Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 328908 | 24 pages | https://doi.org/10.1155/2012/328908

Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences

Academic Editor: Hans Triebel
Received10 Dec 2010
Accepted10 Feb 2011
Published12 Jan 2012

Abstract

We present characterizations of the Besov-type spaces ๐ต๐‘ ,๐œ๐‘,๐‘ž and the Triebel-Lizorkin-type spaces ๐น๐‘ ,๐œ๐‘,๐‘ž by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking ๐œ=0.

1. Introduction

The ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces and ๐น๐‘ ,๐œ๐‘,๐‘ž spaces have been studied extensively in recent years. When ๐œ=0 they coincide with the usual function spaces ๐ต๐‘ ๐‘,๐‘ž and ๐น๐‘ ๐‘,๐‘ž, respectively, studied in detail by Triebel in [1โ€“3]. When ๐‘ โˆˆโ„, ๐œโˆˆ[0,โˆž) and 1โ‰ค๐‘, ๐‘ž<โˆž the ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces were first introduced by El Baraka in [4, 5]. In these papers, El Baraka investigated embeddings as well as Littlewood-Paley characterizations of Campanato spaces. El Baraka showed that the spaces ๐ต๐‘ ,๐œ๐‘,๐‘ž cover certain Campanato spaces, studied in [6, 7]. Later on, Drihem gave in [8] a characterization for ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces by local means and maximal functions. For a complete treatment of ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces and ๐น๐‘ ,๐œ๐‘,๐‘ž spaces we refer the reader the work of Yuan et al. [9]. Yang and Yuan, in [10โ€“12], have introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces ฬ‡๐ต๐‘ ,๐œ๐‘,๐‘ž and ฬ‡๐น๐‘ ,๐œ๐‘,๐‘ž (๐‘โ‰ โˆž), which generalize the homogeneous Besov-Triebel-Lizorkin spaces ฬ‡๐ต๐‘ ,๐œ๐‘,๐‘ž, ฬ‡๐น๐‘ ,๐œ๐‘,๐‘ž and established the relation between ฬ‡๐น๐‘ ,๐œ๐‘,๐‘ž and ๐‘„๐›ผ spaces. See also [13] for further results.

Our main purpose in this paper is to characterize these function spaces by differences. These results are a generalization of some results given in [17], and [9, Chapter 4, Section 4.3]. All these results generalize the existing classical results on Besov spaces and Triebel-Lizorkin spaces by taking ๐œ=0.

The paper is organized as follows. Section 2.1 collects fundamental notation and concepts and Section 2.2 covers results from the theory of these function spaces. Some necessary tools are given in Section 3. These results are used in Section 4 to obtain the characterization of ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces and ๐น๐‘ ,๐œ๐‘,๐‘ž spaces by differences.

2. Preliminaries

2.1. Notation and Conventions

As usual, โ„๐‘› the ๐‘›-dimensional real Euclidean space, โ„• the collection of all natural numbers, and โ„•0=โ„•โˆช{0}. The letter โ„ค stands for the set of all integer numbers. For a multi-index ๐›ผ=(๐›ผ1,โ€ฆ,๐›ผ๐‘›)โˆˆโ„•๐‘›0, we write |๐›ผ|=๐›ผ1+โ‹ฏ+๐›ผ๐‘› and ๐ท๐›ผ=๐œ•|๐›ผ|/๐œ•๐‘ฅ๐›ผ11โ‹ฏ๐œ•๐‘ฅ๐›ผ๐‘›๐‘›. For ๐‘ฃโˆˆโ„ค,let ๐ต๐‘ฃ be the ball of โ„๐‘› with radius 2โˆ’๐‘ฃ and ๐‘ฃ+=max{๐‘ฃ,0}. The Euclidean scalar product of ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›) and ๐‘ฆ=(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘›) is given by ๐‘ฅโ‹…๐‘ฆ=๐‘ฅ1๐‘ฆ1+โ‹ฏ+๐‘ฅ๐‘›๐‘ฆ๐‘›. We denote by |ฮฉ| the ๐‘›-dimensional Lebesgue measure of ฮฉโŠ†โ„๐‘›. For any measurable subset ฮฉโŠ†โ„๐‘› the Lebesgue space ๐ฟ๐‘(ฮฉ), 0<๐‘โ‰คโˆž consists of all measurable functions for whichโ€–๐‘“โˆฃ๐ฟ๐‘โ€–=๎‚ต๎€œ(ฮฉ)ฮฉ||||๐‘“(๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘<โˆž,0<๐‘<โˆž,โ€–๐‘“โˆฃ๐ฟโˆž(ฮฉ)โ€–=ess-sup๐‘ฅโˆˆฮฉ||||๐‘“(๐‘ฅ)<โˆž.(2.1)

By ๐’ฎ(โ„๐‘›) we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on โ„๐‘› and by ๐’ฎ๎…ž(โ„๐‘›) the dual space of all tempered distributions on โ„๐‘›. We define the Fourier transform of a function ๐‘“โˆˆ๐’ฎ(โ„๐‘›) byโ„ฑ(๐‘“)(๐œ‰)=(2๐œ‹)โˆ’๐‘›/2๎€œโ„๐‘›๐‘’โˆ’๐‘–๐‘ฅโ‹…๐œ‰๐‘“(๐‘ฅ)๐‘‘๐‘ฅ.(2.2) Its inverse is denoted by โ„ฑโˆ’1๐‘“. Both โ„ฑ and โ„ฑโˆ’1 are extended to the dual Schwartz space ๐’ฎ๎…ž(โ„๐‘›) in the usual way.

Let ๐œโˆˆ[0,โˆž) and ๐‘โˆˆ(0,โˆž]. Let ๐ฟ๐‘๐œ(โ„๐‘›) be the collection of functions ๐‘“โˆˆ๐ฟ๐‘loc(โ„๐‘›) such thatโ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–=sup๐ต๐ฝ1||๐ต๐ฝ||๐œ๎‚ต๎€œ๐ต๐ฝ||||๐‘“(๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘<โˆž,(2.3) where the supremum is taken over all ๐ฝโˆˆโ„คโงตโ„• and all balls ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ. Obviously, when ๐œ=0, then ๐ฟ๐‘๐œ(โ„๐‘›)=๐ฟ๐‘(โ„๐‘›). Furthermore,๐ฟ๐‘๐œ(โ„๐‘›)โ†ช๐’ฎ๎…ž(โ„๐‘›),(2.4) (see [9, page 46]).

If ๐‘ โˆˆโ„, 0<๐‘žโ‰คโˆž and ๐ฝโˆˆโ„ค, then โ„“๐‘ ๐‘ž,๐ฝ+ is the set of all sequences {๐‘“๐‘˜}๐‘˜โ‰ฅ๐ฝ+ of complex numbers such thatโ€–โ€–๎€ฝ๐‘“๐‘˜๎€พ๐‘˜โ‰ฅ๐ฝ+โˆฃโ„“๐‘ ๐‘ž,๐ฝ+โ€–โ€–=๎ƒฉ๎“๐‘˜โ‰ฅ๐ฝ+2๐‘˜๐‘ ๐‘ž||๐‘“๐‘˜||๐‘ž๎ƒช1/๐‘ž<โˆž,(2.5) with the obvious modification if ๐‘ž=โˆž. We recall that for any 0<๐œƒโ‰ค1 and any ๐ฝโˆˆโ„ค๎ƒฉ๎“๐‘˜โ‰ฅ๐ฝ+||๐‘“๐‘˜||๎ƒช๐œƒโ‰ค๎“๐‘˜โ‰ฅ๐ฝ+||๐‘“๐‘˜||๐œƒ,(2.6)(๐‘ฆ+๐‘ง)๐‘‘๎€ทโ‰คmax1,2๐‘‘โˆ’1๐‘ฆ๎€ธ๎€ท๐‘‘+๐‘ง๐‘‘๎€ธ,๐‘ฆ,๐‘งโ‰ฅ0,๐‘‘>0.(2.7)

Let ๐‘“ be an arbitrary function on โ„๐‘› and ๐‘ฅ, โ„Žโˆˆโ„๐‘›. Thenฮ”โ„Ž๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ),ฮ”โ„Ž๐‘€+1๐‘“(๐‘ฅ)=ฮ”โ„Ž๎€ทฮ”๐‘€โ„Ž๐‘“๎€ธ(๐‘ฅ),๐‘€โˆˆโ„•.(2.8) These are the well-known differences of functions which play an important role in the theory of function spaces. Using mathematical induction one can show the explicit formulaฮ”๐‘€โ„Ž๐‘“(๐‘ฅ)=๐‘€๎“๐‘—=0(โˆ’1)๐‘—๐ถ๐‘€๐‘—๐‘“(๐‘ฅ+(๐‘€โˆ’๐‘—)โ„Ž),(2.9) where ๐ถ๐‘€๐‘— are the binomial coefficients.

