๐‘›๐‘(๐‘ฅ),esssup๐‘ฅโˆˆโ„๐‘›๐‘(๐‘ฅ)=๐‘+<โˆž.(2.3) For a complex-valued locally Lebesgue-integrable function ๐‘“(โ‹…) on โ„๐‘›, let (1โ„ณ๐‘“)(๐‘ฅ)=sup||๐ต||๎€œ๐ต||||๐‘“(๐‘ฆ)d๐‘ฆ(2.4) denote the Hardy-Littlewood maximal function, where the supremum is taken over all balls ๐ต centered at ๐‘ฅ. There exists ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›) such that the Hardy-Littlewood maximal operator โ„ณ is not bounded on ๐ฟ๐‘(โ‹…)(โ„๐‘›) [11], although the operator โ„ณis bounded on ๐ฟ๐‘(โ„๐‘›) for ๐‘>1. Some sufficient conditions on ๐‘(โ‹…) for maximal operator โ„ณ to be bounded on ๐ฟ๐‘(โ‹…)(โ„๐‘›) are known. Let โ„ฌ(โ„๐‘›) be the set of ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›) such that the Hardy-Littlewood maximal operator โ„ณ is bounded on ๐ฟ๐‘(โ‹…)(โ„๐‘›).

Let ๐’ฎ(โ„๐‘›) be the Schwartz space of all complex-valued rapidly decreasing and infinitely differentiable functions on โ„๐‘›. Let ๐’ฎ๎…ž(โ„๐‘›) be the set of all the tempered distribution on โ„๐‘›. For ๐œ‘โˆˆ๐’ฎ(โ„๐‘›), let โ„ฑ๐œ‘ denote the Fourier transform of ๐œ‘ and โ„ฑโˆ’1๐œ‘ the inverse Fourier transform of ๐œ‘. We write โ„ฑโˆ’1๐‘šโ„ฑ๐‘“=โ„ฑโˆ’1[๐‘šโ‹…โ„ฑ๐‘“] for the sake of simplicity.

Let ๐‘ โ‰ฅ0 and ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›). The variable exponent Bessel potential space ๐ฟ๐‘ ,๐‘(โ‹…)(โ„๐‘›) is the collection of ๐‘“โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›) such that the norm: โ€–๐‘“โ€–๐ฟ๐‘ ,๐‘(โ‹…)=โ€–โ€–โ„ฑโˆ’1๎€ท1+|โ‹…|2๎€ธ๐‘ /2โ€–โ€–โ„ฑ๐‘“(โ‹…)๐ฟ๐‘(โ‹…)<โˆž.(2.5)

Let ๐‘˜โˆˆโ„• and ๐‘โˆˆ๐’ซ(โ„๐‘›). The variable exponent Sobolev space ๐‘Š๐‘˜,๐‘(โ‹…)(โ„๐‘›) is the collection of ๐‘“โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›) such that the derivatives (in the sense of distribution) up to the order ๐‘˜ belong to ๐ฟ๐‘(โ‹…)(โ„๐‘›) and the norm: โ€–๐‘“โ€–๐‘Š๐‘˜,๐‘(โ‹…)=๎“|๐›ผ|โ‰ค๐‘˜โ€–๐ท๐›ผ๐‘“โ€–๐ฟ๐‘(โ‹…)<โˆž,(2.6) where ๐›ผ is a multi-index and |๐›ผ|=๐›ผ1+โ‹ฏ+๐›ผ๐‘›.

Let ฮฆโˆˆ๐’ฎ(โ„๐‘›) be a function such that suppโ„ฑฮฆโŠ‚๐ต(0,1) and โ„ฑฮฆ=1 on ๐ต(0,1/2), where ๐ต(0,๐œ†) is an open ball centered at 0 with radius ๐œ†. Set ฮฆ๐‘—(๐‘ฅ)=2๐‘›๐‘—ฮฆ(2๐‘—๐‘ฅ) for ๐‘ฅโˆˆโ„๐‘› and ๐‘—โˆˆโ„ค. We also set ๐œƒ๐‘—(๐‘ฅ)=ฮฆ๐‘—(๐‘ฅ)โˆ’ฮฆ๐‘—โˆ’1(๐‘ฅ).(2.7) It follows that โˆž๎“๐‘—=0โ„ฑ๐œƒ๐‘—(๐œ‰)โ‰ก1,(2.8) where ๐œƒ0=ฮฆ0.

Definition 2.1. Let ๐‘ โˆˆโ„, 0<๐‘ž<โˆž, and ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›). Let ๐œƒ๐‘—, ๐‘—โˆˆโ„•0=โ„•โˆช{0} as above. The variable exponent Triebel-Lizorkin space ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) is the collection of ๐‘“โˆˆ๐’ฎโ€ฒ(โ„๐‘›) such that โ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž=โ€–โ€–๎€ฝ2๐‘ ๐‘—๐œƒ๐‘—๎€พโˆ—๐‘“โˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)<โˆž.(2.9) Let ๐‘ โˆˆโ„, 0<๐‘žโ‰คโˆž, and ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›). The variable exponent Besov space ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) is the collection of ๐‘“โˆˆ๐’ฎโ€ฒ(โ„๐‘›) such that โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),๐‘ž=โ€–โ€–๎€ฝ2๐‘ ๐‘—๐œƒ๐‘—๎€พโˆ—๐‘“โˆž0โ€–โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…))<โˆž.(2.10)

Here, ๐ฟ๐‘(โ‹…)(โ„“๐‘ž) and โ„“๐‘ž(๐ฟ๐‘(โ‹…)) are the spaces of all sequences {๐‘”๐‘—} of measurable functions on โ„๐‘› such that quasi-norms: โ€–โ€–๎€ฝ๐‘”๐‘—๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)=โ€–โ€–โ€–โ€–{๐‘”๐‘—}โˆž0โ€–โ€–โ„“๐‘žโ€–โ€–๐ฟ๐‘(โ‹…)=โ€–โ€–โ€–โ€–๎ƒฉโˆž๎“๐‘—=0||๐‘”๐‘—||(โ‹…)๐‘ž๎ƒช1/๐‘žโ€–โ€–โ€–โ€–๐ฟ๐‘(โ‹…),โ€–โ€–๎€ฝ๐‘”๐‘—๎€พโˆž0โ€–โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…))=โ€–โ€–โ€–๐‘”โ€–๎€ฝ๐‘—โ€–โ€–๐ฟ๐‘(โ‹…)๎€พโˆž0โ€–โ€–โ„“๐‘ž=๎ƒฉโˆž๎“๐‘—=0โ€–โ€–๐‘”๐‘—โ€–โ€–(โ‹…)๐‘ž๐ฟ๐‘(โ‹…)๎ƒช1/๐‘ž(2.11) are finite, respectively.

Definition 2.2. Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›) and 0<๐‘žโ‰คโˆž. For a sequence of compact subsets ฮฉ={ฮฉ๐‘˜}โˆž๐‘˜=0 of โ„๐‘›, ๐ฟฮฉ๐‘(โ‹…)(โ„“๐‘ž) denotes the space of all sequences {๐‘“๐‘˜}โˆž๐‘˜=0 of ๐’ฎ๎…ž(โ„๐‘›) such that suppโ„ฑ๐‘“๐‘˜โŠ‚ฮฉ๐‘˜for๐‘˜=0,1,2,โ€ฆ,(2.12) and โ€–๐‘“๐‘˜โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)<โˆž. For a compact subset ฮ“ of โ„๐‘›, ๐ฟฮ“๐‘(โ‹…)(โ„๐‘›) denotes the space of all elements ๐‘“โˆˆ๐’ฎโ€ฒ(โ„๐‘›) such that suppโ„ฑ๐‘“โŠ‚ฮ“,(2.13) and โ€–๐‘“โ€–๐ฟ๐‘(โ‹…)<โˆž.

3. Preliminaries

We need the following fundamental properties of ๐ฟ๐‘(โ‹…)(โ„๐‘›).

