Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

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

Douadi Drihem, "Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences", Journal of Function Spaces, vol. 2012, Article ID 328908, 24 pages, 2012. 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 𝐵𝑠,𝜏𝑝,𝑞 and the Triebel-Lizorkin-type spaces 𝐹𝑠,𝜏𝑝,𝑞 by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking 𝜏=0.

1. Introduction

The 𝐵𝑠,𝜏𝑝,𝑞 spaces and 𝐹𝑠,𝜏𝑝,𝑞 spaces have been studied extensively in recent years. When 𝜏=0 they coincide with the usual function spaces 𝐵𝑠𝑝,𝑞 and 𝐹𝑠𝑝,𝑞, respectively, studied in detail by Triebel in [13]. When 𝑠, 𝜏[0,) and 1𝑝, 𝑞< the 𝐵𝑠,𝜏𝑝,𝑞 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 𝐵𝑠,𝜏𝑝,𝑞 cover certain Campanato spaces, studied in [6, 7]. Later on, Drihem gave in [8] a characterization for 𝐵𝑠,𝜏𝑝,𝑞 spaces by local means and maximal functions. For a complete treatment of 𝐵𝑠,𝜏𝑝,𝑞 spaces and 𝐹𝑠,𝜏𝑝,𝑞 spaces we refer the reader the work of Yuan et al. [9]. Yang and Yuan, in [1012], have introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces ̇𝐵𝑠,𝜏𝑝,𝑞 and ̇𝐹𝑠,𝜏𝑝,𝑞 (𝑝), which generalize the homogeneous Besov-Triebel-Lizorkin spaces ̇𝐵𝑠,𝜏𝑝,𝑞, ̇𝐹𝑠,𝜏𝑝,𝑞 and established the relation between ̇𝐹𝑠,𝜏𝑝,𝑞 and 𝑄𝛼 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 𝜏=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 𝐵𝑠,𝜏𝑝,𝑞 spaces and 𝐹𝑠,𝜏𝑝,𝑞 spaces by differences.

2. Preliminaries

2.1. Notation and Conventions

As usual, 𝑛 the 𝑛-dimensional real Euclidean space, the collection of all natural numbers, and 0={0}. The letter stands for the set of all integer numbers. For a multi-index 𝛼=(𝛼1,,𝛼𝑛)𝑛0, we write |𝛼|=𝛼1++𝛼𝑛 and 𝐷𝛼=𝜕|𝛼|/𝜕𝑥𝛼11𝜕𝑥𝛼𝑛𝑛. For 𝑣,let 𝐵𝑣 be the ball of 𝑛 with radius 2𝑣 and 𝑣+=max{𝑣,0}. The Euclidean scalar product of 𝑥=(𝑥1,,𝑥𝑛) and 𝑦=(𝑦1,,𝑦𝑛) is given by 𝑥𝑦=𝑥1𝑦1++𝑥𝑛𝑦𝑛. We denote by |Ω| the 𝑛-dimensional Lebesgue measure of Ω𝑛. For any measurable subset Ω𝑛 the Lebesgue space 𝐿𝑝(Ω), 0<𝑝 consists of all measurable functions for which𝑓𝐿𝑝=(Ω)Ω||||𝑓(𝑥)𝑝𝑑𝑥1/𝑝<,0<𝑝<,𝑓𝐿(Ω)=ess-sup𝑥Ω||||𝑓(𝑥)<.(2.1)

By 𝒮(𝑛) we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on 𝑛 and by 𝒮(𝑛) the dual space of all tempered distributions on 𝑛. We define the Fourier transform of a function 𝑓𝒮(𝑛) by(𝑓)(𝜉)=(2𝜋)𝑛/2𝑛𝑒𝑖𝑥𝜉𝑓(𝑥)𝑑𝑥.(2.2) Its inverse is denoted by 1𝑓. Both and 1 are extended to the dual Schwartz space 𝒮(𝑛) in the usual way.

Let 𝜏[0,) and 𝑝(0,]. Let 𝐿𝑝𝜏(𝑛) be the collection of functions 𝑓𝐿𝑝loc(𝑛) such that𝑓𝐿𝑝𝜏(𝑛)=sup𝐵𝐽1||𝐵𝐽||𝜏𝐵𝐽||||𝑓(𝑥)𝑝𝑑𝑥1/𝑝<,(2.3) where the supremum is taken over all 𝐽 and all balls 𝐵𝐽 of 𝑛 with radius 2𝐽. Obviously, when 𝜏=0, then 𝐿𝑝𝜏(𝑛)=𝐿𝑝(𝑛). Furthermore,𝐿𝑝𝜏(𝑛)𝒮(𝑛),(2.4) (see [9, page 46]).

If 𝑠, 0<𝑞 and 𝐽, then 𝑠𝑞,𝐽+ is the set of all sequences {𝑓𝑘}𝑘𝐽+ of complex numbers such that𝑓𝑘𝑘𝐽+𝑠𝑞,𝐽+=𝑘𝐽+2𝑘𝑠𝑞||𝑓𝑘||𝑞1/𝑞<,(2.5) with the obvious modification if 𝑞=. We recall that for any 0<𝜃1 and any 𝐽𝑘𝐽+||𝑓𝑘||𝜃𝑘𝐽+||𝑓𝑘||𝜃,(2.6)(𝑦+𝑧)𝑑max1,2𝑑1𝑦𝑑+𝑧𝑑,𝑦,𝑧0,𝑑>0.(2.7)

Let 𝑓 be an arbitrary function on 𝑛 and 𝑥, 𝑛. ThenΔ𝑓(𝑥)=𝑓(𝑥+)𝑓(𝑥),Δ𝑀+1𝑓(𝑥)=ΔΔ𝑀𝑓(𝑥),𝑀.(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 formulaΔ𝑀𝑓(𝑥)=𝑀𝑗=0(1)𝑗𝐶𝑀𝑗𝑓(𝑥+(𝑀𝑗)),(2.9) where 𝐶𝑀𝑗 are the binomial coefficients.