Recall that ๐œ‚๐‘—,๐‘(๐‘ฅ)=2๐‘—๐‘›(1+2๐‘—|๐‘ฅ|)โˆ’๐‘, for any ๐‘ฅโˆˆโ„๐‘›, ๐‘—โˆˆโ„•0 and ๐‘>0. By ๐‘ we denote generic positive constants, which may have different values at different occurrences.

2.2. The ๐ต๐‘ ,๐œ๐‘,๐‘ž Spaces and ๐น๐‘ ,๐œ๐‘,๐‘ž Spaces

In this subsection we present the Fourier analytical definition of ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces, ๐น๐‘ ,๐œ๐‘,๐‘ž spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.

Definition 2.1. Let ฮจ be a function in ๐’ฎ(โ„๐‘›) satisfying ฮจ(๐‘ฅ)=1 for |๐‘ฅ|โ‰ค1 and ฮจ(๐‘ฅ)=0 for |๐‘ฅ|โ‰ฅ3/2. We put ๐œ‘0(๐‘ฅ)=ฮจ(๐‘ฅ), ๐œ‘1(๐‘ฅ)=ฮจ(๐‘ฅ/2)โˆ’ฮจ(๐‘ฅ) and ๐œ‘๐‘—(๐‘ฅ)=๐œ‘1๎€ท2โˆ’๐‘—+1๐‘ฅ๎€ธfor๐‘—=2,3,โ€ฆ.(2.10) Then we have supp๐œ‘๐‘—โŠ‚{๐‘ฅโˆˆโ„๐‘›โˆถ2๐‘—โˆ’1โ‰ค|๐‘ฅ|โ‰ค3โ‹…2๐‘—โˆ’1}, ๐œ‘๐‘—(๐‘ฅ)=1 for 3โ‹…2๐‘—โˆ’2โ‰ค|๐‘ฅ|โ‰ค2๐‘— and โˆ‘ฮจ(๐‘ฅ)+๐‘—โ‰ฅ1๐œ‘๐‘—(๐‘ฅ)=1 for all ๐‘ฅโˆˆโ„๐‘›. The system of functions {๐œ‘๐‘—} is called a smooth dyadic resolution of unity. We define the convolution operators ฮ”๐‘— by the following: ฮ”๐‘—๐‘“=โ„ฑโˆ’1๐œ‘๐‘—โˆ—๐‘“,๐‘—โˆˆโ„•,ฮ”0๐‘“=โ„ฑโˆ’1ฮจโˆ—๐‘“,๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›).(2.11) Thus we obtain the Littlewood-Paley decomposition ๎“๐‘“=๐‘—โ‰ฅ0ฮ”๐‘—๐‘“,(2.12) of all ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›)(convergencein๐’ฎ๎…ž(โ„๐‘›)).

The ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces and ๐น๐‘ ,๐œ๐‘,๐‘ž spaces are defined in the following way.

Definition 2.2. (i) Let ๐‘ โˆˆโ„, ๐œโˆˆ[0,โˆž) and 0<๐‘, ๐‘žโ‰คโˆž. The space ๐ต๐‘ ,๐œ๐‘,๐‘ž is the collection of all ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) such that โ€–โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–โ€–=sup๐ต๐ฝ1||๐ต๐ฝ||๐œโŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘žโ€–โ€–ฮ”๐‘—๐‘“โˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž<โˆž,(2.13) where the supremum is taken over all ๐ฝโˆˆโ„ค and all balls ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ.
(ii) Let ๐‘ โˆˆโ„, ๐œโˆˆ[0,โˆž), 0<๐‘<โˆž and 0<๐‘žโ‰คโˆž. The space ๐น๐‘ ,๐œ๐‘,๐‘ž is the collection of all ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) such that โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–=sup๐ต๐ฝ1||๐ต๐ฝ||๐œโ€–โ€–โ€–โ€–โ€–โŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž||ฮ”๐‘—๐‘“||๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–โ€–<โˆž,(2.14) where the supremum is taken over all ๐ฝโˆˆโ„ค and all balls ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ.

Remark 2.3. The spacesโ€‰โ€‰๐ต๐‘ ,๐œ๐‘,๐‘žโ€‰โ€‰andโ€‰โ€‰๐น๐‘ ,๐œ๐‘,๐‘žโ€‰โ€‰are independent of the particular choice of the smooth dyadic resolution of unityโ€‰โ€‰{๐œ‘๐‘—}โ€‰โ€‰appearing in their definitions. They are quasi-Banach spaces (Banach spaces ifโ€‰โ€‰๐‘โ‰ฅ1, ๐‘žโ‰ฅ1).โ€‰โ€‰In particular, ๐ต๐‘ ,0๐‘,๐‘ž=๐ต๐‘ ๐‘,๐‘ž,๐น๐‘ ,0๐‘,๐‘ž=๐น๐‘ ๐‘,๐‘ž,(2.15) where ๐ต๐‘ ๐‘,๐‘ž and ๐น๐‘ ๐‘,๐‘ž are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls ๐ต๐ฝ by dyadic cubes ๐‘ƒ (with side length 2โˆ’๐ฝ) we obtain equivalent norms.

The full treatment of both scales of spaces can be found in [9]. Let ฮ”๐‘—๐‘“ (๐‘—โˆˆโ„•0) be the functions introduced in Definition 2.1. For any ๐‘Ž>0, any ๐‘ฅโˆˆโ„๐‘› and any ๐ฝโˆˆโ„ค we denote (Peetreโ€™s maximal functions)ฮ”โˆ—,๐‘Ž๐‘—,๐ฝ๐‘“(๐‘ฅ)=sup๐‘ฆโˆˆ๐ต๐ฝ||ฮ”๐‘—||๐‘“(๐‘ฆ)๎€ท1+2๐‘—||||๎€ธ๐‘ฅโˆ’๐‘ฆ๐‘Ž๐‘—โˆˆโ„•0,๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›).(2.16)

We now present a fundamental characterization of ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces and ๐น๐‘ ,๐œ๐‘,๐‘ž spaces.

Theorem 2.4. Let ๐‘ โˆˆโ„, ๐œโˆˆ[0,โˆž), 0<๐‘, ๐‘žโ‰คโˆž and ๐‘Ž>๐‘›/๐‘. Then โ€–โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–โ€–โˆ—=sup๐ต๐ฝ1||๐ต๐ฝ||๐œโŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘žโ€–โ€–ฮ”โˆ—,๐‘Ž๐‘—,๐ฝ๐‘“โˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž,(2.17) is an equivalent quasinorm in ๐ต๐‘ ,๐œ๐‘,๐‘ž.

Theorem 2.5. Let ๐‘ โˆˆโ„, ๐œโˆˆ[0,โˆž), 0<๐‘<โˆž, 0<๐‘žโ‰คโˆž and ๐‘Ž>๐‘›/min(๐‘,๐‘ž). Then โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โˆ—=sup๐ต๐ฝ(1/|๐ต๐ฝ|๐œโˆ‘)โ€–(๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž|ฮ”โˆ—,๐‘Ž๐‘—,๐ฝ๐‘“|๐‘ž)1/๐‘žโˆฃ๐ฟ๐‘(๐ต๐ฝ)โ€–, is an equivalent quasinorm in ๐น๐‘ ,๐œ๐‘,๐‘ž.

Remark 2.6. Theorem 2.4 forโ€‰โ€‰0<๐‘, ๐‘ž<โˆžโ€‰โ€‰is given in [8, Theorem 4.5]. For Theorem 2.5 see [12, Theorem 1.1]. In addition ifโ€‰โ€‰๐‘Ž>๐‘›max(1/๐‘,๐œ), then in Theorem 2.4โ€‰โ€‰ฮ”โˆ—,๐‘Ž๐‘—,๐ฝ๐‘“โ€‰โ€‰can be replaced byโ€‰โ€‰ฮ”๐‘—โˆ—,๐‘Ž๐‘“, where ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฅ)=sup๐‘ฆโˆˆโ„๐‘›||ฮ”๐‘—||๐‘“(๐‘ฆ)๎€ท1+2๐‘—||||๎€ธ๐‘ฅโˆ’๐‘ฆ๐‘Ž๐‘—โˆˆโ„•0,๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›).(2.18)

3. Some Technical Lemmas

To prove our results, we need some technical lemmas. The following lemma for ฮ”๐‘—๐‘“, in place of ฮ”๐‘—โˆ—,๐‘Ž๐‘“, is given in [14, pages 87โ€“89] (for the ๐ต๐‘ ,๐œ๐‘,๐‘ž spaces and 1โ‰ค๐‘<โˆž). Further results, can be found in [12, Lemma 2.4].