Theorem 3.1 (see [5, Theorem 8.1]). Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›). Then, the following conditions are equivalent:
(a)๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›),
(b)๐‘(โ‹…)/๐‘กโˆˆโ„ฌ(โ„๐‘›) for some 1<๐‘ก<๐‘โˆ’,
(c)๐‘โ€ฒ(โ‹…)โˆˆโ„ฌ(โ„๐‘›), where ๐‘๎…ž(โ‹…)=๐‘(โ‹…).๐‘(โ‹…)โˆ’1(3.1)

In [1, 5], some other conditions equivalent to the above are given. Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›) and 0<๐‘ž<โˆž. Then we write ๐‘Ÿ๐‘โˆ’,๐‘ž๎‚ป๎€ท๐‘=sup๐‘Ÿโˆˆโ„โˆถ0<๐‘Ÿ<minโˆ’๎€ธ,,๐‘ž๐‘(โ‹…)๐‘Ÿโˆˆโ„ฌ(โ„๐‘›)๎‚ผ.(3.2) If ๐‘ž=๐‘โˆ’, then, we write ๐‘Ÿ๐‘โˆ’,๐‘โˆ’=๐‘Ÿ๐‘โˆ’.

Remark 3.2. Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›) and ๐‘ก as in Theorem 3.1. Then, for any ๐‘คโˆˆ(1,๐‘ก], we have ๐‘(โ‹…)/๐‘คโˆˆโ„ฌ(โ„๐‘›). Furthermore, for any ๐‘คโˆˆ(0,1], we have also ๐‘(โ‹…)/๐‘คโˆˆโ„ฌ(โ„๐‘›) by Jensen inequality. The proofs are found in [5]. Consequently, if ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›), then, ๐‘(โ‹…)/๐‘Ÿโˆˆโ„ฌ(โ„๐‘›) for any ๐‘Ÿโˆˆ(0,๐‘Ÿ๐‘โˆ’). For any ๐‘Ÿโˆˆ(0,๐‘Ÿ๐‘โˆ’,๐‘ž), we have also ๐‘(โ‹…)/๐‘Ÿโˆˆโ„ฌ(โ„๐‘›).

The next theorem gives a generalized Hรถlder inequality, which is shown in [9, 13].

Theorem 3.3 (see [9, 13]). Let ๐‘โˆˆ๐’ซ(โ„๐‘›). Then, ๎€œโ„๐‘›||||๎‚ต1๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)d๐‘ฅโ‰ค1+๐‘โˆ’โˆ’1๐‘+๎‚ถโ€–๐‘“โ€–๐ฟ๐‘(โ‹…)โ€–๐‘”โ€–๐ฟ๐‘โ€ฒ(โ‹…),(3.3) for every ๐‘“โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›) and ๐‘”โˆˆ๐ฟ๐‘โ€ฒ(โ‹…)(โ„๐‘›).

The next theorem is shown in [6].

Theorem 3.4. Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›) and ๐‘“โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›). Then, ๎ƒฏ๎‚ต๎€œminโ„๐‘›|๐‘“(๐‘ฅ)|๐‘(๐‘ฅ)๎‚ถd๐‘ฅ1/๐‘โˆ’,๎‚ต๎€œโ„๐‘›||||๐‘“(๐‘ฅ)๐‘(๐‘ฅ)๎‚ถd๐‘ฅ1/๐‘+๎ƒฐโ‰คโ€–๐‘“โ€–๐ฟ๐‘(โ‹…)๎ƒฏ๎‚ต๎€œโ‰คmaxโ„๐‘›||||๐‘“(๐‘ฅ)๐‘(๐‘ฅ)๎‚ถd๐‘ฅ1/๐‘โˆ’,๎‚ต๎€œโ„๐‘›||||๐‘“(๐‘ฅ)๐‘(๐‘ฅ)๎‚ถd๐‘ฅ1/๐‘+๎ƒฐ.(3.4)

The next theorem is shown in [9].

Theorem 3.5. (i) Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›). Then, ๐ฟ๐‘(โ‹…)(โ„๐‘›) is a Banach space.
(ii) If ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›), then ๐ถโˆž0(โ„๐‘›) is dense in ๐ฟ๐‘(โ‹…)(โ„๐‘›).

Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›) such that ๐‘(โ‹…) is not a constant function. Then, according to [9, Example 2.9, Theorem 2.10], for every ๐ฟ๐‘(โ‹…), there exists a function ๐‘“(โ‹…)โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›) such that its translation ๐‘“(๐›ฝ+โ‹…)โˆ‰๐ฟ๐‘(โ‹…)(โ„๐‘›) if ๐›ฝโˆˆโ„๐‘›โงต{(0,0,โ€ฆ,0)}. However, we can prove that all elements of ๐’ฎ(โ„๐‘›) belong to ๐ฟ๐‘(โ‹…)(โ„๐‘›) for every ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›). Let ๐‘“(โ‹…)โˆˆ๐’ฎ(โ„๐‘›). Then, we have sup๐›ฝโˆˆโ„๐‘›โ€–๐‘“(๐›ฝ+๐›ผโ‹…)โ€–๐ฟ๐‘(โ‹…)โ‰คโŽงโŽชโŽจโŽชโŽฉ|๐›ผ|โˆ’๐‘›/๐‘+๐œ‹๐‘›/๐‘โˆ’sup๐‘ฅโˆˆโ„๐‘›๎‚†๎€ท1+|๐‘ฅ|2๎€ธ๐‘›||||๎‚‡๐‘“(๐‘ฅ)if|๐›ผ|โ‰ฅ1,|๐›ผ|โˆ’๐‘›/๐‘โˆ’๐œ‹๐‘›/๐‘โˆ’sup๐‘ฅโˆˆโ„๐‘›๎‚†๎€ท1+|๐‘ฅ|2๎€ธ๐‘›||||๎‚‡๐‘“(๐‘ฅ)if|๐›ผ|<1,(3.5) where ๐›ผโˆˆโ„โงต{0}.

Hence, by the same arguments of Triebel [20], we have the following two theorems.

Theorem 3.6. Let ฮฉ be a compact subset of โ„๐‘› and ๐›ผ an arbitrary multi-index. Let ๐‘(โ‹…), ๐‘ž(โ‹…)โˆˆโ„ฌ(โ„๐‘›) such that 1<๐‘(๐‘ฅ)โ‰ค๐‘ž(๐‘ฅ)<โˆž. Then, there exists a positive constant ๐‘ such that โ€–๐ท๐›ผ๐‘“โ€–๐ฟ๐‘ž(โ‹…)โ‰ค๐‘โ€–๐‘“โ€–๐ฟ๐‘(โ‹…),โ€–๐ท(3.6)๐›ผ๐‘“โ€–๐ฟโˆžโ‰ค๐‘โ€–๐‘“โ€–๐ฟ๐‘(โ‹…),(3.7) for all ๐‘“โˆˆ๐ฟฮฉ๐‘(โ‹…)(โ„๐‘›).

Let ๐‘“(โ‹…)โˆˆ๐ฟฮฉ๐‘(โ‹…)(โ„๐‘›) and โ„ฑโˆ’1๐‘€โˆˆ๐ฟ1(โ„๐‘›). Then, ๎€ทโ„ฑโˆ’1๎€ธ(๎€œ๐‘€โ„ฑ๐‘“๐‘ฅ)=๐‘โ„๐‘›๎€ทโ„ฑโˆ’1๐‘€๎€ธ(๐‘ฅโˆ’๐‘ฆ)๐‘“(๐‘ฆ)d๐‘ฆ(3.8) makes sense for any ๐‘ฅโˆˆโ„๐‘› by the classical Hรถlder inequality and (3.7).

Let ๐‘  be a real number and ๐ป๐‘ 2(โ„๐‘›) the spaces of ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) such that โ€–๐‘“โ€–๐ป๐‘ 2=โ€–โ€–๎€ท1+|โ‹…|2๎€ธ๐‘ /2โ€–โ€–(โ„ฑ๐‘“)(โ‹…)๐ฟ2<โˆž.(3.9)

Theorem 3.7. Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›) and 0<๐‘ž<โˆž. Let ฮฉ={ฮฉ๐‘˜}โˆž๐‘˜=0 be a sequence of compact subsets of โ„๐‘›. Let ๐‘‘๐‘˜>0 be the diameter of ฮฉ๐‘˜. If ๐‘ฃ>๐‘›/2+๐‘›/๐‘Ÿ๐‘โˆ’,๐‘ž, then there exists a number ๐‘ such that โ€–โ€–โ„ฑโˆ’1๐‘€๐‘˜โ„ฑ๐‘“๐‘˜โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค๐‘sup๐‘™โ€–โ€–๐‘€๐‘™๎€ท๐‘‘๐‘™โ‹…๎€ธโ€–โ€–๐ป๐‘ฃ2โ€–โ€–๐‘“๐‘˜โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)(3.10) for {๐‘“๐‘˜(๐‘ฅ)}โˆž๐‘˜=0โˆˆ๐ฟฮฉ๐‘(โ‹…)(โ„“๐‘ž) and {๐‘€๐‘˜(๐‘ฅ)}โˆž๐‘˜=0โˆˆ๐ป๐‘ฃ2(โ„๐‘›).