Recall that 𝜂𝑗,𝑁(𝑥)=2𝑗𝑛(1+2𝑗|𝑥|)𝑁, for any 𝑥𝑛, 𝑗0 and 𝑁>0. By 𝑐 we denote generic positive constants, which may have different values at different occurrences.

2.2. The 𝐵𝑠,𝜏𝑝,𝑞 Spaces and 𝐹𝑠,𝜏𝑝,𝑞 Spaces

In this subsection we present the Fourier analytical definition of 𝐵𝑠,𝜏𝑝,𝑞 spaces, 𝐹𝑠,𝜏𝑝,𝑞 spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.

Definition 2.1. Let Ψ be a function in 𝒮(𝑛) satisfying Ψ(𝑥)=1 for |𝑥|1 and Ψ(𝑥)=0 for |𝑥|3/2. We put 𝜑0(𝑥)=Ψ(𝑥), 𝜑1(𝑥)=Ψ(𝑥/2)Ψ(𝑥) and 𝜑𝑗(𝑥)=𝜑12𝑗+1𝑥for𝑗=2,3,.(2.10) Then we have supp𝜑𝑗{𝑥𝑛2𝑗1|𝑥|32𝑗1}, 𝜑𝑗(𝑥)=1 for 32𝑗2|𝑥|2𝑗 and Ψ(𝑥)+𝑗1𝜑𝑗(𝑥)=1 for all 𝑥𝑛. The system of functions {𝜑𝑗} is called a smooth dyadic resolution of unity. We define the convolution operators Δ𝑗 by the following: Δ𝑗𝑓=1𝜑𝑗𝑓,𝑗,Δ0𝑓=1Ψ𝑓,𝑓𝒮(𝑛).(2.11) Thus we obtain the Littlewood-Paley decomposition 𝑓=𝑗0Δ𝑗𝑓,(2.12) of all 𝑓𝒮(𝑛)(convergencein𝒮(𝑛)).

The 𝐵𝑠,𝜏𝑝,𝑞 spaces and 𝐹𝑠,𝜏𝑝,𝑞 spaces are defined in the following way.

Definition 2.2. (i) Let 𝑠, 𝜏[0,) and 0<𝑝, 𝑞. The space 𝐵𝑠,𝜏𝑝,𝑞 is the collection of all 𝑓𝒮(𝑛) such that 𝑓𝐵𝑠,𝜏𝑝,𝑞=sup𝐵𝐽1||𝐵𝐽||𝜏𝑗𝐽+2𝑗𝑠𝑞Δ𝑗𝑓𝐿𝑝𝐵𝐽𝑞1/𝑞<,(2.13) where the supremum is taken over all 𝐽 and all balls 𝐵𝐽 of 𝑛 with radius 2𝐽.
(ii) Let 𝑠, 𝜏[0,), 0<𝑝< and 0<𝑞. The space 𝐹𝑠,𝜏𝑝,𝑞 is the collection of all 𝑓𝒮(𝑛) such that 𝑓𝐹𝑠,𝜏𝑝,𝑞=sup𝐵𝐽1||𝐵𝐽||𝜏𝑗𝐽+2𝑗𝑠𝑞||Δ𝑗𝑓||𝑞1/𝑞𝐿𝑝𝐵𝐽<,(2.14) where the supremum is taken over all 𝐽 and all balls 𝐵𝐽 of 𝑛 with radius 2𝐽.

Remark 2.3. The spaces  𝐵𝑠,𝜏𝑝,𝑞  and  𝐹𝑠,𝜏𝑝,𝑞  are independent of the particular choice of the smooth dyadic resolution of unity  {𝜑𝑗}  appearing in their definitions. They are quasi-Banach spaces (Banach spaces if  𝑝1, 𝑞1).  In particular, 𝐵𝑠,0𝑝,𝑞=𝐵𝑠𝑝,𝑞,𝐹𝑠,0𝑝,𝑞=𝐹𝑠𝑝,𝑞,(2.15) where 𝐵𝑠𝑝,𝑞 and 𝐹𝑠𝑝,𝑞 are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls 𝐵𝐽 by dyadic cubes 𝑃 (with side length 2𝐽) we obtain equivalent norms.

The full treatment of both scales of spaces can be found in [9]. Let Δ𝑗𝑓 (𝑗0) be the functions introduced in Definition 2.1. For any 𝑎>0, any 𝑥𝑛 and any 𝐽 we denote (Peetre’s maximal functions)Δ,𝑎𝑗,𝐽𝑓(𝑥)=sup𝑦𝐵𝐽||Δ𝑗||𝑓(𝑦)1+2𝑗||||𝑥𝑦𝑎𝑗0,𝑓𝒮(𝑛).(2.16)

We now present a fundamental characterization of 𝐵𝑠,𝜏𝑝,𝑞 spaces and 𝐹𝑠,𝜏𝑝,𝑞 spaces.

Theorem 2.4. Let 𝑠, 𝜏[0,), 0<𝑝, 𝑞 and 𝑎>𝑛/𝑝. Then 𝑓𝐵𝑠,𝜏𝑝,𝑞=sup𝐵𝐽1||𝐵𝐽||𝜏𝑗𝐽+2𝑗𝑠𝑞Δ,𝑎𝑗,𝐽𝑓𝐿𝑝𝐵𝐽𝑞1/𝑞,(2.17) is an equivalent quasinorm in 𝐵𝑠,𝜏𝑝,𝑞.

Theorem 2.5. Let 𝑠, 𝜏[0,), 0<𝑝<, 0<𝑞 and 𝑎>𝑛/min(𝑝,𝑞). Then 𝑓𝐹𝑠,𝜏𝑝,𝑞=sup𝐵𝐽(1/|𝐵𝐽|𝜏)(𝑗𝐽+2𝑗𝑠𝑞|Δ,𝑎𝑗,𝐽𝑓|𝑞)1/𝑞𝐿𝑝(𝐵𝐽), is an equivalent quasinorm in 𝐹𝑠,𝜏𝑝,𝑞.