Lemma 3.1. Let ฮ”๐‘—๐‘“ be as in Definition 2.1 and let ๐‘ โˆˆโ„,โ€‰โ€‰๐‘Ž>0,โ€‰โ€‰๐œโˆˆ[0,โˆž) and 0<๐‘, ๐‘žโ‰คโˆž(0<๐‘<โˆž for the space ๐น๐‘ ,๐œ๐‘,๐‘ž). Then there is a constant ๐‘>0, independent of ๐‘—, such that for any ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›)โ€–โ€–ฮ”๐‘—โˆ—,๐‘Ž๐‘“โ€–โ€–โˆžโ‰ค๐‘2๐‘—(๐‘›/๐‘โˆ’๐‘ โˆ’๐‘›๐œ)โ€–โ€–๐‘“โˆฃ๐ด๐‘ ,๐œ๐‘,๐‘žโ€–โ€–,๐‘—โˆˆโ„•0.(3.1) Here one uses ๐ด๐‘ ,๐œ๐‘,๐‘ž to denote either ๐ต๐‘ ,๐œ๐‘,๐‘ž or ๐น๐‘ ,๐œ๐‘,๐‘ž.

Proof. Let ๐œ“,๐œ“0โˆˆ๐’ฎ(โ„๐‘›) be two functions such that โ„ฑ๐œ“=1 and โ„ฑ๐œ“0=1 on supp๐œ‘1 and supp๐œ‘0 respectively. Then ||ฮ”๐‘—||=||๐œ“๐‘“(๐‘ฆ)๐‘—โˆ—ฮ”๐‘—||๐‘“(๐‘ฆ),๐‘ฆโˆˆโ„๐‘›,(3.2) with ๐œ“๐‘—=2(๐‘—โˆ’1)๐‘›๐œ“(2๐‘—โˆ’1) if ๐‘—โˆˆโ„•. Since ๐œ‘โˆˆ๐’ฎ(โ„๐‘›), the right-hand side is bounded by ๐‘๐œ‚๐‘—โˆ’1,๐‘โˆ—|ฮ”๐‘—๐‘“|(๐‘ฆ), for any ๐‘>0. Hence we get for all ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) and any ๐‘ฅโˆˆโ„๐‘›ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฅ)โ‰ค๐‘sup๐‘ฆโˆˆโ„๐‘›๐œ‚๐‘—โˆ’1,๐‘โˆ—||ฮ”๐‘—๐‘“||(๐‘ฆ).(3.3) Using the same method given in [9,Proposition 2.6] we obtain for any ๐‘ฆโˆˆโ„๐‘›๐œ‚๐‘—โˆ’1,๐‘โˆ—||ฮ”๐‘—๐‘“||(๐‘ฆ)โ‰ค๐‘2๐‘—(๐‘›/๐‘โˆ’๐‘ โˆ’๐‘›๐œ)โ€–โ€–๐‘“โˆฃ๐ด๐‘ ,๐œ๐‘,๐‘žโ€–โ€–.(3.4) The proof is completed.

Lemma 3.2. Let ๐‘€โˆˆโ„•, ๐ฝโˆˆโ„คโงตโ„•, ๐ด>0, ๐œโˆˆ[0,โˆž) and 1โ‰ค๐‘โ‰คโˆž. Then there is a constant ๐‘>0, independent of ๐ฝ and ๐ด, such that โ€–โ€–โ€–๎€œ|โ„Ž|โ‰ค๐ด||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘‘โ„Žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ‰ค๐‘๐ด๐‘›||๐ต๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œโ€–โ€–,(3.5) for any ball ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ and any function ๐‘“ such that โ€–๐‘“โˆฃ๐ฟ๐‘๐œโ€–<โˆž.

Proof. Since 1โ‰ค๐‘โ‰คโˆž, the left-hand side is bounded by ๎€œ|โ„Ž|โ‰ค๐ดโ€–โ€–ฮ”๐‘€โ„Ž๐‘“โˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐‘‘โ„Ž.(3.6) From the definition of ฮ”๐‘€โ„Ž๐‘“ we have ||ฮ”๐‘€โ„Ž||โ‰ค๐‘“(๐‘ฅ)๐‘€๎“๐‘š=0๐ถ๐‘€๐‘š||||๐‘“(๐‘ฅ+(๐‘€โˆ’๐‘š)โ„Ž).(3.7) Take the ๐ฟ๐‘(๐ต๐ฝ)-norm to estimate (3.6) form above by ๐‘€๎“๐‘š=0๐ถ๐‘€๐‘š๎€œ|โ„Ž|โ‰ค๐ดโ€–โ€–๐‘“โˆฃ๐ฟ๐‘๎‚€๎‚๐ต๐ฝ๎‚โ€–โ€–๐‘‘โ„Ž,(3.8) where if ๐‘ฅ0 the centre of ๐ต๐ฝ then ๐‘ฅ0+(๐‘€โˆ’๐‘š)โ„Ž is the centre of ๎‚๐ต๐ฝ. Using the fact that |๎‚๐ต๐ฝ|=|๐ต๐ฝ| to estimate (3.8) from above by ๐‘๐ด๐‘›|๐ต๐ฝ|๐œโ€–๐‘“โˆฃ๐ฟ๐‘๐œโ€–.
The lemma is proved.

Remark 3.3. Let ๐‘€, ๐ด, ๐œ, and ๐‘ be as in Lemma 3.2. Let ๐ฝโˆˆโ„•0. By the embedding ๐ฟ๐‘(๐ต0)โ†ช๐ฟ๐‘(๐ต๐ฝ) there is a constant ๐‘>0, independent of ๐ฝ and ๐ด, such that โ€–โ€–โ€–๎€œ|โ„Ž|โ‰ค๐ด||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘‘โ„Žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ‰ค๐‘๐ด๐‘›โ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œโ€–โ€–,(3.9) for any ball ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ and any function ๐‘“ such that โ€–๐‘“โˆฃ๐ฟ๐‘๐œโ€–<โˆž.
For ๐‘ >0, ๐‘€โˆˆโ„•, ๐œโˆˆ[0,โˆž), 1โ‰ค๐‘<โˆž and 0<๐‘žโ‰คโˆž, we set โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๐‘€=โ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–+sup๐ต๐ฝ1||๐ต๐ฝ||๐œโ€–โ€–โ€–โ€–๎ƒฉ๎€œ2+โˆ’๐ฝ+10๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–.(3.10)

Here the supremum is taken over all ๐ฝโˆˆโ„ค and all balls ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ.

Lemma 3.4. Let ๐‘ >0, ๐‘€โˆˆโ„•, ๐ฝโˆˆโ„ค, ๐œโˆˆ[0,โˆž), 1โ‰ค๐‘<โˆž and 0<๐‘žโ‰คโˆž. Then there is a constant ๐‘>0, independent of ๐ฝ, such that โ€–โ€–โ€–โ€–โ€–โŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž๎‚ต๎€œ|๐‘ฃ|โ‰ค1||ฮ”๐‘€2โˆ’๐‘—๐‘ฃ||๎‚ถ๐‘“(โ‹…)๐‘‘๐‘ฃ๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–โ€–||๐ตโ‰ค๐‘๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๐‘€,โ€–โ€–โ€–โ€–โ€–โŽ›โŽœโŽœโŽ๎“(3.11)๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž๎‚ต๎€œ|๐‘ฃ|>1||ฮ”๐‘€2โˆ’๐‘—๐‘ฃ||๎‚ถ๐‘“(โ‹…)๐œ”(๐‘ฃ)๐‘‘๐‘ฃ๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–โ€–||๐ตโ‰ค๐‘๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๐‘€,(3.12) for any ball ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ, any ๐œ”โˆˆ๐’ฎ(โ„๐‘›) and any function ๐‘“ such that โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–๐‘€<โˆž.

