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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space

Yanqi Yang / Shuangping Tao
  • Corresponding author
  • College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, Gansu, China
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Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/math-2018-0133

Abstract

The aim of this paper is to deal with the boundedness of the θ-type Calderón-Zygmund operators and their commutators on Herz spaces with two variable exponents p(⋅), q(⋅). It is proved that the θ-type Calderón-Zygmund operators are bounded on the homogeneous Herz space with variable exponents K˙p()α,q()(Rn). Furthermore, the boundedness of the corresponding commutators generated by BMO function and Lipschitz function is also obtained respectively.

Keywords: θ-type Calderón-Zygmund operator; commutator; Herz spaces with variable exponent; BMO space

MSC 2010: 42B20; 42B25; 42B35

1 Introduction and Main Results

The theory of Calderón-Zygmund operators, which has played very important roles in modern harmonic analysis with lots of extensive applications in the others fields of mathematics, has been extensively studied (see [1, 2, 3, 4, 5, 6], for instance). In 1985, Yabuta introduced certain θ-type Calderón-Zygmund operators to facilitate his study of certain classes of pseudodifferential operators (see [7]). Following the terminology of Yabuta, we recall the so-called θ-type Calderón-Zygmund operators. Let θ be a non-negative and non-decreasing function on ℝ+ = (0, ∞) satisfying

01θ(t)tdt<.(1.1)

A measurable function K(⋅, ⋅) on ℝn × ℝn∖{(x, x) : x ∈ ℝn} is said to be a θ-type Calderón-Zygmund kernel if it satisfies

|K(x,y)|C|xy|n,xy,(1.2)

and

|K(x,y)K(x,y)|+|K(y,x)K(y,x)|Cθ(|xx||xy|)|xy|n,as|xy|2|xx|,xy.(1.3)

Definition 1.1

[7] Let Tθ be a linear operator from 𝓢 into its dual 𝓢. One can say that Tθ is a θ-type Calderón-Zygmund operator if it satisfies the following conditions:

  1. Tθ can be extended to be a bounded linear operator on L2(ℝn) ;

  2. there is a θ-type Calderón-Zygmund kernel K(x, y) such that

    Tθf(x):=RnK(x,y)f(y)dy,asfCc(Rn)andxsuppf.(1.4)

    It is easy to see that the classical Calderón-Zygmund operator with standard kernel is a special case of θ-type operator Tθ as θ(t) = tδ with 0 < δ ≤ 1. Given a locally integrable function b, the commutator generated by Tθ and b is defined by

    [b,Tθ]f(x)=b(x)Tθf(x)Tθ(bf)(x)=Rn[b(x)b(y)]K(x,y)f(y)dy.(1.5)

Such type of operators are extensively applied in PDE with non-smooth area. Many authors concentrate on the boundedness of these operators on various function spaces, we refer the reader to see [8, 9, 10, 11, 12, 13, 14] for its developments and applications. In [11], Quek-Yang established the boundedness of Tθ on spaces such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. Ri-Zhang obtained the bounedness of Tθ on Hardy spaces with non-doubling measures and non-homogeneous metric measure spaces in [12, 13]. Wang proved the boundedness of Tθ and [b, Tθ] on the generalized weighted Morrey spaces in [14]. Inspired by the results mentioned previously, a natural and interesting problem is to consider whether or not the θ-type Calderón-Zygmund operators Tθ and their commutators [b, Tθ] are bounded on Herz space with variable exponents. The purpose of this paper is to give a positive answer.

The spaces with variable exponent have been widely studied in recent ten years. The results show that they are not only the generalized forms of the classical function spaces with invariable exponent, but also there are some new breakthroughs in the research techniques. These new real variable methods help people further understand the function spaces. Lebesgue spaces with variable exponent Lp(⋅)(ℝn) become one class of important function spaces due to the fundamental paper [15] by Kovóčik Rákosník. The theory of the function spaces with variable exponent have been applied in fluid dynamics, elastlcity dynamics, calculus of variations and differential equations with non-standard growth conditions(for example, see [16, 17, 18, 19, 20]). In [21], authors proved the extrapolation theorem which leads the boundedness of some classical operators including the commutators on Lp(⋅)(ℝn). Karlovich and Lerner also obtained the bundedness of the singular integral commutators in [22]. The boundedness of some typical operators is being studied with keen interest on spaces with variable exponent (see [23, 24, 25, 26]). Recently, Tao and Yang established the boundedness of θ-Type C-Z operators and their commutators generated respectively by BMO functions, Lipschitz functions and Besov functions on variable exponent Lebesgue spaces Lp(⋅)(ℝn) (see [27]), at the same time, the boundedness of singular integrals with variable kernel and fractional differentiations is also obtained.