4. Basic Properties of Variable Exponent Triebel-Lizorkin and Besov Spaces

We will first consider the quasi-norms on ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›).

Definition 4.1. (i) The set ฮจ(โ„๐‘›) is the collection of all systems ๐œ‘={๐œ‘๐‘—}โˆž๐‘—=0โŠ‚๐’ฎ(โ„๐‘›) with โ„ฑ๐œ‘๐‘˜(๐œ‰)=โ„ฑ๐œ‘(2โˆ’๐‘˜๐œ‰) for ๐‘˜=1,2,โ€ฆ, such that suppโ„ฑ๐œ‘0โŠ‚{๐‘ฅโˆถ|๐‘ฅ|โ‰ค2},suppโ„ฑ๐œ‘๐‘—โŠ‚๎€ฝ๐‘ฅโˆถ2๐‘—โˆ’1โ‰ค|๐‘ฅ|โ‰ค2๐‘—+1๎€พfor๐‘—=1,2,โ€ฆ,(4.1) for every multi-index ๐›ผ, there exists a positive number ๐‘๐›ผ such that 2๐‘—|๐›ผ|||๐ท๐›ผโ„ฑ๐œ‘๐‘—||(๐‘ฅ)โ‰ค๐‘๐›ผ,(4.2) for ๐‘—=0,1,โ€ฆ, and ๐‘ฅโˆˆโ„๐‘› and there exists a positive number ๐‘ such that ๐‘โ‰คโˆž๎“๐‘—=0โ„ฑ๐œ‘๐‘—(๐‘ฅ),(4.3) for ๐‘ฅโˆˆโ„๐‘›.
(ii) The set ฮฆ(โ„๐‘›) is the collection of all systems ๐œ‘={๐œ‘๐‘—}โˆž๐‘—=0โŠ‚๐’ฎ(โ„๐‘›) such that supp๐œ‘0โŠ‚{๐‘ฅโˆถ|๐‘ฅ|โ‰ค2},supp๐œ‘๐‘—โŠ‚๎€ฝ๐‘ฅโˆถ2๐‘—โˆ’1โ‰ค|๐‘ฅ|โ‰ค2๐‘—+1๎€พfor๐‘—=1,2,โ€ฆ,(4.4) for every multi-index ๐›ผ, there exists a positive number ๐‘๐›ผ such that 2๐‘—|๐›ผ|||๐ท๐›ผ๐œ‘๐‘—||(๐‘ฅ)โ‰ค๐‘๐›ผ,(4.5) for ๐‘—=0,1,โ€ฆ, and ๐‘ฅโˆˆโ„๐‘›, and โˆž๎“๐‘—=0๐œ‘๐‘—(๐‘ฅ)=1,(4.6) for ๐‘ฅโˆˆโ„๐‘›.

Theorem 4.2. (i) Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›), 0<๐‘žโ‰คโˆž, and ๐‘ โˆˆโ„. Let ๐œƒ๐‘— be as in Definition 2.1 and ๐œ‘๐‘—=โ„ฑ๐œƒ๐‘—. Then, the norm โ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž coincides with โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž) and the norm โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),๐‘ž with โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…)).
(ii) Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›), 0<๐‘žโ‰คโˆž, and ๐‘ โˆˆโ„. Let ๐œ‘, ๐œ“โˆˆฮฆ(โ„๐‘›). Then, โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž) and โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ“๐‘—โ„ฑ๐‘“โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž) are equivalent quasi-norms on ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›). Similarly, โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…)) and โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ“๐‘—โ„ฑ๐‘“โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…)) are equivalent quasi-norms on ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„n).
(iii) Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›), 0<๐‘žโ‰คโˆž, and ๐‘ โˆˆโ„. Let ๐œ‘โˆˆฮฆ(โ„๐‘›) and ๐œ“โˆˆฮจ(โ„๐‘›). Then, โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž) and โ€–2๐‘—๐‘ ๐œ“๐‘—โˆ—๐‘“โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž) are equivalent quasi-norms on ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›). Similarly, โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…)) and โ€–2๐‘—๐‘ ๐œ“๐‘—โˆ—๐‘“โ€–โ„“๐‘ž(L๐‘(โ‹…)) are equivalent quasi norms on ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›).