Proof. Let ๐‘ƒ be a dyadic cube with side length 2โˆ’๐ฝ. This result, for ๐‘ƒ in place of ๐ต๐ฝ, is already known, see [9, Lemmas 4.3, 4.4]. By simple modifications of their arguments we will give another proof of (3.12). The proof is given only when 0<๐‘ž<โˆž. The case ๐‘ž=โˆž is similar.
Before proving this result we note that for any ๐‘ฅโˆˆโ„๐‘› and any ๐‘–โˆˆโ„ค๎ƒฉ๎“๐‘ฃโ‰ฅ๐‘–2(๐‘ +๐‘›)๐‘ฃ๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค2โˆ’๐‘ฃ||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘ž๎ƒช1/๐‘ž๎ƒฉ๎€œโ‰ค๐‘2โˆ’๐‘–+10๐‘กโˆ’(๐‘ +๐‘›)๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘ž.(3.13) Here we will prove that the left-hand side of (3.12) is bounded by โ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–+sup๐ต๐ฝ1||๐ต๐ฝ||๐œโ€–โ€–โ€–โ€–๎ƒฉ๎€œ2+โˆ’๐ฝ0๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–=โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€.(3.14) Obviously, โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–๎…ž๐‘€โ‰คโ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–๐‘€. We write 2๐‘—๐‘ ๎€œ|๐‘ฃ|>1||ฮ”๐‘€2โˆ’๐‘—๐‘ฃ||๐‘“(๐‘ฅ)๐œ”(๐‘ฃ)๐‘‘๐‘ฃ=โˆž๎“๐‘˜=02๐‘—๐‘ ๎€œ2๐‘˜<|๐‘ฃ|โ‰ค2๐‘˜+1||ฮ”๐‘€2โˆ’๐‘—๐‘ฃ||๐‘“(๐‘ฅ)๐œ”(๐‘ฃ)๐‘‘๐‘ฃโ‰ค๐‘โˆž๎“๐‘˜=02(๐‘ +๐‘›)๐‘—โˆ’๐‘๐‘˜๎€œ2๐‘˜โˆ’๐‘—<|โ„Ž|โ‰ค2๐‘˜โˆ’๐‘—+1||ฮ”๐‘€โ„Ž๐‘“||(๐‘ฅ)๐‘‘โ„Ž,(3.15) where ๐‘>0 is at our disposal and we have used the properties of the function ๐œ”, ||||๐œ”(๐‘ฅ)โ‰ค๐‘(1+|๐‘ฅ|)โˆ’๐‘,(3.16) for any ๐‘ฅโˆˆโ„๐‘› and any ๐‘>0. Now the right-hand side of (3.15) in โ„“๐‘ ๐‘ž,๐ฝ+-norm is bounded by (with ๐œŽ=min(1,๐‘ž)) ๐‘โŽ›โŽœโŽœโŽโˆž๎“๐‘˜=02โˆ’๐‘๐œŽ๐‘˜โŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2(๐‘ +๐‘›)๐‘—๐‘ž๎‚ต๎€œ2๐‘˜โˆ’๐‘—<|โ„Ž|โ‰ค2๐‘˜โˆ’๐‘—+1||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘žโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘žโŽžโŽŸโŽŸโŽ 1/๐œŽ=โŽ›โŽœโŽœโŽ๐ฝ+โˆ’1๎“๐‘˜=0๎“โ‹ฏ+๐‘˜โ‰ฅ๐ฝ+โ‹ฏโŽžโŽŸโŽŸโŽ 1/๐œŽ=๎€ท๐ผ๐ฝ(๐‘ฅ)+๐ผ๐ผ๐ฝ๎€ธ(๐‘ฅ)1/๐œŽโ‰ค21/๐œŽโˆ’1๎‚€๎€ท๐ผ๐ฝ๎€ธ(๐‘ฅ)1/๐œŽ+๎€ท๐ผ๐ผ๐ฝ๎€ธ(๐‘ฅ)1/๐œŽ๎‚,(3.17) by (2.7). Here we put โˆ‘๐ฝ+โˆ’1๐‘˜=0โ‹ฏ=0 if ๐ฝ+=0. Take the ๐ฟ๐‘(๐ต๐ฝ)-norm we obtain that the left-hand side of (3.12) is bounded by 21/๐œŽโˆ’1โ€–โ€–๎€ท๐ผ๐ฝ๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–+21/๐œŽโˆ’1โ€–โ€–๎€ท๐ผ๐ผ๐ฝ๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–.(3.18) Let us estimate (๐ผ๐ฝ)1/๐œŽ in ๐ฟ๐‘(๐ต๐ฝ)-norm. After a change of variable ๐‘—โˆ’๐‘˜โˆ’1=๐‘ฃ, we get for any ๐‘ฅโˆˆโ„๐‘› (here ๐ฝ+=๐ฝ and ๐‘˜<๐ฝ) ๐ผ๐ฝ(๐‘ฅ)โ‰ค๐‘๐ฝ+โˆ’1๎“๐‘˜=02(๐‘ +๐‘›โˆ’๐‘)๐œŽ๐‘˜๎ƒฉ๎“๐‘ฃโ‰ฅ๐ฝ+โˆ’๐‘˜โˆ’12(๐‘ +๐‘›)๐‘ฃ๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค2โˆ’๐‘ฃ||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘ž๎ƒช๐œŽ/๐‘ž=๐ฝ+โˆ’2๎“๐‘˜=0โ‹ฏ+๐ฝ+โˆ’1๎“๐‘˜=๐ฝ+โˆ’1โ‹ฏ=๐‘€1(๐‘ฅ)+๐‘€2(๐‘ฅ).(3.19) Here we put โˆ‘๐ฝ+โˆ’2๐‘˜=0โ‹ฏ=0 if ๐ฝ+โ‰ค1. We have ๐‘€1(๐‘ฅ)โ‰ค๐‘๐ฝ+โˆ’2๎“๐‘˜=02(๐‘ +๐‘›โˆ’๐‘)๐œŽ๐‘˜๎ƒฉ๎€œ2โˆ’๐ฝ+๐‘˜+20๐‘กโˆ’(๐‘ +๐‘›)๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช๐œŽ/๐‘ž.(3.20) Since ๐ฟ๐‘/๐œŽ(๐ต๐ฝ) is a normed spaces and ๐ต๐ฝโŠ‚๐ต๐ฝโˆ’๐‘˜โˆ’2, the right-hand side in ๐ฟ๐‘/๐œŽ(๐ต๐ฝ)-norm can be estimated from above by ๐‘๐ฝ+โˆ’2๎“๐‘˜=02(๐‘ +๐‘›โˆ’๐‘)๐œŽ๐‘˜โ€–โ€–โ€–โ€–๎ƒฉ๎€œ2โˆ’๐ฝ+๐‘˜+20๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝโˆ’๐‘˜โˆ’2๎€ธโ€–โ€–โ€–โ€–๐œŽ||๐ตโ‰ค๐‘๐ฝ||๐ฝ๐œ๐œŽ+โˆ’2๎“๐‘˜=02(๐‘ +๐‘›โˆ’๐‘+๐‘›๐œ)๐œŽ๐‘˜๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ.(3.21) We choose ๐‘>2(๐‘ +๐‘›)+๐œ๐‘›. This yields that the last expression is bounded by ๐‘||๐ต๐ฝ||๐œ๐œŽ๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ,(3.22) where ๐‘>0 is independent of ๐ฝ. Now (๐‘€2)1/๐œŽ in ๐ฟ๐‘(๐ต๐ฝ)-norm is bounded by ๐‘2(๐‘ +๐‘›โˆ’๐‘)๐ฝ+โ€–โ€–โ€–โ€–๎ƒฉ๎“๐‘ฃโ‰ฅ02(๐‘ +๐‘›)๐‘ฃ๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค2โˆ’๐‘ฃ||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(โ‹…)๐‘‘โ„Ž๐‘ž๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–โ‰ค๐‘2(๐‘ +๐‘›โˆ’๐‘)๐ฝ+โ€–โ€–โ€–โ€–๎ƒฉ๎“๐‘ฃโ‰ฅ12(๐‘ +๐‘›)๐‘ฃ๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค2โˆ’๐‘ฃ||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(โ‹…)๐‘‘โ„Ž๐‘ž๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–+๐‘2(๐‘ +๐‘›โˆ’๐‘)๐ฝ+โ€–โ€–โ€–๎€œ|โ„Ž|โ‰ค1||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘‘โ„Žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–,(3.23) where we have used (2.7). Using the embedding ๐ฟ๐‘(๐ต0)โ†ช๐ฟ๐‘(๐ต๐ฝ) and Remark 3.3, we obtain โ€–โ€–๐‘€2โˆฃ๐ฟ๐‘/๐œŽ๎€ท๐ต๐ฝ๎€ธโ€–โ€–=โ€–โ€–๎€ท๐‘€2๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐œŽโ‰ค๐‘2(๐‘ +๐‘›โˆ’๐‘)๐œŽ๐ฝ+๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ=๐‘2(๐‘ +๐‘›โˆ’๐‘+๐‘›๐œ)๐œŽ๐ฝ+||๐ต๐ฝ||๐œ๐œŽ๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ||๐ตโ‰ค๐‘๐ฝ||๐œ๐œŽ๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ,(3.24) because of ๐‘>๐‘ +๐‘›+๐‘›๐œ. Therefore, โ€–โ€–๎€ท๐ผ๐ฝ๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐œŽ=โ€–โ€–๐ผ๐ฝโˆฃ๐ฟ๐‘/๐œŽ๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘€1โˆฃ๐ฟ๐‘/๐œŽ๎€ท๐ต๐ฝ๎€ธโ€–โ€–+โ€–โ€–๐‘€2โˆฃ๐ฟ๐‘/๐œŽ๎€ท๐ต๐ฝ๎€ธโ€–โ€–||๐ตโ‰ค๐‘๐ฝ||๐œ๐œŽ๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ.(3.25) Now let us estimate (๐ผ๐ผ๐ฝ)1/๐œŽ in ๐ฟ๐‘(๐ต๐ฝ)-norm. We write ๐ผ๐ผ๐ฝ(๎“โ‹…)=๐‘๐‘˜โ‰ฅ๐ฝ+2โˆ’๐‘๐œŽ๐‘˜โŽ›โŽœโŽœโŽ๐‘˜+๐ฝ++1๎“๐‘—=๐ฝ+โ‹ฏ+โˆž๎“๐‘—=๐‘˜+๐ฝ++2โ‹ฏโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘ž๎“โ‰ค๐‘๐‘˜โ‰ฅ๐ฝ+2โˆ’๐‘๐œŽ๐‘˜โŽ›โŽœโŽœโŽโŽ›โŽœโŽœโŽ๐‘˜+๐ฝ++1๎“๐‘—=๐ฝ+โ‹ฏโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘ž+โŽ›โŽœโŽœโŽโˆž๎“๐‘—=๐‘˜+๐ฝ++2โ‹ฏโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘žโŽžโŽŸโŽŸโŽ ๎“=๐‘๐‘˜โ‰ฅ๐ฝ+2โˆ’๐‘๐œŽ๐‘˜๎€ท๐‘†1๐‘˜(โ‹…)+๐‘†2๐‘˜๎€ธ.(โ‹…)(3.26) After a change of variable ๐‘—โˆ’๐‘˜โˆ’1=๐‘ฃ, we get ๎€ท๐‘†1๐‘˜(๎€ธ๐‘ฅ)1/๐œŽโ‰ค๐‘2(๐‘ +๐‘›)๐‘˜โŽ›โŽœโŽœโŽ๐ฝ+๎“๐‘ฃ=๐ฝ+โˆ’๐‘˜โˆ’12(๐‘ +๐‘›)๐‘ฃ๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค2โˆ’๐‘ฃ||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘žโ‰ค๐‘2(๐‘ +๐‘›)๐‘˜๎ƒฉ๎€œ2+๐‘˜โˆ’๐ฝ+22+โˆ’๐ฝ๐‘กโˆ’(๐‘ +๐‘›)๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘žโ‰ค๐‘2(๐‘ +๐‘›)๐‘˜๎€œ|โ„Ž|โ‰ค2+๐‘˜โˆ’๐ฝ+2||ฮ”๐‘€โ„Ž๐‘“||๎ƒฉ๎€œ(๐‘ฅ)๐‘‘โ„Ž2+๐‘˜โˆ’๐ฝ+22+โˆ’๐ฝ๐‘กโˆ’(๐‘ +๐‘›)๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘žโ‰ค๐‘2(๐‘ +๐‘›)๐ฝ++(๐‘ +๐‘›)๐‘˜๎€œ|โ„Ž|โ‰ค2+๐‘˜โˆ’๐ฝ+2||ฮ”๐‘€โ„Ž||๐‘“(๐‘ฅ)๐‘‘โ„Ž.(3.27) Therefore there exists a constant ๐‘>0 independent of ๐ฝ and ๐‘˜ such that โ€–โ€–๎€ท๐‘†1๐‘˜๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ‰ค๐‘2(๐‘ +๐‘›)๐ฝ++(๐‘ +๐‘›)๐‘˜โ€–โ€–โ€–๎€œ|โ„Ž|โ‰ค2+๐‘˜โˆ’๐ฝ+2ฮ”๐‘€โ„Ž๐‘“(โ‹…)๐‘‘โ„Žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ‰ค๐‘2(๐‘ +๐œ๐‘›)๐ฝ++(๐‘ +2๐‘›)๐‘˜||๐ต๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œโ€–โ€–โ‰ค๐‘2(2๐‘ +2๐‘›+๐œ๐‘›)๐‘˜||๐ต๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€,(3.28) by Lemma 3.2 (combined with Remark 3.3) and the fact that ๐‘˜โ‰ฅ๐ฝ+. Now let us estimate (๐‘†2๐‘˜)1/๐œŽin ๐ฟ๐‘(๐ต๐ฝ)-norm. We have ๎€ท๐‘†2๐‘˜๎€ธ(๐‘ฅ)1/๐œŽโ‰ค๐‘2(๐‘›+๐‘ )๐‘˜๎ƒฉโˆž๎“๐‘ฃ=๐ฝ++12(๐‘ +๐‘›)๐‘ฃ๐‘ž๎‚ต๎€œ|โ„Ž|โ‰ค2โˆ’๐‘ฃ||ฮ”๐‘€โ„Ž||๎‚ถ๐‘“(๐‘ฅ)๐‘‘โ„Ž๐‘ž๎ƒช1/๐‘žโ‰ค๐‘2(๐‘›+๐‘ )๐‘˜๎ƒฉ๎€œ2+โˆ’๐ฝ0๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๐‘“(๐‘ฅ)๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘ž.(3.29) Therefore, โ€–โ€–๎€ท๐‘†2๐‘˜๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ‰ค๐‘2(๐‘›+๐‘ )๐‘˜โ€–โ€–โ€–โ€–๎ƒฉ๎€œ2+โˆ’๐ฝ0๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๐‘“(โ‹…)๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–โ‰ค๐‘2(๐‘›+๐‘ )๐‘˜||๐ต๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€.(3.30) Consequently, โ€–โ€–๎€ท๐ผ๐ผ๐ฝ๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐œŽ=โ€–โ€–๐ผ๐ผ๐ฝโˆฃ๐ฟ๐‘/๐œŽ๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ‰ค๎“๐‘˜โ‰ฅ๐ฝ+2โˆ’๐‘๐œŽ๐‘˜๎‚€โ€–โ€–๎€ท๐‘†1๐‘˜๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐œŽ+โ€–โ€–๎€ท๐‘†2๐‘˜๎€ธ1/๐œŽโˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐œŽ๎‚๎“โ‰ค๐‘๐‘˜โ‰ฅ๐ฝ+2(2๐‘ +2๐‘›+๐œ๐‘›โˆ’๐‘)๐œŽ๐‘˜||๐ต๐ฝ||๐œ๐œŽ๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ||๐ตโ‰ค๐‘๐ฝ||๐œ๐œŽ๎€ทโ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๎…ž๐‘€๎€ธ๐œŽ,(3.31) since ๐‘>2(๐‘ +๐‘›)+๐œ๐‘›. This finishes the proof of Lemma 3.4.