It is well known that Herz spaces play an important role in harmonic analysis. After the Herz spaces with one exponent p(⋅) were introduced in [28], the boundedness of some operators and some characterizations of these spaces were studied widely (see [29, 30, 31, 32]). In this paper, we will study the boundedness of the θ-type Calderón-Zygmund operators Tθ and their commutators [b, Tθ] on Herz spaces with two variable exponents p(⋅), q(⋅). In order to do this, we need to recall some notations and definitions.

Denote 𝓟(ℝn) to be the set of the all measurable functions p(x) with p =: ess infxRn p(x) > 1 and p+ =: ess supxRn p(x) < ∞ and 𝓑(ℝn) to be the set of all functions p(⋅) ∈ 𝓟(ℝn) satisfying the condition that the Hardy-Littlewood maximal operator M is bounded on Lp(⋅)(ℝn), 𝓟0(ℝn) the set of all measurable functions p(x) with p > 0 and p+ < ∞.

Given a function p(x) ∈ 𝓟(ℝn), the space Lp(x)(ℝn) is now defined by[15]

fLp()(Rn)=infη>0:Rn|f(x)|ηp(x)dx1.

Given a function p(⋅) ∈ 𝓟0(ℝn), the space Lp(⋅)(ℝn) is defined by

Lp()(Rn)=f:|f|p0Lq()(Rn)forsomep0with0<p0<pandq(x)=p(x)p0.

It is easy to see that the above Luxemburg-Nakano quasi-norm is also equivalent with[21]

fLp()(Rn)=|f|p0Lq()(Rn)1/p0,

where q(x)=p(x)p0 and for some p0 with 0 < p0 < p.

Before recalling the Herz spaces with two variable exponents, we will introduce the following function space, which is named as the mixed Lebesgue sequence space (see [33]).

Definition 1.2

[24, 33] Let p(⋅), q(⋅) ∈ 𝓟0(ℝn). The mixed Lebesgue sequence space with variable exponent lq(⋅)(Lp(⋅)) is the collection of all sequences {fj}j=0 of measurable functions on ℝn such that

{fj}j=0lq()(Lp())=infμ>0:Λlq()(Lp())fj/μj=01<,

where

Λlq()(Lp())({fj}j=0)=j=0infμj>0:Rn|fj(x)|/μj1q(x)p(x)dx1.

Noting q+ < ∞, we see that

Λlq()(Lp())({fj}j=0)=j=0|fj|q()Lp()q().

Let k ∈ ℤ, Bk = {x ∈ ℝn : |x| ≤ 2k}, Ck = BkBk−1, χk = χCk.

Definition 1.3

[24] Let α ∈ ℝ, p(⋅), q(⋅) ∈ 𝓟0(ℝn). The homogeneous Herz space with variable exponent K˙p()α,q()(Rn) is defined by

K˙p()α,q()(Rn)={fLlocp()(Rn{0}):fK˙p()α,q()(Rn)<},

where

fK˙p()α,q()(Rn)={2kα|fχk|}k=lq()(Lp())=infη>0,k=2kα|fχk|/ηq()Lp()q()1.

Remark 1.1

  1. It is easy to see that K˙p()0,p()(Rn) = Lp(⋅)(ℝn) and if p(x) = p0 and q(x) = q0 are constants, then K˙p()α,q()(Rn)=K˙p0α,q0(Rn) is just the usual Herz spaces (see [34]).

  2. If q1(⋅), q2(⋅) ∈ 𝓟0(ℝn) satisfying (q1)+ ≤ (q2), then (see [24])

    K˙p()α,q1()(Rn)K˙p()α,q2()(Rn).

Definition 1.5

[35] When 0 < β ≤ 1, the Homogeneous Lipschitz space Lipβ(ℝn) is the space of functions such that

fLipβ=supx,hRn;h0|f(x+h)f(x)||h|β<.

Our main results in this paper are fomulated as follows.

Theorem 1.1

Suppose that Tθ is a θ-type Calderón-Zygmund operators with θ satisfies (1, 1), p(⋅) ∈ 𝓑(ℝn), q1(⋅), q2(⋅) ∈ 𝓟0(ℝn) with (q2) ≥ (q1)+ and p(⋅)/q2(⋅) ∈ 𝓟(ℝn). If −12 < α < 11, where δ11, δ12 are the constants in Lemma 2.2, then the operator Tθ is bounded from K˙p()α,q1()(Rn)toK˙p()α,q2()(Rn).

Theorem 1.2

Let b ∈ BMO, m ∈ ℕ and p(⋅), q1(⋅), q2(⋅) and α are the same as in Theorem 1.1. Suppose that [b, Tθ] is defined by (1.5) with θ satisfying

01θ(t)t|logt|dt<.(1.6)

Then the commutator [b, Tθ] is bounded from K˙p()α,q1()(Rn)toK˙p()α,q2()(Rn).