Proof. Let ๐œƒ๐‘— be as in Definition 2.1. It is obvious that ๐œƒ๐‘—โˆ—๐‘“=๐‘‘โ„ฑโ„ฑโˆ’1(โ„ฑ๐œƒ๐‘—โ‹…โ„ฑ๐‘“) for some positive number ๐‘‘โ„ฑ, which depends only on the definition of Fourier transform โ„ฑ. Let ๐œ‘๐‘—=โ„ฑ๐œƒ๐‘—. Then {๐œ‘๐‘—}โˆž๐‘—=0โˆˆฮฆ(โ„๐‘›). Hence, the norm โ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž coincides with โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž). Similarly, the variable exponent Besov norm โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),๐‘ž coincides with โ€–2๐‘—๐‘ โ„ฑโˆ’1๐œ‘๐‘—โ„ฑ๐‘“โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…)). This proves (i). The proof of (ii) is done as in the proof of [20, page 46, Proposition 1] with using Theorem 3.7, the scalar case of Theorem 3.7, and Definition 4.1.
We will prove (iii). Let ๐‘“โˆˆ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›), ๐‘”๐‘—=โ„ฑโˆ’1(๐œ‘๐‘—โ„ฑ๐‘“), and ๐œ“โˆˆฮจ(โ„๐‘›). Let ๐œ“๐‘˜โ‰ก๐‘”๐‘˜โ‰ก0 for ๐‘˜<0. Then, ๐‘“โˆ—๐œ“๐‘˜(๐‘ฅ)=๐’ฎโ€ฒโˆž๎“๐‘—=0๐‘”๐‘—โˆ—๐œ“๐‘˜(๐‘ฅ)=๐‘˜+2๎“๐‘—=๐‘˜โˆ’2๎€ท๐‘”๐‘—โˆ—๐œ“๐‘˜๎€ธ(๐‘ฅ),(4.7) because ๐‘”๐‘—โˆ—๐œ“๐‘˜=โ„ฑโˆ’1โ„ฑ๎€ท๐‘”๐‘—โˆ—๐œ“๐‘˜๎€ธ=๐‘‘โ„ฑโ„ฑโˆ’1๎€ทโ„ฑ๐œ“๐‘˜โ‹…โ„ฑ๐‘”๐‘—๎€ธ=0,(4.8) for ๐‘—<๐‘˜โˆ’2 and ๐‘—>๐‘˜+2. Here, ๐‘“=๐’ฎโ€ฒโˆ‘โˆž0๐‘Ž๐‘— means โˆ‘๐‘0๐‘Ž๐‘— converges to ๐‘“ in ๐’ฎ๎…ž(โ„๐‘›). It follows that โ€–โ€–๎€ฝ2๐‘ ๐‘˜๐‘“โˆ—๐œ“๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค2๎“๐‘Ÿ=โˆ’2โ€–โ€–๎€ฝ2๐‘ ๐‘˜๐‘”๐‘˜+๐‘Ÿโˆ—๐œ“๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค2๎“๐‘Ÿ=โˆ’2โ€–โ€–๎€ท2โ„ฑ๐œ“๐‘Ÿ+1โ‹…๎€ธโ€–โ€–๐ป๐œ˜2โ€–โ€–๎€ฝ2๐‘ ๐‘˜๐‘”๐‘˜+๐‘Ÿ๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž),(4.9) where we use Theorem 3.7 with ๐‘“๐‘˜=๐‘”๐‘˜+๐‘Ÿ, ๐‘€๐‘˜=โ„ฑ๐œ“๐‘˜, and ๐‘‘๐‘˜=2๐‘˜+1+๐‘Ÿ. Hence, we have โ€–โ€–๎€ฝ2๐‘ ๐‘˜๐‘“โˆ—๐œ“๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค2๎“๐‘Ÿ=โˆ’2โ€–โ€–๎€ท2โ„ฑ๐œ“๐‘Ÿ+1โ‹…๎€ธโ€–โ€–๐ป๐œ˜2โ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž.(4.10) We will prove the opposite inequality. Let ๐œŒ(๐‘ฅ) be a real function with ๐œŒ(๐‘ฅ)=1 for 2โˆ’1โ‰ค|๐‘ฅ|โ‰ค2 and ๐œŒ(โ‹…)โˆˆ๐ถโˆž0๎€ท๎€ฝ๐œ‰โˆถ2โˆ’2โ‰ค||๐œ‰||โ‰ค22.๎€พ๎€ธ(4.11) We set ๐œŒ๐‘˜(๐‘ฅ)=๐œŒ(2โˆ’๐‘˜๐‘ฅ), for ๐‘˜=1,2,โ€ฆ. Furthermore, let ๐œŒ0(๐‘ฅ) be a real function such that ๐œŒ0(๐‘ฅ)=1 for |๐‘ฅ|โ‰ค2 and ๐œŒ0(โ‹…)โˆˆ๐ถโˆž0||๐œ‰||๎€ท๎€ฝ๐œ‰โˆถโ‰ค22.๎€พ๎€ธ(4.12) We define โ„Ž๐‘˜(๐‘ฅ) by โ„ฑโ„Ž๐‘˜๎ƒฉ(๐œ‰)=โˆž๎“๐‘™=0โ„ฑ๐œ“๐‘™๎ƒช(๐œ‰)โˆ’1๐œŒ๐‘˜(๐œ‰),(4.13) for ๐‘˜=0,1,2,โ€ฆ (โ„Ž๐‘˜=0 for ๐‘˜<0). Then, โ„ฑโ„Ž๐‘˜(โ‹…)โˆˆ๐ถโˆž0(โ„๐‘›) and โ„Ž๐‘˜(โ‹…)โˆˆ๐’ฎ(โ„๐‘›). Let ๐›พ๐‘˜=๐œ“๐‘˜โˆ—โ„Ž๐‘˜. It is obvious that {โ„ฑ๐›พ๐‘˜/๐‘‘โ„ฑ}โˆˆฮฆ(โ„๐‘›) and {2๐‘ ๐‘˜๐œ“๐‘˜โˆ—๐‘“}โˆž0โˆˆ๐ฟฮฉ๐‘(โ‹…)(โ„“๐‘ž) by Theorem 4.8 and (4.10), where ฮฉ is some sequence of compact subsets of โ„๐‘›. Then, โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘“โˆ—๐›พ๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)=โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘“โˆ—๐œ“๐‘˜โˆ—โ„Ž๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰คsup๐‘˜โ€–โ€–โ„ฑโ„Ž๐‘˜๎€ท2๐‘˜+1โ‹…๎€ธโ€–โ€–๐ป๐œ˜2โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘“โˆ—๐œ“๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž),โ„ฑโ„Ž๐‘˜๎€ท2๐‘˜+1๐‘ฅ๎€ธ=๐œŒ๐‘˜๎€ท2๐‘˜+1๐‘ฅ๎€ธโˆ‘โˆž๐‘™=0โ„ฑ๐œ“๐‘™๎€ท2๐‘˜+1๐‘ฅ๎€ธ=๐œŒ๐‘˜๎€ท2๐‘˜+1๐‘ฅ๎€ธโˆ‘๐‘˜+3๐‘™=๐‘˜โˆ’3โ„ฑ๐œ“๐‘™๎€ท2๐‘˜+1๐‘ฅ๎€ธ=๐œŒ(2๐‘ฅ)โˆ‘4๐‘Ÿ=โˆ’4โ„ฑ๐œ“(2๐‘Ÿ,๐‘ฅ)(4.14) for any ๐‘˜=1,2,โ€ฆ. This means that โ„ฑโ„Ž๐‘˜(2๐‘˜+1๐‘ฅ) is independent of ๐‘˜โˆˆโ„• and in ๐ถโˆž0(โ„๐‘›). Hence, we have sup๐‘˜โ€–โ„ฑโ„Ž๐‘˜(2๐‘˜+1โ‹…)โ€–๐ป๐œ˜2<โˆž. This implies that โ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘žโ‰คsup๐‘˜โ€–โ€–โ„ฑโ„Ž๐‘˜๎€ท2๐‘˜+1โ‹…๎€ธโ€–โ€–๐ป๐œ˜2โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘“โˆ—๐œ“๐‘˜๎€พโˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž).(4.15) This proves the ๐น๐‘ ๐‘(โ‹…),๐‘ž case of (iii). The proof of the ๐ต๐‘ ๐‘(โ‹…),๐‘ž case of (iii) is essentially the same as one of the ๐น๐‘ ๐‘(โ‹…),๐‘ž case.

Corollary 4.3. Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›), ๐‘ โˆˆโ„, and 1<๐‘ž<โˆž.
(i)๐‘“โˆˆ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) if and only if there exist continuous functions {๐‘Ž๐‘—}๐‘—โˆˆโ„•0 such that ๐‘“=๐’ฎ๎…žโˆ‘โˆž๐‘—=0๐‘Ž๐‘—, โ€–{2๐‘—๐‘ ๐‘Ž๐‘—}โˆž0โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)<โˆž, suppโ„ฑ๐‘Ž0โŠ‚๎€ฝ||๐œ‰||๎€พ๐œ‰โˆถโ‰ค2,suppโ„ฑ๐‘Ž๐‘—โŠ‚๎€ฝ๐œ‰โˆถ2๐‘—โˆ’1โ‰ค||๐œ‰||โ‰ค2๐‘—+1๎€พ,for๐‘—โˆˆโ„•0.(4.16) In particular, โ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž coincides with infโˆ‘๐‘Ž๐‘“=jโ€–{2๐‘—๐‘ ๐‘Ž๐‘—}โˆž0โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž), where the infimum is taken over all representation โˆ‘๐‘Ž๐‘“=๐‘—.
(ii)๐‘“โˆˆ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) if and only if, there exist continuous functions {๐‘Ž๐‘—}๐‘—โˆˆโ„•0 such that ๐‘“=๐’ฎ๎…žโˆ‘โˆž๐‘—=0๐‘Ž๐‘—, โ€–{2๐‘—๐‘ ๐‘Ž๐‘—}โˆž0โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…))<โˆž, suppโ„ฑ๐‘Ž0โŠ‚{๐œ‰โˆถ|๐œ‰|โ‰ค2}, and suppโ„ฑ๐‘Ž๐‘—โŠ‚{๐œ‰โˆถ2๐‘—โˆ’1โ‰ค|๐œ‰|โ‰ค2๐‘—+1} for ๐‘—โˆˆโ„•0. In particular, the norm โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),๐‘ž coincides with infโˆ‘๐‘Ž๐‘“=๐‘—โ€–{2๐‘—๐‘ ๐‘Ž๐‘—}โˆž0โ€–โ„“๐‘ž(๐ฟ๐‘(โ‹…)).