For ๐‘ >0, ๐‘€โˆˆโ„•, ๐œโˆˆ[0,โˆž), 1โ‰ค๐‘โ‰คโˆž, and 0<๐‘žโ‰คโˆž, we setโ€–โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๐‘€=โ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–+sup๐ต๐ฝ1||๐ต๐ฝ||๐œ๎ƒฉ๎€œ2+โˆ’๐ฝ+10๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘กโ€–โ€–ฮ”๐‘€โ„Ž๐‘“โˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๐‘ž๐‘‘๐‘ก๐‘ก๎ƒช1/๐‘ž.(3.32) Here the supremum is taken over all ๐ฝโˆˆโ„ค and all balls ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ. Similar arguments yield.

Lemma 3.5. Let ๐‘ >0, ๐‘€โˆˆโ„•, ๐ฝโˆˆโ„ค, ๐œโˆˆ[0,โˆž), 1โ‰ค๐‘โ‰คโˆž and 0<๐‘žโ‰คโˆž. Then there is a constant ๐‘>0, independent of ๐ฝ, such that โŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž๎‚ต๎€œ|๐‘ฃ|โ‰ค1โ€–โ€–ฮ”๐‘€2โˆ’๐‘—๐‘ฃ๐‘“โˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–๎‚ถ๐‘‘๐‘ฃ๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž||๐ตโ‰ค๐‘๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๐‘€,โŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž๎‚ต๎€œ|๐‘ฃ|>1โ€–โ€–ฮ”๐‘€2โˆ’๐‘—๐‘ฃ๐‘“โˆฃ๐ฟ๐‘๎€ท๐ต๐ฝ๎€ธโ€–โ€–||||๎‚ถ๐œ”(๐‘ฃ)๐‘‘๐‘ฃ๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž||๐ตโ‰ค๐‘๐ฝ||๐œโ€–โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–โ€–๐‘€,(3.33) for any ball ๐ต๐ฝ of โ„๐‘› with radius 2โˆ’๐ฝ, any ๐œ”โˆˆ๐’ฎ(โ„๐‘›) and any function ๐‘“ such that โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–๐‘€<โˆž.

Now we recall the following lemma which is useful for us.

Lemma 3.6. Let 0<๐‘Ž<1, ๐ฝโˆˆโ„ค and 0<๐‘žโ‰คโˆž. Let {๐œ€๐‘˜} be a sequences of positive real numbers, such that โ€–โ€–๎€ฝ๐œ€๐‘˜๎€พ๐‘˜โ‰ฅ๐ฝ+โˆฃโ„“0๐‘ž,๐ฝ+โ€–โ€–=๐ผ<โˆž.(3.34) The sequences {๐›ฟ๐‘˜โˆถ๐›ฟ๐‘˜=โˆ‘๐‘˜๐‘—=๐ฝ+๐‘Ž๐‘˜โˆ’๐‘—๐œ€๐‘—}๐‘˜โ‰ฅ๐ฝ+๐‘Ž๐‘›๐‘‘{๐œ‚๐‘˜โˆถ๐œ‚๐‘˜=โˆ‘โˆž๐‘—=๐‘˜๐‘Ž๐‘—โˆ’๐‘˜๐œ€๐‘—}๐‘˜โ‰ฅ๐ฝ+, are in โ„“0๐‘ž,๐ฝ+ with โ€–โ€–๎€ฝ๐›ฟ๐‘˜๎€พ๐‘˜โ‰ฅ๐ฝ+โˆฃโ„“0๐‘ž,๐ฝ+โ€–โ€–+โ€–โ€–๎€ฝ๐œ‚๐‘˜๎€พ๐‘˜โ‰ฅ๐ฝ+โˆฃโ„“0๐‘ž,๐ฝ+โ€–โ€–โ‰ค๐‘๐ผ,(3.35)๐‘depends only on ๐‘Ž and ๐‘ž.

4. Characterizations with Differences

We are able to state the main results of this paper.

Theorem 4.1. Let 1โ‰ค๐‘<โˆž, 0<๐‘žโ‰คโˆž, ๐œโˆˆ[0,โˆž)โ€‰โ€‰andโ€‰โ€‰๐‘€โˆˆโ„•. Assume ๐‘›1min(๐‘,๐‘ž)<๐‘ <๐‘€,0โ‰ค๐œ<๐‘(4.1) or ๐‘›๐‘›min(๐‘,๐‘ž)<๐‘ <๐‘€โˆ’๐‘›๐œ+๐‘1,๐œโ‰ฅ๐‘.(4.2) Then โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–๐‘€ is an equivalent quasinorm in ๐น๐‘ ,๐œ๐‘,๐‘ž.

Theorem 4.2. Let 1โ‰ค๐‘โ‰คโˆž, 0<๐‘žโ‰คโˆž, ๐œโˆˆ[0,โˆž) and ๐‘€โˆˆโ„•. Assume 10<๐‘ <๐‘€,0โ‰ค๐œ<๐‘(4.3) or ๐‘›0<๐‘ <๐‘€โˆ’๐‘›๐œ+๐‘1,๐œโ‰ฅ๐‘.(4.4) Then โ€–๐‘“โˆฃ๐ต๐‘ ,๐œ๐‘,๐‘žโ€–๐‘€ is an equivalent quasinorm in ๐ต๐‘ ,๐œ๐‘,๐‘ž.

Remark 4.3. Theorems 4.1 and 4.2 forโ€‰โ€‰0โ‰ค๐œ<๐‘ /๐‘›+1/๐‘โ€‰โ€‰andโ€‰โ€‰0โ‰ค๐œ<๐‘ /๐‘›+1/๐‘โˆ’max(1/min(๐‘,๐‘ž)โˆ’1,0),โ€‰โ€‰respectively, are given in [9] Theorems 4.6 and 4.7, respectively.