Theorem 1.3

Let b ∈ Lipβ(ℝn)(0 < β < 1) and [b, Tθ] be defined by (1.5) with θ satisfying (1.6). Suppose that q1(⋅), q2(⋅) ∈ 𝓟0(ℝn) with (q2) ≥ (q1)+ and p1(⋅), p2(⋅) ∈ 𝓑(ℝn) is such that (p1)+ < nβ, 1/p1(x)−1/p2(x) = β/n, (1−β/n)p2(⋅) ∈ 𝓑(ℝn) and p2(⋅)/q2(⋅) ∈ 𝓟(ℝn). If −12 < α < 11, then the commutator [b, Tθ] is bounded from K˙p1()α,q1()(Rn)toK˙p2()α,q2()(Rn).

We make some conventions. In what follows, C always denotes a positive constant which is independent of the main parameters involved but whose value may differ in different occurrences, |E| denotes the Lebesgue measure of E ∈ ℝn. Given a function f, we denote the mean value of f on E by fE =: 1|E| Ef(x)dx. p(⋅) means the conjugate exponent of p(⋅), namely, 1/p(x) + 1/p(x) = 1 holds.

2 Preliminary Lemmas

Before proving the main results, we need the following lemmas.

Lemma 2.1

[15] (Generalized Hölder’s Inequality) Let p(⋅), p1(⋅), p2(⋅) ∈ p(⋅) ∈ 𝓟(ℝn).

  1. For any fLp(⋅)(ℝn) and gLp(⋅)(ℝn),

    Rn|f(x)g(x)|dxCpfLp()gLp(),

    where Cp = 1 + 1/p − 1/p+.

  2. For any fLp1(⋅)(ℝn) and gLp2()(Rn), when 1/p(x) = 1/p1(x) + 1/p2(x), we have

    f(x)g(x)Lp()(Rn)Cp1,p2fLp1()gLp2(),

    where Cp1, p2 = (1 + 1/p1 − 1/p1+)1/p.

Lemma 2.2

[22] If pi ∈ 𝓑(ℝn) (i = 1, 2), then there exist constants pi, 1, pi, 2, C > 0, such that for all balls B ∈ ℝn and all measurable subsets SB,

χBLpi()(Rn)χSLpi()(Rn)|B||S|,χSLpi()(Rn)χBLpi()(Rn)|S||B|δi,1,χSLpi()(Rn)χBLpi()(Rn)|S||B|δi,2.

Lemma 2.3

[30] If pi(⋅) ∈ 𝓑(ℝn), then there exists a constant C > 0 such that for all balls B ∈ ℝn,

1|B|χBLp()(Rn)χBLp()(Rn)C.

Lemma 2.4

[24] Let p(⋅), q1(⋅) ∈ 𝓟(ℝn). If fLp(⋅)q(⋅)(ℝn), then

min(fLp()q()q+,fLp()q()q)|f|q()Lp()max(fLp()q()q+,fLp()q()q).

Lemma 2.5

[30] Let b ∈ BMO and m be a positive integer. There exists a constant C > 0, such that for any k, j ∈ ℤ with k > j,

  1. C−1 bmsupB1χBLp()(Rn) ∥(bbB)mχBLp(⋅)(ℝn)C bm;

  2. ∥(bbBj)mχBkLp(⋅)(ℝn)C(kj)m bmχBkLp(⋅)(ℝn).

Lemma 2.6

[36] If p(⋅) ∈ 𝓑(ℝn), then there exist constants 0 < δ < 1, C > 0, such that for all Y ∈ 𝓣, all nonnegative numbers tQ and all fLloc1(ℝn) with fQ ≠ 0 (QY),

QYtQffQδχQLp()(Rn)CQYtQχQLp()(Rn),

where 𝓣 denotes all families of disjoint and open cube in ℝn.

Lemma 2.7

[24] Let b ∈ Lipβ(ℝn) and m be a positive integer. There exists a constant C > 0, such that for any k, j ∈ ℤ with k > j

  1. C−1 bLipβm1|B|mβ/nχBLp()(Rn)1∥(bbB)mχBLp(⋅)(ℝn)C bLipβm;

  2. ∥(bbBj)mχBkLp(⋅)(ℝn)C|Bk|/n bLipβmχBLp(⋅)(ℝn).

Lemma 2.8

[37] Let f ∈ Lipβ(ℝn), 0 < β < 1, 1 ≤ p < ∞, B1B2. We have

  1. fLipβ ≈ supB 1|B|β/n(1|B|B|f(x)fB|pdx)1/p;

  2. |fB1fB2| ≤ CfLipβ|B2|β/n.

Lemma 2.9

[24] Let k ∈ ℕ, ak ≥ 0, 1 ≤ pk < ∞. Then

k=0akpk(k=0ak)p,wherep=minkNpk,k=0ak1;maxkNpk,k=0ak>1.