Proposition 4.4. (i) Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›), ๐‘ โˆˆโ„, and 0<๐‘ž1โ‰ค๐‘ž2โ‰คโˆž. Then, ๐ต๐‘ ๐‘(โ‹…),๐‘ž1(โ„๐‘›)โŠ‚๐ต๐‘ ๐‘(โ‹…),๐‘ž2(โ„๐‘›),๐น๐‘ ๐‘(โ‹…),๐‘ž1(โ„๐‘›)โŠ‚๐น๐‘ ๐‘(โ‹…),๐‘ž2(โ„๐‘›).(4.17)
(ii) Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›), ๐‘ โˆˆโ„, 0<๐‘ž1โ‰คโˆž, 0<๐‘ž2โ‰คโˆž, and ๐œ–>0. Then ๐ต๐‘ +๐œ–๐‘(โ‹…),๐‘ž1(โ„๐‘›)โŠ‚๐ต๐‘ ๐‘(โ‹…),๐‘ž2(โ„๐‘›๐น),๐‘ +๐œ–๐‘(โ‹…),๐‘ž1(โ„๐‘›)โŠ‚๐น๐‘ ๐‘(โ‹…),๐‘ž2(โ„๐‘›).(4.18)
(iii) Let ๐‘(โ‹…)โˆˆ๐’ซ(โ„๐‘›) and ๐‘ โˆˆโ„. If 0<๐‘žโ‰คโˆž, then ๐ต๐‘ ๐‘(โ‹…),min{๐‘โˆ’,๐‘ž}(โ„๐‘›)โŠ‚๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›)โŠ‚๐ต๐‘ ๐‘(โ‹…),max{๐‘+,๐‘ž}(โ„๐‘›).(4.19)

The proof is almost same as in [20]. Furthermore, Almeida and Hรคstรถ [14] proved the above inclusion for ๐น๐›ผ(โ‹…)๐‘(โ‹…),๐‘ž(โ‹…)(โ„๐‘›) and ๐ต๐›ผ(โ‹…)๐‘(โ‹…),๐‘ž(โ‹…)(โ„๐‘›).

Theorem 4.5. Let 0<๐‘ž<โˆž, ๐‘ โˆˆโ„, and ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). Let ๐ด๐‘ ๐‘(โ‹…),๐‘ž be either ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) or ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›). Then, ๐’ฎ(โ„๐‘›)โŠ‚๐ด๐‘ ๐‘(โ‹…),๐‘žโŠ‚๐’ฎ๎…ž(โ„๐‘›). Furthermore, ๐‘†(โ„๐‘›) is dense in ๐ด๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and ๐ด๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) is a quasi Banach space (a Banach space if 1โ‰ค๐‘žโ‰คโˆž).

Proof. To prove the inclusion, we may restrict ourselves to ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) with ๐‘ž=โˆž by Proposition 4.4. Let ๐‘“โˆˆ๐’ฎ(โ„๐‘›) and ๐œ‘={๐œ‘๐‘˜(๐‘ฅ)}โˆž๐‘˜=0โˆˆฮฆ(โ„๐‘›). We recall that the topology of the complete locally convex space ๐’ฎ(โ„๐‘›) is generated by seminorms: ๐‘๐‘(๐‘“)=sup๐‘ฅโˆˆโ„๐‘›๎“|๐›ผ|+๐‘˜โ‰ค๐‘๎€ท1+|๐‘ฅ|2๎€ธ๐‘˜||๐ท๐›ผ||๐‘“(๐‘ฅ),๐‘โˆˆโ„•,(4.20)โ„ฑ yields a one-to-one mapping from ๐’ฎ(โ„๐‘›) onto itself, and, in particular, ๐‘๐‘(โ„ฑ๐œ‘) with ๐‘=1,2,โ€ฆ generates the topology of ๐’ฎ(โ„๐‘›). If ๐‘ is a sufficiently large natural number, then โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),๐‘ž=sup๐‘˜2๐‘˜๐‘ โ€–โ€–โ„ฑโˆ’1๐œ‘๐‘˜โ€–โ€–โ„ฑ๐‘“๐ฟ๐‘(โ‹…)โ‰ค๐‘๐‘๐‘(โ„ฑ๐‘“),(4.21) where ๐‘ is a positive number. This proves the left-hand side of the inclusion with ๐‘ž=โˆž. We prove the right-hand side of the inclusion with ๐‘ž=โˆž. Let ๐œ‘โˆˆฮฆ(โ„๐‘›) be the above system. It is well known that ๐‘“(๐œ“)=โˆž๎“๐‘˜=0โ„ฑโˆ’1๐œ‘๐‘˜โ„ฑ๐‘“(๐œ“)(4.22) in ๐’ฎ๎…ž(โ„๐‘›) for all ๐‘“โˆˆ๐’ฎโ€ฒ(โ„๐‘›) and ๐œ“โˆˆ๐’ฎ(โ„๐‘›). By the Paley-Wiener-Schwartz Theorem, the definition of ๐œ‘๐‘˜ implies that โ„ฑโˆ’1๐œ‘๐‘˜โ„ฑ๐‘“ are analytic functions, and that these functions are regular distributions. We put ๐œ’๐‘˜(๐‘ฅ)=๐œ‘๐‘˜โˆ’1(๐‘ฅ)+๐œ‘๐‘˜(๐‘ฅ)+๐œ‘๐‘˜+1(๐‘ฅ) if ๐‘˜=0,1,โ€ฆ, with ๐œ‘โˆ’1=0. If ๐‘“โˆˆ๐ต๐‘ ๐‘(โ‹…),โˆž(โ„๐‘›) and ๐œ“โˆˆ๐’ฎ(โ„๐‘›), then ๐‘“(๐œ“) denotes the value of the functional ๐‘“ of ๐’ฎโ€ฒ(โ„๐‘›) of the test function ๐œ“. We obtain ||||โ‰ค|||||๐‘“(๐œ“)โˆž๎“๐‘˜=0โ„ฑโˆ’1๐œ‘๐‘˜๎€ทโ„ฑ๐‘“โ„ฑ๐œ’๐‘˜โ„ฑโˆ’1๐œ“๎€ธ|||||โ‰คโˆž๎“๐‘˜=0โ€–โ€–โ„ฑโˆ’1๐œ‘๐‘˜โ€–โ€–โ„ฑ๐‘“๐ฟโˆžโ€–โ€–โ„ฑ๐œ’๐‘˜โ„ฑโˆ’1๐œ“โ€–โ€–๐ฟ1(4.23) by the classical Hรถlder inequality. Because both โ„ฑโˆ’1๐œ‘๐‘˜โ„ฑ๐‘“ and โ„ฑ๐œ’๐‘˜โ„ฑโˆ’1๐œ“ are analytic functions, the last estimate makes sense. By applying (3.7) with ๐›ผ=(0,0,โ€ฆ,0) to โ„ฑโˆ’1๐œ‘๐‘˜โ„ฑ๐‘“, we have ||||๐‘“(๐œ“)โ‰ค๐‘โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),โˆžโˆž๎“๐‘˜=02โˆ’๐‘ ๐‘˜โ€–โ€–โ„ฑ๐œ’๐‘˜โ„ฑโˆ’1๐œ“โ€–โ€–๐ฟ1โ‰ค๐‘โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),โˆžโ€–๐œ“โ€–๐ตโˆ’๐‘ 1,1.(4.24) If ๐‘ is a sufficiently large natural number, then, for any ๐œ“โˆˆ๐’ฎ(โ„๐‘›), we have ||||๐‘“(๐œ“)โ‰ค๐‘โ€–๐‘“โ€–๐ต๐‘ ๐‘(โ‹…),โˆž๐‘๐‘(๐œ“)(4.25) by the left-hand side of the inclusion for ๐ตโˆ’๐‘ 1,1(โ„๐‘›). Hence, we have the inclusion for ๐ด๐‘ ๐‘(โ‹…),๐‘ž.
The proof of the completeness and the density of ๐’ฎ(โ„๐‘›) is given by the same arguments of step 4 and step 5 of the proof of [20, page 48, Theorem] with ๐‘ replaced by ๐‘(โ‹…).

As we mentioned in Section 1, the next theorem is found in [15, 16].

Theorem 4.6. Let ๐‘ โ‰ฅ0 and ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). Then, ๐น๐‘ ๐‘(โ‹…),2(โ„๐‘›) coincides with ๐ฟ๐‘ ,๐‘(โ‹…)(โ„๐‘›). Let ๐‘˜โˆˆโ„• and ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). Then, ๐น๐‘˜๐‘(โ‹…),2(โ„๐‘›) coincides with ๐‘Š๐‘˜,๐‘(โ‹…)(โ„๐‘›).