Proof of Theorem 4.1. Let ๐ต๐ฝ be any ball centered at ๐‘ฅ0โˆˆโ„๐‘› and of radius 2โˆ’๐ฝ, ๐ฝโˆˆโ„ค. We will do the proof in three steps. Step 1. We have with ๐‘ >0, ๎“๐‘—โ‰ฅ0||ฮ”๐‘—๐‘“||=๎“๐‘—โ‰ฅ02โˆ’๐‘—๐‘ 2๐‘—๐‘ ||ฮ”๐‘—๐‘“||โ‰ค๐‘sup๐‘—โˆˆโ„•02๐‘—๐‘ ||ฮ”๐‘—๐‘“||๎ƒฉ๎“โ‰ค๐‘๐‘—โ‰ฅ02๐‘—๐‘ ๐‘ž||ฮ”๐‘—๐‘“||๐‘ž๎ƒช1/๐‘ž.(4.5) Let ๐‘“โˆˆ๐น๐‘ ,๐œ๐‘,๐‘ž. Then, โ€–โ€–๐‘“โˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–โ‰คโ€–โ€–โ€–โ€–๎“๐‘—โ‰ฅ0||ฮ”๐‘—๐‘“||โˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๎ƒฉ๎“โ‰ค๐‘๐‘—โ‰ฅ02๐‘—๐‘ ๐‘ž||ฮ”๐‘—๐‘“||๐‘ž๎ƒช1/๐‘žโˆฃ๐ฟ๐‘๐œ(โ„๐‘›)โ€–โ€–โ€–โ€–โ€–โ€–โ‰ค๐‘๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–.(4.6)Step 2. For any ๐‘ฅโˆˆ๐ต๐ฝ we put ๐ป๐ฝ(๎€œ๐‘ฅ)=2+โˆ’๐ฝ+10๐‘กโˆ’๐‘ ๐‘žsup||โ„Ž||โ‰ค๐‘ก||ฮ”๐‘€โ„Ž||๐‘“(๐‘ฅ)๐‘ž๐‘‘๐‘ก๐‘ก.(4.7) After a change of variable ๐‘ก=2โˆ’๐‘ฆ, we get ๐ป๐ฝ๎€œ(๐‘ฅ)=ln2๐ฝ+โˆž+โˆ’12๐‘ฆ๐‘ ๐‘žsup||โ„Ž||โ‰ค2โˆ’๐‘ฆ||ฮ”๐‘€โ„Ž||๐‘“(๐‘ฅ)๐‘ž๐‘‘๐‘ฆ.(4.8) Then ๐ป๐ฝ(๎“๐‘ฅ)โ‰ค๐‘๐‘˜โ‰ฅ๐ฝ+2๐‘˜๐‘ ๐‘žsup||โ„Ž||โ‰ค2โˆ’๐‘˜+1||ฮ”๐‘€โ„Ž||๐‘“(๐‘ฅ)๐‘ž.(4.9)Let ๐œ“,๐œ“0โˆˆ๐’ฎ(โ„๐‘›) be two functions such that โ„ฑ๐œ“=1 and โ„ฑ๐œ“0=1 on supp๐œ‘1 and suppฮจ respectively. Using the mean value theorem we obtain for any ๐‘ฅโˆˆ๐ต๐ฝ, ๐‘—โˆˆโ„•0, and |โ„Ž|โ‰ค2โˆ’๐‘˜+1||ฮ”1โ„Žฮ”๐‘—||=||ฮ”๐‘“(๐‘ฅ)1โ„Ž๎€ท๐œ“๐‘—โˆ—ฮ”๐‘—๐‘“๎€ธ(||๐‘ฅ)โ‰ค2โˆ’๐‘˜sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐‘2โˆ’๐‘˜๎“|๐›ผ|=1||๐ท๐›ผ๐œ“๐‘—โˆ—ฮ”๐‘—||,๐‘“(๐‘ฆ)(4.10) with some positive constant ๐‘, independent of ๐‘— and ๐‘˜, and ๐œ“๐‘—(โ‹…)=2(๐‘—โˆ’1)๐‘›๐œ“(2๐‘—โˆ’1โ‹…) for ๐‘—=1,2,โ€ฆ. By induction on ๐‘€, we show that ||ฮ”๐‘€โ„Žฮ”๐‘—||๐‘“(๐‘ฅ)โ‰ค2โˆ’๐‘˜๐‘€sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐‘2โˆ’๐‘˜๎“|๐›ผ|=๐‘€||๐ท๐›ผ๐œ“๐‘—โˆ—ฮ”๐‘—||๐‘“(๐‘ฆ).(4.11) We see that if |๐›ผ|=๐‘€ and ๐‘Ž>0||๐ท๐›ผ๐œ“๐‘—โˆ—ฮ”๐‘—๐‘“||(๐‘ฆ)=2(๐‘—โˆ’1)๐‘›||||๎€œโ„๐‘›๐ท๐›ผ๎€ท๐œ“๎€ท2๐‘—โˆ’1ฮ”(๐‘ฆโˆ’๐‘ง)๎€ธ๎€ธ๐‘—๐‘“||||(๐‘ง)๐‘‘๐‘งโ‰ค2(๐‘—โˆ’1)(๐‘›+๐‘€)๎€œโ„๐‘›||(๐ท๐›ผ๐œ“)๎€ท2๐‘—โˆ’1๎€ธ||||ฮ”(๐‘ฆโˆ’๐‘ง)๐‘—๐‘“||(๐‘ง)๐‘‘๐‘ง.(4.12) Suppose that 0โ‰ค๐‘—โ‰ค๐ฝ+โˆ’1. The right-hand side in (4.12) may be estimated as follows: ๐‘2๐‘—(๐‘›+๐‘€)ฮ”๐‘—โˆ—,๐‘Ž๎€œ๐‘“(๐‘ฆ)โ„๐‘›||(๐ท๐›ผ๎€ท2๐œ“)๐‘—โˆ’1(๎€ธ||๎€ท๐‘ฆโˆ’๐‘ง)1+2๐‘—||||๎€ธ๐‘ฆโˆ’๐‘ง๐‘Ž๐‘‘๐‘งโ‰ค๐‘2๐‘—๐‘€ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฆ).(4.13) Then we obtain for any ๐‘ฅโˆˆ๐ต๐ฝ, |โ„Ž|โ‰ค2โˆ’๐‘˜+1 and any ๐‘˜โ‰ฅ๐ฝ+||ฮ”๐‘€โ„Žฮ”๐‘—๐‘“||(๐‘ฅ)โ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘€sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐‘2โˆ’๐‘˜ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฆ)โ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘€๎€ท1+2๐‘—โˆ’๐‘˜๎€ธ๐‘Žsup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐‘2โˆ’๐‘˜ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฆ)๎€ท1+2๐‘—||||๎€ธ๐‘ฅโˆ’๐‘ฆ๐‘Žโ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘€ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฅ),(4.14) if 0โ‰ค๐‘—โ‰ค๐ฝ+โˆ’1.
Suppose now that ๐ฝ+โ‰ค๐‘—โ‰ค๐‘˜. By our assumption on ๐‘ฅ and ๐‘˜ we have ||๐‘ฆโˆ’๐‘ฅ0||โ‰ค||๐‘ฅโˆ’๐‘ฅ0||+||||๐‘ฅโˆ’๐‘ฆ<2โˆ’๐ฝ+๐‘2โˆ’๐‘˜โ‰ค(๐‘+1)2โˆ’๐ฝ,(4.15) which implies that ๐‘ฆ is located in some ball ๎‚๐ต๐ฝ, where ๎‚๐ต๐ฝ={๐‘ฆโˆˆโ„๐‘›โˆถ|๐‘ฆโˆ’๐‘ฅ0|<(๐‘+1)2โˆ’๐ฝ}. Writing the integral in (4.12) as follows ๎€œ๎‚๐ต๐ฝโˆ’1๎“โ‹ฏ๐‘‘๐‘ง+๐‘–โ‰ฅ0๎€œ๎‚๐ต๐ฝโˆ’๐‘–โˆ’2โงต๎‚๐ต๐ฝโˆ’๐‘–โˆ’1โ‹ฏ๐‘‘๐‘ง=๐ผ๐‘—,๐ฝ(๎“๐‘ฆ)+๐‘–โ‰ฅ0๐ผ๐ผ๐‘—,๐ฝโˆ’๐‘–(๐‘ฆ).(4.16) We recall that ฮ”โˆ—,๐‘Ž๐‘—,๐‘™๐‘“(๐‘ฆ)=sup๐‘งโˆˆ๎‚๐ต๐‘™||ฮ”๐‘—||๐‘“(๐‘ง)๎€ท1+2๐‘—||||๎€ธ๐‘ฆโˆ’๐‘ง๐‘Ž,(4.17) for any ๐‘—โˆˆโ„•0, ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) and any ๐‘™โˆˆโ„ค. We have ๐ผ๐‘—,๐ฝ(๐‘ฆ)โ‰คฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’1๎€œ๎‚๐ต๐‘“(๐‘ฆ)๐ฝโˆ’1||(๐ท๐›ผ๎€ท2๐œ“)๐‘—โˆ’1(๎€ธ||๎€ท๐‘ฆโˆ’๐‘ง)1+2๐‘—||||๎€ธ๐‘ฆโˆ’๐‘ง๐‘Ž๐‘‘๐‘งโ‰ค๐‘2โˆ’๐‘—๐‘›ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’1๐‘“(๐‘ฆ)โ‰ค๐‘2โˆ’๐‘—๐‘›ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’2๐‘“(๐‘ฆ).(4.18) Let us estimate ๐ผ๐ผ๐‘—,๐ฝโˆ’๐‘–. Since ๐œ“โˆˆ๐’ฎ(โ„๐‘›), we have ||๐ท๐›ผ||๐œ“(๐‘ฅ)โ‰ค๐‘(1+|๐‘ฅ|)โˆ’2๐‘,(4.19) for any ๐‘ฅโˆˆโ„๐‘› and any ๐‘>0. Then for any ๐‘ large enough, ๐ผ๐ผ๐‘—,๐ฝโˆ’๐‘–(๐‘ฆ) does not exceed ๐‘ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๎€œ๎‚๐ต๐‘“(๐‘ฆ)๐ฝโˆ’๐‘–โˆ’2โงต๎‚๐ต๐ฝโˆ’๐‘–โˆ’1๎€ท1+2๐‘—โˆ’1||||๎€ธ๐‘ฆโˆ’๐‘งโˆ’2๐‘+๐‘Ž๐‘‘๐‘งโ‰ค๐‘2โˆ’๐‘–๐‘ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๎€œ๐‘“(๐‘ฆ)โ„๐‘›๎€ท1+2๐‘—โˆ’1||||๎€ธ๐‘ฆโˆ’๐‘งโˆ’๐‘+๐‘Ž๐‘‘๐‘งโ‰ค๐‘2โˆ’๐‘–๐‘โˆ’๐‘—๐‘›ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๐‘“(๐‘ฆ),(4.20) where we have used 2๐‘—โˆ’1|๐‘ฆโˆ’๐‘ง|>(๐‘+1)2๐‘—โˆ’๐ฝ+๐‘–โˆ’1โ‰ฅ(๐‘+1)2๐‘–โˆ’1. Therefore, ||๐ท๐›ผ๐œ“๐‘—โˆ—ฮ”๐‘—||๐‘“(๐‘ฆ)โ‰ค๐‘2๐‘—๐‘€๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๐‘“(๐‘ฆ).