Remark 2.6. Theorem 2.4 for  0<𝑝, 𝑞<  is given in [8, Theorem 4.5]. For Theorem 2.5 see [12, Theorem 1.1]. In addition if  𝑎>𝑛max(1/𝑝,𝜏), then in Theorem 2.4  Δ,𝑎𝑗,𝐽𝑓  can be replaced by  Δ𝑗,𝑎𝑓, where Δ𝑗,𝑎𝑓(𝑥)=sup𝑦𝑛||Δ𝑗||𝑓(𝑦)1+2𝑗||||𝑥𝑦𝑎𝑗0,𝑓𝒮(𝑛).(2.18)

3. Some Technical Lemmas

To prove our results, we need some technical lemmas. The following lemma for Δ𝑗𝑓, in place of Δ𝑗,𝑎𝑓, is given in [14, pages 87–89] (for the 𝐵𝑠,𝜏𝑝,𝑞 spaces and 1𝑝<). Further results, can be found in [12, Lemma 2.4].

Lemma 3.1. Let Δ𝑗𝑓 be as in Definition 2.1 and let 𝑠,  𝑎>0,  𝜏[0,) and 0<𝑝, 𝑞(0<𝑝< for the space 𝐹𝑠,𝜏𝑝,𝑞). Then there is a constant 𝑐>0, independent of 𝑗, such that for any 𝑓𝒮(𝑛)Δ𝑗,𝑎𝑓𝑐2𝑗(𝑛/𝑝𝑠𝑛𝜏)𝑓𝐴𝑠,𝜏𝑝,𝑞,𝑗0.(3.1) Here one uses 𝐴𝑠,𝜏𝑝,𝑞 to denote either 𝐵𝑠,𝜏𝑝,𝑞 or 𝐹𝑠,𝜏𝑝,𝑞.

Proof. Let 𝜓,𝜓0𝒮(𝑛) be two functions such that 𝜓=1 and 𝜓0=1 on supp𝜑1 and supp𝜑0 respectively. Then ||Δ𝑗||=||𝜓𝑓(𝑦)𝑗Δ𝑗||𝑓(𝑦),𝑦𝑛,(3.2) with 𝜓𝑗=2(𝑗1)𝑛𝜓(2𝑗1) if 𝑗. Since 𝜑𝒮(𝑛), the right-hand side is bounded by 𝑐𝜂𝑗1,𝑁|Δ𝑗𝑓|(𝑦), for any 𝑁>0. Hence we get for all 𝑓𝒮(𝑛) and any 𝑥𝑛Δ𝑗,𝑎𝑓(𝑥)𝑐sup𝑦𝑛𝜂𝑗1,𝑁||Δ𝑗𝑓||(𝑦).(3.3) Using the same method given in [9,Proposition 2.6] we obtain for any 𝑦𝑛𝜂𝑗1,𝑁||Δ𝑗𝑓||(𝑦)𝑐2𝑗(𝑛/𝑝𝑠𝑛𝜏)𝑓𝐴𝑠,𝜏𝑝,𝑞.(3.4) The proof is completed.

Lemma 3.2. Let 𝑀, 𝐽, 𝐴>0, 𝜏[0,) and 1𝑝. Then there is a constant 𝑐>0, independent of 𝐽 and 𝐴, such that ||𝐴||Δ𝑀||𝑓()𝑑𝐿𝑝𝐵𝐽𝑐𝐴𝑛||𝐵𝐽||𝜏𝑓𝐿𝑝𝜏,(3.5) for any ball 𝐵𝐽 of 𝑛 with radius 2𝐽 and any function 𝑓 such that 𝑓𝐿𝑝𝜏<.

Proof. Since 1𝑝, the left-hand side is bounded by ||𝐴Δ𝑀𝑓𝐿𝑝𝐵𝐽𝑑.(3.6) From the definition of Δ𝑀𝑓 we have ||Δ𝑀||𝑓(𝑥)𝑀𝑚=0𝐶𝑀𝑚||||𝑓(𝑥+(𝑀𝑚)).(3.7) Take the 𝐿𝑝(𝐵𝐽)-norm to estimate (3.6) form above by 𝑀𝑚=0𝐶𝑀𝑚||𝐴𝑓𝐿𝑝𝐵𝐽𝑑,(3.8) where if 𝑥0 the centre of 𝐵𝐽 then 𝑥0+(𝑀𝑚) is the centre of 𝐵𝐽. Using the fact that |𝐵𝐽|=|𝐵𝐽| to estimate (3.8) from above by 𝑐𝐴𝑛|𝐵𝐽|𝜏𝑓𝐿𝑝𝜏.
The lemma is proved.

Remark 3.3. Let 𝑀, 𝐴, 𝜏, and 𝑝 be as in Lemma 3.2. Let 𝐽0. By the embedding 𝐿𝑝(𝐵0)𝐿𝑝(𝐵𝐽) there is a constant 𝑐>0, independent of 𝐽 and 𝐴, such that ||𝐴||Δ𝑀||𝑓()𝑑𝐿𝑝𝐵𝐽𝑐𝐴𝑛𝑓𝐿𝑝𝜏,(3.9) for any ball 𝐵𝐽 of 𝑛 with radius 2𝐽 and any function 𝑓 such that 𝑓𝐿𝑝𝜏<.
For 𝑠>0, 𝑀, 𝜏[0,), 1𝑝< and 0<𝑞, we set 𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀=𝑓𝐿𝑝𝜏(𝑛)+sup𝐵𝐽1||𝐵𝐽||𝜏2+𝐽+10𝑡𝑠𝑞sup||||𝑡||Δ𝑀||𝑓()𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝐵𝐽.(3.10)

Here the supremum is taken over all 𝐽 and all balls 𝐵𝐽 of 𝑛 with radius 2𝐽.