Definition 4.7. Let ๐œ‘โˆˆ๐’ฎ(โ„๐‘›) such that (โ„ฑ๐œ‘)(๐œ‰)=1 for โˆš1/โˆš2โ‰ค|๐œ‰|โ‰ค2 and 0โ‰คโ„ฑ๐œ‘โˆˆ๐ถโˆž0๎ƒฉ๎ƒฏ๐œ‰โˆˆโ„๐‘›โˆถ1โˆš2||๐œ‰||<โˆšโˆ’๐œ–<2+๐œ–๎ƒฐ๎ƒช,(4.26) and let ๐œŒโˆˆ๐’ฎ(โ„๐‘›) such that 0โ‰คโ„ฑ๐œŒโˆˆ๐ถโˆž0๎ƒฉ๎ƒฏ๐œ‰โˆˆโ„๐‘›โˆถ1โˆš2||๐œ‰||<โˆš+๐›ฟ<2โˆ’๐›ฟ๎ƒฐ๎ƒช,(4.27) where ๐œ– and ๐›ฟ are positive numbers. We construct {๐œ‘๐‘˜}โˆž๐‘˜=0 and {๐œŒ๐‘˜}โˆž๐‘˜=0 by ๎€ทโ„ฑ๐œ‘๐‘˜๎€ธ๎€ท2(๐œ‰)=(โ„ฑ๐œ‘)โˆ’๐‘˜๐œ‰๎€ธ,๎€ทโ„ฑ๐œŒ๐‘˜๎€ธ๎€ท2(๐œ‰)=(โ„ฑ๐œŒ)โˆ’๐‘˜๐œ‰๎€ธ.(4.28) We choose ๐œ– and ๐›ฟ sufficiently small so that โ„ฑ๐œ‘๐‘˜โ‹…โ„ฑ๐œŒ๐‘—โ‰ก0(4.29) for ๐‘˜โ‰ ๐‘—.

Diening [3] shows that Young inequality โ€–๐‘“โˆ—๐‘”โ€–๐ฟ๐‘(โ‹…)โ‰คโ€–๐‘“โ€–๐ฟ๐‘(โ‹…)โ€–๐‘”โ€–๐ฟ1 holds if and only if ๐‘(โ‹…) is a constant function. However, the next theorem, which is a part of Corollary 3.6 of [3], holds.

Theorem 4.8. Let ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). Let ๐œ™ be an integrable function on โ„๐‘› with range in โ„ and set ๐œ™๐œ–(๐‘ฅ)=๐œ–โˆ’๐‘›๐œ™(๐‘ฅ/๐œ–) for all ๐œ–>0. Assume that the least decreasing radial majorant of ๐œ™ is integrable, that is, โˆซ๐ด=โ„๐‘›sup|๐‘ฆ|โ‰ฅ|๐‘ฅ||๐œ™(๐‘ฆ)|d๐‘ฅ<โˆž. Then, for any ๐‘“โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›), one has sup๐œ–>0||๎€ท๐‘“โˆ—๐œ™๐œ–๎€ธ||(๐‘ฅ)โ‰ค2๐ดโ„ณ๐‘“(๐‘ฅ).(4.30) Hence, there exist a positive number ๐ถ(๐ด,๐‘) such that โ€–โ€–๐‘“โˆ—๐œ™๐œ–โ€–โ€–๐ฟ๐‘(โ‹…)โ‰ค๐ถ(๐ด,๐‘)โ€–๐‘“โ€–๐ฟ๐‘(โ‹…),(4.31) where ๐ถ(๐ด,๐‘) only depends on ๐ด and ๐‘(โ‹…).

The duality pair of ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›))โ€ฒ and ๐‘“โˆˆ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) is denoted by โŸจ๐‘”,๐‘“โŸฉ, although, for ๐œ‘โˆˆ๐’ฎ(โ„๐‘›)โŠ‚๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›), we also write ๐‘”(๐œ‘) as โŸจ๐‘”,๐œ‘โŸฉ. In order to prove the duality of ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) (Theorem 5.1), we will first prove the following two lemmas which correspond to [19, Lemmasโ€‰โ€‰7.1.3, andโ€‰7.1.5].

Lemma 4.9. Let ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒ, {๐œŒ๐‘˜} as in Definition 4.7 and ฬƒ๐‘”=โˆž๎“๐‘˜=0โ„ฑโˆ’1๎€ทโ„ฑ๐œŒ๐‘˜๎€ธโ‹…โ„ฑ๐‘”(4.32) in ๐’ฎ๎…ž(โ„๐‘›). Then, ฬƒ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž)๎…ž and โ€–ฬƒ๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒ,(4.33) where ๐‘ does not depend on ๐‘”.

Proof. We use the same argument of the proof of [19, Lemma 7.1.3]. Let ๐‘“โˆˆ๐’ฎ(โ„๐‘›). Then, it holds that ๎€ทโ„ฑฬƒ๐‘”(๐‘“)=(โ„ฑฬƒ๐‘”)โˆ’1๐‘“๎€ธ=โˆž๎“๐‘˜=0๎€ทโ„ฑ๐‘”โ„ฑ๐œŒ๐‘˜โ„ฑโˆ’1๐‘“๎€ธ=๐‘โˆž๎“๐‘˜=0๐‘”๎€ท๐œŒโˆจ๐‘˜๎€ธ๎ƒฉโˆ—๐‘“=๐‘๐‘”โˆž๎“๐‘˜=0๐œŒโˆจ๐‘˜๎ƒช,โˆ—๐‘“(4.34) where ๐œŒโˆจ๐‘˜(๐‘ฅ)=๐œŒ๐‘˜(โˆ’๐‘ฅ). Using ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒ and Theorem 3.7 with ๐‘€๐‘˜=โ„ฑ(๐œŒโˆจ๐‘˜) and ๐‘‘๐‘˜=2๐‘˜+1, we find that ||||ฬƒ๐‘”(๐‘“)โ‰คโ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ€–โ€–๎€ฝ2๐‘˜๐‘ ๐œŒโˆจ๐‘˜โˆ—๐œ‘๐‘˜๎€พโˆ—๐‘“โˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ€–โ€–๎€ฝ2๐‘˜๐‘ ๐œ‘๐‘˜๎€พโˆ—๐‘“โˆž0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž),(4.35) where {๐œ‘๐‘˜}โˆž๐‘˜=0 is a system of Definition 4.7. Thus, we get ||||ฬƒ๐‘”(๐‘“)โ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž,(4.36) from Definition 2.1 and Theorem 4.2.

Lemma 4.10. Let {๐œŒ๐‘˜} and {๐œ‘๐‘˜} be ones as in Definition 4.7, ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒ, and ฬƒ๐‘” the functional in Lemma 4.9. Let {๐‘๐‘˜}๐‘๐‘˜=0 be a system of functions such that ๐‘๐‘˜โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›). Set ๐‘Ž๐‘˜=โ„ฑโˆ’1๎€ทโ„ฑ๐œŒ๐‘˜๎€ธ,๐‘โ‹…โ„ฑ๐‘”(4.37)๐‘˜=๐‘๐‘˜โˆ—๐œ‘๐‘˜,(4.38) for ๐‘˜=0,1,2,โ€ฆ,๐‘, and ๐‘“=๐‘๎“๐‘˜=0๐‘๐‘˜.(4.39) Then, ๎€œโŸจฬƒ๐‘”,๐‘“โŸฉ=๐‘โ„๐‘›๐‘๎“๐‘˜=0๐‘Ž๐‘˜๐‘๐‘˜d๐‘ฅ,(4.40) where ๐‘ depends only on the definition of Fourier transform.