(4.21) Then we obtain for any ๐‘ฅโˆˆ๐ต๐ฝ any |โ„Ž|โ‰ค2โˆ’๐‘˜+1 and any ๐ฝ+โ‰ค๐‘—โ‰ค๐‘˜||ฮ”๐‘€โ„Žฮ”๐‘—||๐‘“(๐‘ฅ)โ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘€๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐‘2โˆ’๐‘˜ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๐‘“(๐‘ฆ)โ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘€๎€ท1+2๐‘—โˆ’๐‘˜๎€ธ๐‘Ž๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐‘2โˆ’๐‘˜ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๐‘“(๐‘ฆ)๎€ท1+2๐‘—||||๎€ธ๐‘ฅโˆ’๐‘ฆ๐‘Ž.(4.22) Consequently, for any ๐ฝ+โ‰ค๐‘—โ‰ค๐‘˜ there is a constant ๐‘>0 independent of ๐ฝ, ๐‘—, and ๐‘˜ such that ||ฮ”๐‘€โ„Žฮ”๐‘—||๐‘“(๐‘ฅ)โ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘€๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๐‘“(๐‘ฅ).(4.23) Finally for ๐‘—โ‰ฅ๐‘˜+1 we have for ๐‘ฅโˆˆ๐ต๐ฝ and |โ„Ž|โ‰ค2โˆ’๐‘˜+1||ฮ”๐‘€โ„Žฮ”๐‘—||โ‰ค๐‘“(๐‘ฅ)๐‘€๎“๐‘š=0๐ถ๐‘€๐‘š||ฮ”๐‘—||๐‘“(๐‘ฅ+(๐‘€โˆ’๐‘š)โ„Ž)โ‰ค2๐‘€sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐ถ2โˆ’๐‘˜||ฮ”๐‘—||๐‘“(๐‘ฆ)โ‰ค2๐‘€sup||||๐‘ฅโˆ’๐‘ฆโ‰ค๐ถ2โˆ’๐‘˜||ฮ”๐‘—||๐‘“(๐‘ฆ)๎€ท1+2๐‘—||||๎€ธ๐‘ฅโˆ’๐‘ฆ๐‘Ž๎€ท1+2๐‘—||||๎€ธ๐‘ฅโˆ’๐‘ฆ๐‘Ž.(4.24) We remark also that by our assumption on ๐‘ฅ and ๐‘˜ we have ||๐‘ฆโˆ’๐‘ฅ0||โ‰ค||๐‘ฅโˆ’๐‘ฅ0||+||||๐‘ฅโˆ’๐‘ฆ<2โˆ’๐ฝ+๐ถ2โˆ’๐‘˜โ‰ค(๐ถ+1)2โˆ’๐ฝ,(4.25) and this implies that ๐‘ฆ is located in some ball ๎‚๐ต๐ฝ, where ๎‚๐ต๐ฝ={๐‘ฆโˆˆโ„๐‘›โˆถ|๐‘ฆโˆ’๐‘ฅ0|<(๐ถ+1)2โˆ’๐ฝ}. Then, ||ฮ”๐‘€โ„Žฮ”๐‘—๐‘“||(๐‘ฅ)โ‰ค๐‘2(๐‘—โˆ’๐‘˜)๐‘Žฮ”โˆ—,๐‘Ž๐‘—,๐ฝ๐‘“(๐‘ฅ),(4.26) if ๐‘—โ‰ฅ๐‘˜+1, where ฮ”โˆ—,๐‘Ž๐‘—,๐ฝ๐‘“ is given in (4.17) (with ๎‚๐ต๐ฝ a ball centered at ๐‘ฅ0 and of radius (๐ถ+1)2โˆ’๐ฝ).
We write, ฮ”๐‘€โ„Ž๎“๐‘“(๐‘ฅ)=๐‘—โ‰ฅ0ฮ”๐‘€โ„Žฮ”๐‘—=๐‘“(๐‘ฅ)๐ฝ+โˆ’1๎“๐‘—=0โ‹ฏ+๐‘˜๎“๐‘—=๐ฝ+๎“โ‹ฏ+๐‘—โ‰ฅ๐‘˜+1โ‹ฏ.(4.27) Here we put โˆ‘๐ฝ+โˆ’1๐‘—=0โ‹ฏ=0 if ๐ฝ+=0. Let us estimate each term in โ„“๐‘ ๐‘ž,๐ฝ+-norm. We have by (4.14) and Lemma 3.1๐ฝ+โˆ’1๎“๐‘—=0||ฮ”๐‘€โ„Žฮ”๐‘—||โ‰ค๐‘“(๐‘ฅ)๐ฝ+โˆ’1๎“๐‘—=02(๐‘—โˆ’๐‘˜)๐‘€ฮ”๐‘—โˆ—,๐‘Ž๐‘“(๐‘ฅ)โ‰ค๐‘๐ฝ+โˆ’1๎“๐‘—=02(๐‘—โˆ’๐‘˜)๐‘€+๐‘—(๐‘›/๐‘โˆ’๐‘ โˆ’๐‘›๐œ)โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–โ‰ค๐‘2๐ฝโˆ’๐‘˜๐‘€+โˆ’1๎“๐‘—=02๐‘—(๐‘€+๐‘›/๐‘โˆ’๐‘ โˆ’๐‘›๐œ)โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–โ‰ค๐‘2๐ฝ(๐‘€+๐‘›/๐‘โˆ’๐‘ โˆ’๐‘›๐œ)โˆ’๐‘˜๐‘€โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–,(4.28) where the last inequality can be obtained by our assumption on ๐‘  and ๐œ. The last expression in โ„“๐‘ ๐‘ž,๐ฝ+-norm does not exceed ๐‘2๐ฝ๐‘›(1/๐‘โˆ’๐œ)โ€–โ€–โ€–๎‚†2(๐‘˜โˆ’๐ฝ+)(๐‘ โˆ’๐‘€)๎‚‡๐‘˜โ‰ฅ๐ฝ+โˆฃโ„“0๐‘ž,๐ฝ+โ€–โ€–โ€–โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–โ‰ค๐‘2๐ฝ๐‘›(1/๐‘โˆ’๐œ)โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–,(4.29) since ๐‘ <๐‘€. Therefore, ๎€ท๐ป๐ฝ๎€ธ(๐‘ฅ)1/๐‘žโ‰ค๐‘2๐ฝ๐‘›(1/๐‘โˆ’๐œ)โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐œ๐‘,๐‘žโ€–โ€–โŽ›โŽœโŽœโŽ๎“(4.30)+๐‘๐‘˜โ‰ฅ๐ฝ+โŽ›โŽœโŽœโŽ๎“๐‘˜๐‘–โ‰ฅ0๎“๐‘—=๐ฝ+2(๐‘—โˆ’๐‘˜)(๐‘€โˆ’๐‘ )+๐‘ ๐‘—โˆ’๐‘–๐‘|||ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2|||โŽžโŽŸโŽŸโŽ ๐‘“(๐‘ฅ)๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž๎ƒฉ๎“(4.31)+๐‘๐‘˜โ‰ฅ๐ฝ+๎ƒฉ๎“๐‘—โ‰ฅ๐‘˜2(๐‘—โˆ’๐‘˜)(๐‘Žโˆ’๐‘ )+๐‘ ๐‘—|||ฮ”โˆ—,๐‘Ž๐‘—,๐ฝ|||๎ƒช๐‘“(๐‘ฅ)๐‘ž๎ƒช1/๐‘ž.(4.32) The second term can be estimated by (with ๐œŽ=min(1,๐‘ž)) ๐‘โŽ›โŽœโŽœโŽ๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘๐œŽโŽ›โŽœโŽœโŽ๎“๐‘˜โ‰ฅ๐ฝ+โŽ›โŽœโŽœโŽ๐‘˜๎“๐‘—=๐ฝ+2(๐‘—โˆ’๐‘˜)(๐‘€โˆ’๐‘ )+๐‘ ๐‘—|||ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2|||โŽžโŽŸโŽŸโŽ ๐‘“(๐‘ฅ)๐‘žโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘žโŽžโŽŸโŽŸโŽ 1/๐œŽ.(4.33) Since again ๐‘ <๐‘€, then we can apply Lemma 3.6 to estimate the last expression by ๐‘โŽ›โŽœโŽœโŽ๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘๐œŽโŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž|||ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2|||๐‘“(๐‘ฅ)๐‘žโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘žโŽžโŽŸโŽŸโŽ 1/๐œŽ.(4.34) Since ๐ฟ๐‘/๐œŽ(๐ต๐ฝ) is a normed space, so (4.31) in ๐ฟ๐‘(๐ต๐ฝ)-norm is dominated by ๐‘โŽ›โŽœโŽœโŽ๎“๐‘–โ‰ฅ02โˆ’๐‘–๐‘๐œŽโ€–โ€–โ€–โ€–โ€–โŽ›โŽœโŽœโŽ๎“๐‘—โ‰ฅ๐ฝ+2๐‘—๐‘ ๐‘ž|||ฮ”โˆ—,๐‘Ž๐‘—,๐ฝโˆ’๐‘–โˆ’2๐‘“|||๐‘žโŽžโŽŸโŽŸโŽ ๐œŽ/๐‘žโˆฃ๐ฟ๐‘/๐œŽ๎€ท๐ต๐ฝ๎€ธโ€–โ€–โ€–โ€–โ€–โŽžโŽŸโŽŸโŽ 1/๐œŽโŽ›<