Lemma 3.4. Let 𝑠>0, 𝑀, 𝐽, 𝜏[0,), 1𝑝< and 0<𝑞. Then there is a constant 𝑐>0, independent of 𝐽, such that 𝑗𝐽+2𝑗𝑠𝑞|𝑣|1||Δ𝑀2𝑗𝑣||𝑓()𝑑𝑣𝑞1/𝑞𝐿𝑝𝐵𝐽||𝐵𝑐𝐽||𝜏𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀,(3.11)𝑗𝐽+2𝑗𝑠𝑞|𝑣|>1||Δ𝑀2𝑗𝑣||𝑓()𝜔(𝑣)𝑑𝑣𝑞1/𝑞𝐿𝑝𝐵𝐽||𝐵𝑐𝐽||𝜏𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀,(3.12) for any ball 𝐵𝐽 of 𝑛 with radius 2𝐽, any 𝜔𝒮(𝑛) and any function 𝑓 such that 𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀<.

Proof. Let 𝑃 be a dyadic cube with side length 2𝐽. This result, for 𝑃 in place of 𝐵𝐽, 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<𝑞<. The case 𝑞= is similar.
Before proving this result we note that for any 𝑥𝑛 and any 𝑖𝑣𝑖2(𝑠+𝑛)𝑣𝑞||2𝑣||Δ𝑀||𝑓(𝑥)𝑑𝑞1/𝑞𝑐2𝑖+10𝑡(𝑠+𝑛)𝑞||𝑡||Δ𝑀||𝑓(𝑥)𝑑𝑞𝑑𝑡𝑡1/𝑞.(3.13) Here we will prove that the left-hand side of (3.12) is bounded by 𝑓𝐿𝑝𝜏(𝑛)+sup𝐵𝐽1||𝐵𝐽||𝜏2+𝐽0𝑡𝑠𝑞sup||||𝑡||Δ𝑀||𝑓()𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝐵𝐽=𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀.(3.14) Obviously, 𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀. We write 2𝑗𝑠|𝑣|>1||Δ𝑀2𝑗𝑣||𝑓(𝑥)𝜔(𝑣)𝑑𝑣=𝑘=02𝑗𝑠2𝑘<|𝑣|2𝑘+1||Δ𝑀2𝑗𝑣||𝑓(𝑥)𝜔(𝑣)𝑑𝑣𝑐𝑘=02(𝑠+𝑛)𝑗𝑁𝑘2𝑘𝑗<||2𝑘𝑗+1||Δ𝑀𝑓||(𝑥)𝑑,(3.15) where 𝑁>0 is at our disposal and we have used the properties of the function 𝜔, ||||𝜔(𝑥)𝑐(1+|𝑥|)𝑁,(3.16) for any 𝑥𝑛 and any 𝑁>0. Now the right-hand side of (3.15) in 𝑠𝑞,𝐽+-norm is bounded by (with 𝜎=min(1,𝑞)) 𝑐𝑘=02𝑁𝜎𝑘𝑗𝐽+2(𝑠+𝑛)𝑗𝑞2𝑘𝑗<||2𝑘𝑗+1||Δ𝑀||𝑓(𝑥)𝑑𝑞𝜎/𝑞1/𝜎=𝐽+1𝑘=0+𝑘𝐽+1/𝜎=𝐼𝐽(𝑥)+𝐼𝐼𝐽(𝑥)1/𝜎21/𝜎1𝐼𝐽(𝑥)1/𝜎+𝐼𝐼𝐽(𝑥)1/𝜎,(3.17) by (2.7). Here we put 𝐽+1𝑘=0=0 if 𝐽+=0. Take the 𝐿𝑝(𝐵𝐽)-norm we obtain that the left-hand side of (3.12) is bounded by 21/𝜎1𝐼𝐽1/𝜎𝐿𝑝𝐵𝐽+21/𝜎1𝐼𝐼𝐽1/𝜎𝐿𝑝𝐵𝐽.(3.18) Let us estimate (𝐼𝐽)1/𝜎 in 𝐿𝑝(𝐵𝐽)-norm. After a change of variable 𝑗𝑘1=𝑣, we get for any 𝑥𝑛 (here 𝐽+=𝐽 and 𝑘<𝐽) 𝐼𝐽(𝑥)𝑐𝐽+1𝑘=02(𝑠+𝑛𝑁)𝜎𝑘𝑣𝐽+𝑘12(𝑠+𝑛)𝑣𝑞||2𝑣||Δ𝑀||𝑓(𝑥)𝑑𝑞𝜎/𝑞=𝐽+2𝑘=0+𝐽+1𝑘=𝐽+1=𝑀1(𝑥)+𝑀2(𝑥).(3.19) Here we put 𝐽+2𝑘=0=0 if 𝐽+1. We have 𝑀1(𝑥)𝑐𝐽+2𝑘=02(𝑠+𝑛𝑁)𝜎𝑘2𝐽+𝑘+20𝑡(𝑠+𝑛)𝑞||𝑡||Δ𝑀||𝑓(𝑥)𝑑𝑞𝑑𝑡𝑡𝜎/𝑞.(3.20) Since 𝐿𝑝/𝜎(𝐵𝐽) is a normed spaces and 𝐵𝐽𝐵𝐽𝑘2, the right-hand side in 𝐿𝑝/𝜎(𝐵𝐽)-norm can be estimated from above by 𝑐𝐽+2𝑘=02(𝑠+𝑛𝑁)𝜎𝑘2𝐽+𝑘+20𝑡𝑠𝑞sup||||𝑡||Δ𝑀||𝑓()𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝐵𝐽𝑘2𝜎||𝐵𝑐𝐽||𝐽𝜏𝜎+2𝑘=02(𝑠+𝑛𝑁+𝑛𝜏)𝜎𝑘𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎.(3.21) We choose 𝑁>2(𝑠+𝑛)+𝜏𝑛. This yields that the last expression is bounded by 𝑐||𝐵𝐽||𝜏𝜎𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎,(3.22) where 𝑐>0 is independent of 𝐽. Now (𝑀2)1/𝜎 in 𝐿𝑝(𝐵𝐽)-norm is bounded by 𝑐2(𝑠+𝑛𝑁)𝐽+𝑣02(𝑠+𝑛)𝑣𝑞||2𝑣||Δ𝑀||𝑓()𝑑𝑞1/𝑞𝐿𝑝𝐵𝐽𝑐2(𝑠+𝑛𝑁)𝐽+𝑣12(𝑠+𝑛)𝑣𝑞||2𝑣||Δ𝑀||𝑓()𝑑𝑞1/𝑞𝐿𝑝𝐵𝐽+𝑐2(𝑠+𝑛𝑁)𝐽+||1||Δ𝑀||𝑓()𝑑𝐿𝑝𝐵𝐽,(3.23) where we have used (2.7). Using the embedding 𝐿𝑝(𝐵0)𝐿𝑝(𝐵𝐽) and Remark 3.3, we obtain 𝑀2𝐿𝑝/𝜎𝐵𝐽=𝑀21/𝜎𝐿𝑝𝐵𝐽𝜎𝑐2(𝑠+𝑛𝑁)𝜎𝐽+𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎=𝑐2(𝑠+𝑛𝑁+𝑛𝜏)𝜎𝐽+||𝐵𝐽||𝜏𝜎𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎||𝐵𝑐𝐽||𝜏𝜎𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎,(3.24) because of 𝑁>𝑠+𝑛+𝑛𝜏. Therefore, 𝐼𝐽1/𝜎𝐿𝑝𝐵𝐽𝜎=𝐼𝐽𝐿𝑝/𝜎𝐵𝐽𝑀1𝐿𝑝/𝜎𝐵𝐽+𝑀2𝐿𝑝/𝜎𝐵𝐽||𝐵𝑐𝐽||𝜏𝜎𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎.(3.25) Now let us estimate (𝐼𝐼𝐽)1/𝜎 in 𝐿𝑝(𝐵𝐽)-norm. We write 𝐼𝐼𝐽()=𝑐𝑘𝐽+2𝑁𝜎𝑘𝑘+𝐽++1𝑗=𝐽++𝑗=𝑘+𝐽++2𝜎/𝑞𝑐𝑘𝐽+2𝑁𝜎𝑘𝑘+𝐽++1𝑗=𝐽+𝜎/𝑞+𝑗=𝑘+𝐽++2𝜎/𝑞=𝑐𝑘𝐽+2𝑁𝜎𝑘𝑆1𝑘()+𝑆2𝑘.()(3.26) After a change of variable 𝑗𝑘1=𝑣, we get 𝑆1𝑘(𝑥)1/𝜎𝑐2(𝑠+𝑛)𝑘𝐽+𝑣=𝐽+𝑘12(𝑠+𝑛)𝑣𝑞||2𝑣||Δ𝑀||𝑓(𝑥)𝑑𝑞1/𝑞𝑐2(𝑠+𝑛)𝑘2+𝑘𝐽+22+𝐽𝑡(𝑠+𝑛)𝑞||𝑡||Δ𝑀||𝑓(𝑥)𝑑𝑞𝑑𝑡𝑡1/𝑞𝑐2(𝑠+𝑛)𝑘||2+𝑘𝐽+2||Δ𝑀𝑓||(𝑥)𝑑2+𝑘𝐽+22+𝐽𝑡(𝑠+𝑛)𝑞𝑑𝑡𝑡1/𝑞𝑐2(𝑠+𝑛)𝐽++(𝑠+𝑛)𝑘||2+𝑘𝐽+2||Δ𝑀||𝑓(𝑥)𝑑.(3.27) Therefore there exists a constant 𝑐>0 independent of 𝐽 and 𝑘 such that 𝑆1𝑘1/𝜎𝐿𝑝𝐵𝐽𝑐2(𝑠+𝑛)𝐽++(𝑠+𝑛)𝑘||2+𝑘𝐽+2Δ𝑀𝑓()𝑑𝐿𝑝𝐵𝐽𝑐2(𝑠+𝜏𝑛)𝐽++(𝑠+2𝑛)𝑘||𝐵𝐽||𝜏𝑓𝐿𝑝𝜏𝑐2(2𝑠+2𝑛+𝜏𝑛)𝑘||𝐵𝐽||𝜏𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀,(3.28) by Lemma 3.2 (combined with Remark 3.3) and the fact that 𝑘𝐽+. Now let us estimate (𝑆2𝑘)1/𝜎in 𝐿𝑝(𝐵𝐽)-norm. We have 𝑆2𝑘(𝑥)1/𝜎𝑐2(𝑛+𝑠)𝑘𝑣=𝐽++12(𝑠+𝑛)𝑣𝑞||2𝑣||Δ𝑀||𝑓(𝑥)𝑑𝑞1/𝑞𝑐2(𝑛+𝑠)𝑘2+𝐽0𝑡𝑠𝑞sup||||𝑡||Δ𝑀||𝑓(𝑥)𝑞𝑑𝑡𝑡1/𝑞.(3.29) Therefore, 𝑆2𝑘1/𝜎𝐿𝑝𝐵𝐽𝑐2(𝑛+𝑠)𝑘2+𝐽0𝑡𝑠𝑞sup||||𝑡||Δ𝑀||𝑓()𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝐵𝐽𝑐2(𝑛+𝑠)𝑘||𝐵𝐽||𝜏𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀.(3.30) Consequently, 𝐼𝐼𝐽1/𝜎𝐿𝑝𝐵𝐽𝜎=𝐼𝐼𝐽𝐿𝑝/𝜎𝐵𝐽𝑘𝐽+2𝑁𝜎𝑘𝑆1𝑘1/𝜎𝐿𝑝𝐵𝐽𝜎+𝑆2𝑘1/𝜎𝐿𝑝𝐵𝐽𝜎𝑐𝑘𝐽+2(2𝑠+2𝑛+𝜏𝑛𝑁)𝜎𝑘||𝐵𝐽||𝜏𝜎𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎||𝐵𝑐𝐽||𝜏𝜎𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀𝜎,(3.31) since 𝑁>2(𝑠+𝑛)+𝜏𝑛. This finishes the proof of Lemma 3.4.