Proof. We use the same argument of the proof of [19, Lemma 7.1.5]. We first prove that ๐‘Ž๐‘˜โˆˆ๐ฟ๐‘๎…ž(โ‹…), where ๐‘๎…ž(โ‹…) is defined by (3.1). The consideration of the proof of Lemma 4.9 implies that for ๐‘“โˆˆ๐’ฎ(โ„๐‘›), ||๐‘Ž๐‘˜||(๐‘“)โ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ€–โ€–๎€ฝ2๐‘ ๐‘—๐›ฟ๐‘—,๐‘˜๐œ‘๐‘˜๎€พโˆ—๐‘“๐‘—โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค๐‘๎…žโ€–๐‘“โ€–๐ฟ๐‘(โ‹…),(4.41) where ๐›ฟ๐‘˜,๐‘˜=1 and ๐›ฟ๐‘—,๐‘˜=0 if ๐‘—โ‰ ๐‘˜. This implies ๐‘Ž๐‘˜โˆˆ๐ฟ๐‘โ€ฒ(โ‹…) and the right-hand side of (4.40) has a sense. We assume ๐‘๐‘˜โˆˆ๐’ฎ(โ„๐‘›). Then, ๐‘๐‘˜โˆˆ๐’ฎ(โ„๐‘›) and ๐‘“โˆˆ๐’ฎ(โ„๐‘›). By (4.32), we have โŸจฬƒ๐‘”,๐‘“โŸฉ=ฬƒ๐‘”(๐‘“)=โˆž๎“๐‘˜=0๐‘Ž๐‘˜(๐‘“)(4.42) in ๐’ฎ๎…ž(โ„๐‘›) and ๐‘Ž๐‘˜(๐‘“)=๐‘๎“๐‘™=0๐‘Ž๐‘˜๎€ท๐‘๐‘™๎€ธ=๐‘๎“๐‘™=0โ„ฑ๐‘Ž๐‘˜๎€ทโ„ฑโˆ’1๐‘๐‘™๎€ธ=๐‘๐‘๎“๐‘™=0๎€ทโ„ฑ๐‘Ž๐‘˜โ‹…โ„ฑโˆ’1๐œ‘๐‘™โ„ฑ๎€ธ๎€ทโˆ’1๐‘๐‘™๎€ธ.(4.43) Since (โ„ฑโˆ’1โ„Ž)(๐œ‰)=(โ„ฑโ„Ž)(โˆ’๐œ‰), (4.29), (4.37), and (4.38) imply that ๐‘Ž๐‘˜(๐‘“)=๐‘โ„ฑ๐‘Ž๐‘˜๎€ทโ„ฑโˆ’1๐‘๐‘˜๎€ธ๎€œ=๐‘โ„๐‘›๐‘Ž๐‘˜๐‘๐‘˜d๐‘ฅ.(4.44) From this, (4.40) follows for ๐‘๐‘˜โˆˆ๐’ฎ(โ„๐‘›). Let ๐‘๐‘˜โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›), ๐‘๐‘˜,๐‘—โˆˆ๐’ฎ(โ„๐‘›) such that ๐‘๐‘˜,๐‘—โŸถ๐‘๐‘˜(4.45) in ๐ฟ๐‘(โ‹…)(โ„๐‘›) as ๐‘—โ†’โˆž and ๐‘๐‘˜,๐‘—=๐‘๐‘˜,๐‘—โˆ—๐œ‘๐‘˜,๐‘“๐‘—=๐‘๎“๐‘˜=0๐‘๐‘˜,๐‘—.(4.46) Thus, (4.40) holds for ๐‘“๐‘— and ๐‘๐‘˜,๐‘—. For any โ„Žโˆˆ๐’ฎ(โ„๐‘›), the least decreasing radial majorant of โ„Ž is integrable. By the definition of ๐œ‘๐‘˜(๐‘ฅ), ๐œ‘๐‘˜(๐‘ฅ)=2๐‘˜๐‘›๐œ‘๎€ท2๐‘˜๐‘ฅ๎€ธ.(4.47) Hence, ๐‘๐‘˜,๐‘— converges to ๐‘๐‘˜ in ๐ฟ๐‘(โ‹…)(โ„๐‘›) as ๐‘—โ†’โˆž by Theorem 4.8 and ๐‘“๐‘— converges to ๐‘“ in ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) as ๐‘—โ†’โˆž. Equation (4.40) follows from the last relation.

5. Duality and Reflexivity of Variable Exponent Function Spaces

Theorem 5.1. Let 1<๐‘ž<โˆž, ๐‘ โˆˆโ„, and ๐‘(โ‹…)โˆˆโ„ฌ(โ„๐‘›). Let ๐ด๐‘ ๐‘(โ‹…),๐‘ž be either ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) or ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›). Then, ๎‚€๐ด๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›)๎‚๎…ž=๐ด๐‘โˆ’๐‘ โ€ฒ(โ‹…),๐‘žโ€ฒ(โ„๐‘›),(5.1) where ๐‘๎…ž(โ‹…)=๐‘(โ‹…)๐‘(โ‹…)โˆ’1,(5.2) and 1/๐‘ž+1/๐‘ž๎…ž=1. The spaces ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) and ๐ต๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›) are reflexive.

Proof. We will only prove (5.1) because the reflexivity of ๐ด๐‘ (โ‹…)๐‘(โ‹…),๐‘ž(โ‹…) follows from Theorem 3.1.
Step 1. We will first prove the ๐น๐‘ ๐‘(โ‹…),๐‘ž case. We use the arguments of the proof of [19, Theoremsโ€‰โ€‰7.1.7,โ€‰andโ€‰7.2.2]. Let ๐‘“โˆˆ๐’ฎ(โ„๐‘›), ๐‘”โˆˆ๐น๐‘โˆ’๐‘ โ€ฒ(โ‹…),๐‘žโ€ฒ(โ„๐‘›), and {๐œ“๐‘˜}โˆž๐‘˜=0 replaced by {๐œƒ๐‘˜}โˆž๐‘˜=0 as in Definition 2.1. Then, โˆ‘๐‘“=โˆž๐‘˜=0๐‘“โˆ—๐œ“๐‘˜ in ๐’ฎ(โ„๐‘›), ๐‘“โˆ—๐œ“๐‘˜โˆˆ๐’ฎ(โ„๐‘›), and โˆ‘๐‘”=โˆž๐‘˜=0๐‘”โˆ—๐œ“๐‘˜ in ๐’ฎ๎…ž(โ„๐‘›). It follows from Definition 2.1 that ๐‘”(๐‘“)=โˆž๎“๐‘˜=0๎€ท๐‘”โˆ—๐œ“๐‘˜๎€ธ(๐‘“)=โˆž๎“๐‘˜=0๎ƒฉโˆž๎“๐‘™=0๎€ท๐‘”โˆ—๐œ“๐‘˜๎€ธ๎€ท๐‘“โˆ—๐œ“๐‘™๎€ธ๎ƒช=โˆž๎“๐‘˜=0๎ƒฉ2๐‘๎“๐‘Ÿ=โˆ’2๐‘๎€ท๐‘”โˆ—๐œ“๐‘˜๎€ธ๎€ท๐‘“โˆ—๐œ“๐‘˜+๐‘Ÿ๎€ธ๎ƒช(5.3) in ๐’ฎ๎…ž(โ„๐‘›). Here, ๐œ“๐‘—=0 for ๐‘—<0. Using Hรถlder inequality, we get ||||๎€œ๐‘”(๐‘“)โ‰ค๐‘โ„๐‘›โ€–โ€–๎€ฝ2โˆ’๐‘˜๐‘ ๐‘”โˆ—๐œ“๐‘˜๎€พโ€–โ€–โ„“๐‘žโ€ฒโ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘“โˆ—๐œ“๐‘˜๎€พโ€–โ€–โ„“๐‘žโ€–โ€–๎€ฝ2d๐‘ฅโ‰ค๐‘โˆ’๐‘˜๐‘ ๐‘”โˆ—๐œ“๐‘˜๎€พโ€–โ€–๐ฟ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ)โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘“โˆ—๐œ“๐‘˜๎€พโ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค๐‘๎…žโ€–๐‘”โ€–๐น๐‘โ€ฒโˆ’๐‘ (โ‹…),๐‘žโ€ฒโ€–๐‘“โ€–๐น๐‘ ๐‘(โ‹…),๐‘ž,(5.4) where ๐‘๎…ž does not depend on ๐‘”, ๐‘“ and ๐œ“. Hence, we see ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž)๎…ž and โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)๎…žโ‰ค๐‘โ€–๐‘”โ€–๐นโˆ’๐‘ ๐‘โ€ฒ(โ‹…),๐‘žโ€ฒ since ๐’ฎ(โ„๐‘›) is dense in ๐น๐‘ ๐‘(โ‹…),๐‘ž(โ„๐‘›).Step 2. Let ๐‘”โˆˆ(๐น๐‘ ๐‘(โ‹…),๐‘ž)๎…ž. We assume that ฬƒ๐‘” and the functions ๐œ‘๐‘˜ and ๐œŒ๐‘˜ have the same sense as in Definition 4.7, Lemmas 4.9, and 4.10. We show that ฬƒ๐‘”โˆˆ๐นโˆ’๐‘ ๐‘๎…ž(โ‹…),๐‘ž๎…ž(โ„๐‘›). For this purpose, we construct functions ๐‘Ž๐‘˜ as (4.37) and set ๐‘๐‘˜(๐‘ฅ)=sgn๐‘Ž๐‘˜โ‹…||๐‘Ž๐‘˜||(๐‘ฅ)๐‘žโ€ฒโˆ’12โˆ’๐‘ ๐‘˜โ€ฒ๐‘žโ€ฒโ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“๐‘žโ€ฒ,(5.5) for ๐‘˜=0,1,โ€ฆ,๐‘. If ๐‘Ž๐‘˜=0, we set ๐‘๐‘˜=0. We have โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘๐‘˜๎€พ(๐‘ฅ)๐‘๐‘˜=0โ€–โ€–โ„“๐‘ž=๎ƒฏ๐‘๎“๐‘˜=0๎‚ต||๐‘Ž๐‘˜||๐‘žโ€ฒโˆ’12๐‘˜๐‘ (1โˆ’๐‘žโ€ฒ)โ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“๐‘žโ€ฒ๎‚ถ๐‘ž๎ƒฐ1/๐‘ž=โ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“๐‘žโ€ฒ๎ƒฏ๐‘๎“๐‘˜=0||๐‘Ž๐‘˜||๐‘žโ€ฒ2โˆ’๐‘˜๐‘ ๐‘žโ€ฒ๎ƒฐ1/๐‘ž=โ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–๐‘žโ€ฒ/๐‘ž+๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“๐‘žโ€ฒ,(5.6) where we used (๐‘ž๎…žโˆ’1)๐‘ž=๐‘ž๎…ž. We set โŽงโŽชโŽจโŽชโŽฉ๐›ผ=essinf๐‘ฅโˆˆโ„๐‘›๐‘๎…ž(๐‘ฅ)โ€–โ€–๎€ฝ2๐‘(๐‘ฅ),ifโˆ’๐‘˜๐‘ ๐‘Ž๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐ฟ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ)โ‰ค1,esssup๐‘ฅโˆˆโ„๐‘›๐‘๎…ž(๐‘ฅ)โ€–โ€–๎€ฝ2๐‘(๐‘ฅ),ifโˆ’๐‘˜๐‘ ๐‘Ž๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐ฟ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ)>1,(5.7) and ๐œ†=โ€–{2โˆ’๐‘˜๐‘ ๐‘Ž๐‘˜}๐‘๐‘˜=0โ€–๐ฟ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ). Then, ๎€œโ„๐‘›โŽ›โŽœโŽœโŽโ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–โ„“๐‘ž๐œ†๐›ผโŽžโŽŸโŽŸโŽ ๐‘(๐‘ฅ)๎€œd๐‘ฅ=โ„๐‘›โŽ›โŽœโŽœโŽœโŽโ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–๐‘žโ€ฒ/๐‘ž+๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“๐‘žโ€ฒ๐œ†๐›ผโŽžโŽŸโŽŸโŽŸโŽ ๐‘(๐‘ฅ)=๎€œd๐‘ฅโ„๐‘›โ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–(๐‘žโ€ฒ/๐‘ž+๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“)๐‘(๐‘ฅ)๐‘žโ€ฒ๐œ†๐›ผ๐‘(๐‘ฅ)=๎€œd๐‘ฅโ„๐‘›๎ƒฉโ€–โ€–{2โˆ’๐‘—๐‘ ๐‘Žj}๐‘๐‘—=0โ€–โ€–โ„“๐‘žโ€ฒ๐œ†๐›ผ(๐‘(๐‘ฅ)/๐‘โ€ฒ(๐‘ฅ))๎ƒช๐‘โ€ฒ(๐‘ฅ)โ‰ค๎€œd๐‘ฅโ„๐‘›โŽ›โŽœโŽœโŽโ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–โ„“๐‘žโ€ฒ๐œ†โŽžโŽŸโŽŸโŽ ๐‘โ€ฒ(๐‘ฅ)d๐‘ฅโ‰ค1,(5.8) where we used ๐‘ž๎…ž๐‘ž๎€ท๐‘๐‘(๐‘ฅ)+๐‘(๐‘ฅ)๎…ž(๐‘ฅ)โˆ’๐‘ž๎…ž๎€ธ=๐‘๎…ž(๐‘ฅ),esssup๐‘ฅโˆˆโ„๐‘›๐‘๎…ž(๐‘ฅ)๐‘=๎‚ต(๐‘ฅ)essinf๐‘ฅโˆˆโ„๐‘›๐‘(๐‘ฅ)๐‘๎…ž๎‚ถ(๐‘ฅ)โˆ’1,essinf๐‘ฅโˆˆโ„๐‘›๐‘๎…ž(๐‘ฅ)=๎‚ต๐‘(๐‘ฅ)esssup๐‘ฅโˆˆโ„๐‘›๐‘(๐‘ฅ)๐‘๎…ž๎‚ถ(๐‘ฅ)โˆ’1.(5.9) Hence, we have โ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰คโ€–โ€–๎€ฝ2โˆ’๐‘˜๐‘ ๐‘Ž๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐›ผ๐ฟ๐‘โ€ฒ(โ‹…)๎‚€โ„“๐‘žโ€ฒ๎‚.(5.10) In particular, ๐‘๐‘˜โˆˆ๐ฟ๐‘(โ‹…)(โ„๐‘›). Now, we construct the same function ๐‘“ by (4.39). Using Lemma 4.10, (4.37) and Theorem 4.8, we have ๎€œโ„๐‘›โ€–โ€–๎€ฝ2โˆ’๐‘˜๐‘ ๐‘Ž๐‘˜๎€พ๐‘0โ€–โ€–๐‘โ€ฒโ„“(๐‘ฅ)๐‘žโ€ฒ๎€œd๐‘ฅ=โ„๐‘›๐‘๎“๐‘˜=02โˆ’๐‘ ๐‘˜๐‘žโ€ฒ||๐‘Ž๐‘˜||๐‘žโ€ฒโ€–โ€–๎€ฝ2โˆ’๐‘—๐‘ ๐‘Ž๐‘—๎€พ๐‘๐‘—=0โ€–โ€–๐‘โ€ฒ(๐‘ฅ)โˆ’๐‘žโ€ฒโ„“๐‘žโ€ฒ=๎€œd๐‘ฅโ„๐‘›๐‘๎“๐‘˜=0๐‘Ž๐‘˜๐‘๐‘˜d๐‘ฅ=โŸจฬƒ๐‘”,๐‘“โŸฉโ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘๐‘˜โˆ—๐œ‘๐‘˜๎€พ๐‘0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž)โ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒโ€–โ€–๎€ฝ2๐‘˜๐‘ ๐‘๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐ฟ๐‘(โ‹…)(โ„“๐‘ž).(5.11) We set โŽงโŽชโŽจโŽชโŽฉ๐›ฝ=esssup๐‘ฅโˆˆโ„๐‘›๐‘๎…žโ€–โ€–๎€ฝ2(๐‘ฅ),ifโˆ’๐‘˜๐‘ ๐‘Ž๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐ฟ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ)โ‰ค1,essinf๐‘ฅโˆˆโ„๐‘›๐‘๎…žโ€–โ€–(๐‘ฅ),if{2โˆ’๐‘˜๐‘ ๐‘Ž๐‘˜}๐‘๐‘˜=0โ€–โ€–๐ฟ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ)>1.(5.12) Then, it follows from Theorem 3.4, (5.11), and (5.10) that โ€–โ€–๎€ฝ2โˆ’๐‘˜๐‘ ๐‘Ž๐‘˜๎€พ๐‘๐‘˜=0โ€–โ€–๐ฟ๐›ฝโˆ’๐›ผ๐‘โ€ฒ(โ‹…)(โ„“๐‘žโ€ฒ)โ‰ค๐‘โ€–๐‘”โ€–(๐น๐‘ ๐‘(โ‹…),๐‘ž)โ€ฒ.(5.13) If ๐‘๎…ž(๐‘ฅ)