For 𝑠>0, 𝑀, 𝜏[0,), 1𝑝, and 0<𝑞, we set𝑓𝐵𝑠,𝜏𝑝,𝑞𝑀=𝑓𝐿𝑝𝜏(𝑛)+sup𝐵𝐽1||𝐵𝐽||𝜏2+𝐽+10𝑡𝑠𝑞sup||||𝑡Δ𝑀𝑓𝐿𝑝𝐵𝐽𝑞𝑑𝑡𝑡1/𝑞.(3.32) Here the supremum is taken over all 𝐽 and all balls 𝐵𝐽 of 𝑛 with radius 2𝐽. Similar arguments yield.

Lemma 3.5. Let 𝑠>0, 𝑀, 𝐽, 𝜏[0,), 1𝑝 and 0<𝑞. Then there is a constant 𝑐>0, independent of 𝐽, such that 𝑗𝐽+2𝑗𝑠𝑞|𝑣|1Δ𝑀2𝑗𝑣𝑓𝐿𝑝𝐵𝐽𝑑𝑣𝑞1/𝑞||𝐵𝑐𝐽||𝜏𝑓𝐵𝑠,𝜏𝑝,𝑞𝑀,𝑗𝐽+2𝑗𝑠𝑞|𝑣|>1Δ𝑀2𝑗𝑣𝑓𝐿𝑝𝐵𝐽||||𝜔(𝑣)𝑑𝑣𝑞1/𝑞||𝐵𝑐𝐽||𝜏𝑓𝐵𝑠,𝜏𝑝,𝑞𝑀,(3.33) for any ball 𝐵𝐽 of 𝑛 with radius 2𝐽, any 𝜔𝒮(𝑛) and any function 𝑓 such that 𝑓𝐵𝑠,𝜏𝑝,𝑞𝑀<.

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

Lemma 3.6. Let 0<𝑎<1, 𝐽 and 0<𝑞. Let {𝜀𝑘} be a sequences of positive real numbers, such that 𝜀𝑘𝑘𝐽+0𝑞,𝐽+=𝐼<.(3.34) The sequences {𝛿𝑘𝛿𝑘=𝑘𝑗=𝐽+𝑎𝑘𝑗𝜀𝑗}𝑘𝐽+𝑎𝑛𝑑{𝜂𝑘𝜂𝑘=𝑗=𝑘𝑎𝑗𝑘𝜀𝑗}𝑘𝐽+, are in 0𝑞,𝐽+ with 𝛿𝑘𝑘𝐽+0𝑞,𝐽++𝜂𝑘𝑘𝐽+0𝑞,𝐽+𝑐𝐼,(3.35)𝑐depends only on 𝑎 and 𝑞.

4. Characterizations with Differences

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

Theorem 4.1. Let 1𝑝<, 0<𝑞, 𝜏[0,)  and  𝑀. Assume 𝑛1min(𝑝,𝑞)<𝑠<𝑀,0𝜏<𝑝(4.1) or 𝑛𝑛min(𝑝,𝑞)<𝑠<𝑀𝑛𝜏+𝑝1,𝜏𝑝.(4.2) Then 𝑓𝐹𝑠,𝜏𝑝,𝑞𝑀 is an equivalent quasinorm in 𝐹𝑠,𝜏𝑝,𝑞.

Theorem 4.2. Let 1𝑝, 0<𝑞, 𝜏[0,) and 𝑀. Assume 10<𝑠<𝑀,0𝜏<𝑝(4.3) or 𝑛0<𝑠<𝑀𝑛𝜏+𝑝1,𝜏𝑝.(4.4) Then 𝑓𝐵𝑠,𝜏𝑝,𝑞𝑀 is an equivalent quasinorm in 𝐵𝑠,𝜏𝑝,𝑞.

Remark 4.3. Theorems 4.1 and 4.2 for  0𝜏<𝑠/𝑛+1/𝑝  and  0𝜏<𝑠/𝑛+1/𝑝max(1/min(𝑝,𝑞)1,0),  respectively, are given in [9] Theorems 4.6 and 4.7, respectively.

Proof of Theorem 4.1. Let 𝐵𝐽 be any ball centered at 𝑥0𝑛 and of radius 2𝐽, 𝐽. We will do the proof in three steps. Step 1. We have with 𝑠>0, 𝑗0||Δ𝑗𝑓||=𝑗02𝑗𝑠2𝑗𝑠||Δ𝑗𝑓||𝑐sup𝑗02𝑗𝑠||Δ𝑗𝑓||𝑐𝑗02𝑗𝑠𝑞||Δ𝑗𝑓||𝑞1/𝑞.(4.5) Let 𝑓𝐹𝑠,𝜏𝑝,𝑞. Then, 𝑓𝐿𝑝𝜏(𝑛)𝑗0||Δ𝑗𝑓||𝐿𝑝𝜏(𝑛)𝑐𝑗02𝑗𝑠𝑞||Δ𝑗𝑓||𝑞1/𝑞𝐿𝑝𝜏(𝑛)𝑐𝑓𝐹𝑠,𝜏𝑝,𝑞.(4.6)Step 2. For any 𝑥𝐵𝐽 we put 𝐻𝐽(𝑥)=2+𝐽+10𝑡𝑠𝑞sup||||𝑡||Δ𝑀||𝑓(𝑥)𝑞𝑑𝑡𝑡.(4.7) After a change of variable 𝑡=2𝑦, we get 𝐻𝐽(𝑥)=ln2𝐽++12𝑦𝑠𝑞sup||||2𝑦||Δ𝑀||𝑓(𝑥)𝑞𝑑𝑦.(4.8) Then 𝐻𝐽(𝑥)𝑐𝑘𝐽+2𝑘𝑠𝑞sup||||2𝑘+1||Δ𝑀||𝑓(𝑥)𝑞.(4.9)Let 𝜓,𝜓0𝒮(𝑛) be two functions such that 𝜓=1 and 𝜓0=1 on supp𝜑1 and suppΨ respectively. Using the mean value theorem we obtain for any 𝑥𝐵𝐽, 𝑗0, and ||2𝑘+1||Δ1Δ𝑗||=||Δ𝑓(𝑥)1𝜓𝑗Δ𝑗𝑓(||𝑥)2𝑘sup||||𝑥𝑦𝑐2𝑘|𝛼|=1||𝐷𝛼𝜓𝑗Δ𝑗||,𝑓(𝑦)(4.10) with some positive constant 𝑐, independent of 𝑗 and 𝑘, and 𝜓𝑗()=2(𝑗1)𝑛𝜓(2𝑗1) for 𝑗=1,2,. By induction on 𝑀, we show that ||Δ𝑀Δ𝑗||𝑓(𝑥)2𝑘𝑀sup||||𝑥𝑦𝑐2𝑘|𝛼|=𝑀||𝐷𝛼𝜓𝑗Δ𝑗||𝑓(𝑦).(4.11) We see that if |𝛼|=𝑀 and 𝑎>0||𝐷𝛼𝜓𝑗Δ𝑗𝑓||(𝑦)=2(𝑗1)𝑛||||𝑛𝐷𝛼𝜓2𝑗1Δ(𝑦𝑧)𝑗𝑓||||(𝑧)𝑑𝑧2(𝑗1)(𝑛+𝑀)𝑛||(𝐷𝛼𝜓)2𝑗1||||Δ(𝑦𝑧)𝑗𝑓||(𝑧)𝑑𝑧.(4.12) Suppose that 0𝑗𝐽+1. The right-hand side in (4.12) may be estimated as follows: 𝑐2𝑗(𝑛+𝑀)Δ𝑗,𝑎𝑓(𝑦)𝑛||(𝐷𝛼2𝜓)𝑗1(||𝑦𝑧)1+2𝑗||||𝑦𝑧𝑎𝑑𝑧𝑐2𝑗𝑀Δ𝑗,𝑎𝑓(𝑦).(4.13) Then we obtain for any 𝑥𝐵𝐽, ||2𝑘+1 and any 𝑘𝐽+||Δ𝑀Δ𝑗𝑓||(𝑥)𝑐2(𝑗𝑘)𝑀sup||||𝑥𝑦𝑐2𝑘Δ𝑗,𝑎𝑓(𝑦)𝑐2(𝑗𝑘)𝑀1+2𝑗𝑘𝑎sup||||𝑥𝑦𝑐2𝑘Δ𝑗,𝑎𝑓(𝑦)1+2𝑗||||𝑥𝑦𝑎𝑐2(𝑗𝑘)𝑀Δ𝑗,𝑎𝑓(𝑥),(4.14) if 0𝑗𝐽+1.
Suppose now that 𝐽+𝑗𝑘. By our assumption on 𝑥 and 𝑘 we have ||𝑦𝑥0||||𝑥𝑥0||+||||𝑥𝑦<2𝐽+𝑐2𝑘(𝑐+1)2𝐽,(4.15) which implies that 𝑦 is located in some ball 𝐵𝐽, where 𝐵𝐽={𝑦𝑛|𝑦𝑥0|<(𝑐+1)2𝐽}. Writing the integral in (4.12) as follows 𝐵𝐽1𝑑𝑧+𝑖0𝐵𝐽𝑖2𝐵𝐽𝑖1𝑑𝑧=𝐼𝑗,𝐽(𝑦)+𝑖0𝐼𝐼𝑗,𝐽𝑖(𝑦).(4.16) We recall that Δ,𝑎𝑗,𝑙𝑓(𝑦)=sup𝑧𝐵𝑙||Δ𝑗||𝑓(𝑧)1+2𝑗||||𝑦𝑧𝑎,(4.17) for any 𝑗0, 𝑓𝒮(𝑛) and any 𝑙. We have 𝐼𝑗,𝐽(𝑦)Δ,𝑎𝑗,𝐽1𝐵𝑓(𝑦)𝐽1||(𝐷𝛼2𝜓)𝑗1(||𝑦𝑧)1+2𝑗||||𝑦𝑧𝑎𝑑𝑧𝑐2𝑗𝑛Δ,𝑎𝑗,𝐽1𝑓(𝑦)𝑐2𝑗𝑛Δ,𝑎𝑗,𝐽2𝑓(𝑦).(4.18) Let us estimate 𝐼𝐼𝑗,𝐽𝑖. Since 𝜓𝒮(𝑛), we have ||𝐷𝛼||𝜓(𝑥)𝑐(1+|𝑥|)2𝑁,(4.19) for any 𝑥𝑛 and any 𝑁>0. Then for any 𝑁 large enough, 𝐼𝐼𝑗,𝐽𝑖(𝑦) does not exceed 𝑐Δ,𝑎𝑗,𝐽𝑖2𝐵𝑓(𝑦)𝐽𝑖2𝐵𝐽𝑖11+2𝑗1||||𝑦𝑧2𝑁+𝑎𝑑𝑧𝑐2𝑖𝑁Δ,𝑎𝑗,𝐽𝑖2𝑓(𝑦)𝑛1+2𝑗1||||𝑦𝑧𝑁+𝑎𝑑𝑧𝑐2𝑖𝑁𝑗𝑛Δ,𝑎𝑗,𝐽𝑖2𝑓(𝑦),(4.20) where we have used 2𝑗1|𝑦𝑧|>(𝑐+1)2𝑗𝐽+𝑖1(𝑐+1)2𝑖1. Therefore, ||𝐷𝛼𝜓𝑗Δ𝑗||𝑓(𝑦)𝑐2𝑗𝑀𝑖02𝑖𝑁Δ,𝑎𝑗,𝐽𝑖2𝑓(𝑦).(4.21) Then we obtain for any 𝑥𝐵𝐽 any ||2𝑘+1 and any 𝐽+𝑗𝑘||Δ𝑀Δ𝑗||𝑓(𝑥)𝑐2(𝑗𝑘)𝑀𝑖02𝑖𝑁sup||||𝑥𝑦𝑐2𝑘Δ,𝑎𝑗,𝐽𝑖2𝑓(𝑦)𝑐2(𝑗𝑘)𝑀1+2𝑗𝑘𝑎𝑖02𝑖𝑁sup||||𝑥𝑦𝑐2𝑘Δ,𝑎𝑗,𝐽𝑖2𝑓(𝑦)1+2𝑗||||𝑥𝑦𝑎.(4.22) Consequently, for any 𝐽+𝑗𝑘 there is a constant 𝑐>0 independent of 𝐽, 𝑗, and 𝑘 such that ||Δ𝑀Δ𝑗||𝑓(𝑥)𝑐2(𝑗𝑘)𝑀𝑖02𝑖𝑁Δ,𝑎𝑗,𝐽𝑖2𝑓(𝑥).(4.23) Finally for 𝑗𝑘+1 we have for 𝑥𝐵𝐽 and ||2𝑘+1||Δ𝑀Δ𝑗||𝑓(𝑥)𝑀𝑚=0𝐶𝑀𝑚||Δ𝑗||𝑓(𝑥+(𝑀𝑚))2𝑀sup||||𝑥𝑦𝐶2𝑘||Δ𝑗||𝑓(𝑦)2𝑀sup||||𝑥𝑦𝐶2𝑘||Δ𝑗||𝑓(𝑦)1+2𝑗||||𝑥𝑦𝑎1+2𝑗||||𝑥𝑦𝑎.(4.24) We remark also that by our assumption on 𝑥 and 𝑘 we have ||𝑦𝑥0||||𝑥𝑥0||+||||𝑥𝑦<2𝐽+𝐶2𝑘(𝐶+1)2𝐽,(4.25) and this implies that 𝑦 is located in some ball 𝐵𝐽, where 𝐵𝐽={𝑦𝑛|𝑦𝑥0|<(𝐶+1)2𝐽}. Then, ||Δ𝑀Δ𝑗𝑓||(𝑥)𝑐2(𝑗𝑘)𝑎Δ,𝑎𝑗,𝐽𝑓(𝑥),(4.26) if 𝑗𝑘+1, where Δ,𝑎𝑗,𝐽𝑓 is given in (4.17) (with 𝐵𝐽 a ball centered at 𝑥0 and of radius (𝐶+1)2𝐽).
We write, Δ𝑀𝑓(𝑥)=𝑗0Δ𝑀Δ𝑗=𝑓(𝑥)𝐽+1𝑗=0+𝑘𝑗=𝐽++𝑗𝑘+1.(4.27) Here we put 𝐽+1𝑗=0=0 if 𝐽+=0. Let us estimate each term in 𝑠𝑞,𝐽+-norm. We have by (4.14) and Lemma 3.1𝐽+1𝑗=0||Δ𝑀Δ𝑗||𝑓(𝑥)𝐽+1𝑗=02(𝑗𝑘)𝑀Δ𝑗,𝑎𝑓(𝑥)𝑐𝐽+1𝑗=02(𝑗𝑘)𝑀+𝑗(𝑛/𝑝𝑠𝑛𝜏)𝑓𝐹𝑠,𝜏𝑝,𝑞𝑐2𝐽𝑘𝑀+1𝑗=02𝑗(𝑀+𝑛/𝑝𝑠𝑛𝜏)𝑓𝐹𝑠,𝜏𝑝,𝑞𝑐2𝐽(𝑀+𝑛/𝑝𝑠𝑛𝜏)𝑘𝑀𝑓𝐹𝑠,𝜏𝑝,𝑞,(4.28) where the last inequality can be obtained by our assumption on 𝑠 and 𝜏. The last expression in 𝑠𝑞,𝐽+-norm does not exceed 𝑐2𝐽𝑛(1/𝑝𝜏)2(𝑘𝐽+)(𝑠𝑀)𝑘𝐽+0𝑞,𝐽+𝑓𝐹𝑠,𝜏𝑝,𝑞𝑐2𝐽𝑛(1/𝑝𝜏)𝑓𝐹𝑠,𝜏𝑝,𝑞,(4.29) since 𝑠<𝑀. Therefore, 𝐻𝐽(𝑥)1/𝑞𝑐2𝐽𝑛(1/𝑝𝜏)𝑓𝐹𝑠,𝜏𝑝,𝑞(4.30)+𝑐𝑘𝐽+𝑘𝑖0𝑗=𝐽+2(𝑗𝑘)(𝑀𝑠)+𝑠𝑗𝑖𝑁|||Δ,𝑎𝑗,𝐽𝑖2|||𝑓(𝑥)𝑞1/𝑞(4.31)+𝑐𝑘𝐽+𝑗𝑘2(𝑗𝑘)(𝑎𝑠)+𝑠𝑗|||Δ,𝑎𝑗,𝐽|||𝑓(𝑥)𝑞1/𝑞.(4.32) The second term can be estimated by (with 𝜎=min(1,𝑞)) 𝑐𝑖02𝑖𝑁𝜎𝑘𝐽+𝑘𝑗=𝐽+2(𝑗𝑘)(𝑀𝑠)+𝑠𝑗|||Δ,𝑎𝑗,𝐽𝑖2|||𝑓(𝑥)𝑞𝜎/𝑞1/𝜎.(4.33) Since again 𝑠<𝑀, then we can apply Lemma 3.6 to estimate the last expression by 𝑐𝑖02𝑖𝑁𝜎𝑗𝐽+2𝑗𝑠𝑞|||Δ,𝑎𝑗,𝐽𝑖2|||𝑓(𝑥)𝑞𝜎/𝑞1/𝜎.(4.34) Since 𝐿𝑝/𝜎(𝐵𝐽) is a normed space, so (4.31) in 𝐿𝑝(𝐵𝐽)-norm is dominated by 𝑐𝑖02𝑖𝑁𝜎𝑗𝐽+2𝑗𝑠𝑞|||Δ,𝑎𝑗,𝐽𝑖2𝑓|||𝑞𝜎/𝑞𝐿𝑝/𝜎𝐵𝐽1/𝜎=𝑐𝑖02𝑖𝑁𝜎𝑗𝐽+2𝑗𝑠𝑞|||Δ,𝑎𝑗,𝐽𝑖2𝑓|||𝑞1/𝑞𝐿𝑝𝐵𝐽𝜎1/𝜎.(4.35)