Lemma 2.10

[27] Suppose that Tθ is a θ-type Calderón-Zygmund operator and θ satisfies (1, 1). Let p(⋅) ∈ 𝓑(ℝn), then there exits a constant C independent of f such that

Tθ(f)Lp()(Rn)CfLp()(Rn).

Lemma 2.11

[27] Let b ∈ BMO(ℝn). Suppose that p(⋅) ∈ 𝓑(ℝn) and θ satisfies (1.6), then there exists a constant C independent of f such that

[b,Tθ](f)Lp()(Rn)CbfLp()(Rn).

Lemma 2.12

[27] Let b ∈ Lipβ(ℝn) for 0 < β < 1. Suppose that θ satisfies (1, 1) and p(⋅) ∈ 𝓟(ℝn) be such that p+ < nβ. Define q(⋅) by

1p(x)1q(x)=βn.

If q(⋅)(1 − βn) ∈ 𝓑(ℝn), then there exists a constant C independent of f such that

[b,Tθ](f)Lq()(Rn)CbLipβ(Rn)fLp()(Rn).

3 Proof of Main Theorems

Proof of Theorem 1.1

Let fK˙p()α,q1()(Rn). Write

f(x)=j=f(x)χj(x)j=fj(x).

Due to p(⋅)/q2(⋅) ∈ 𝓟(ℝn), which implies inf(p(⋅)/q2(⋅)) ≥ 1, then we have

Tθ(f)K˙p()α,q2()(Rn)=infη>0:k=2kα|Tθ(f)χk|ηq2()Lp()q2()1.

Since

2kα|Tθ(f)χk|ηq2()Lp()q2()C2kα|j=k3Tθ(fj)χk|η11q2()Lp()q2()+C2kα|j=k2k+2Tθ(fj)χk|η12q2()Lp()q2()+C2kα|j=k+3Tθ(fj)χk|η13q2()Lp()q2(),

where

η11=2kα|j=k3Tθ(fj)χk|k=lq2()(Lp()),η12=2kα|j=k2k+2Tθ(fj)χk|k=lq2()(Lp()),η13=2kα|j=k+3Tθ(fj)χk|k=lq2()(Lp())andη=η11+η12+η13.

Hence,

Tθ(f)K˙p()α,q2()(Rn)C(η11+η12+η13).

This implies that, in order to prove our theorem, we only need to show η11, η12, η13CfK˙p()α,q1()(Rn). For simplicity, we denote λ0 = fK˙p()α,q1()(Rn).

First, we consider η12. By Lemma 2.4 and using Aoki-Rolewicz’s theorem, we have

k=2kα|j=k2k+2Tθ(fj)χk|λ0q2()Lp()q2()k=2kα|j=k2k+2Tθ(fj)χk|λ0Lp()(q21)kCk=j=k2k+22kα|Tθ(fj)χk|λ0Lp()(q21)k,

where

(q21)k=(q2),2kα|j=k2k+2Tθ(fj)χk|λ0q2()Lp()q2()1,(q2)+,2kα|j=k2k+2Tθ(fj)χk|λ0q2()Lp()q2()>1.

By using Lemma 2.10, it follows

k=2kα|j=k2k+2Tθ(fj)χk|λ0q2()Lp()q2()Ck=j=k2k+22kα|fj|λ0Lp()(q21)kCk=2kα|fχk|λ0Lp()(q21)k.

Since fK˙p()α,q1()(Rn), we can easily see 2kα|fχk|λ0Lp() ≤ 1 and k=2kα|fχk|λ0q1()Lp()q1() ≤ 1. Hence, by Lemma 2.4 and Lemma 2.9, we have

k=2kα|j=k2k+2Tθ(fj)χk|λ0q2()Lp()q2()Ck=2kα|fχk|λ0q1()Lp()q1()(q21)k(q1)+Ck=2kα|fχk|λ0q1()Lp()q1()qC.

Here (q1)+ ≤ (q2)(q21)kandq=minkZ(q21)k(q1)+. Consequently, we have η120.

Now let us turn to estimate η11.

|Tθ(fj)(x)|=|RnK(x,y)fj(y)dy|CRn1|xy|n|fj(y)|dy.

Let xCk, yCj, jk − 3, then we have |xy| ≥ |x| − |y| ≥ 38|x|. Hence, we have

|Tθ(fj)(x)|C|x|nfjL1(Rn).

Thus, By Lemmas 2.1-2.4 and the fact that 2jα|fχj|λ0q1()Lp()q1() ≤ 1, it follows that

k=2kα|j=k3Tθ(fj)χk|λ0q2()Lp()q2()Ck=2kα|j=k3|x|nfjL1(Rn)χk|λ0q2()Lp()q2()Ck=2kα|j=k3|x|nfjL1(Rn)χk|λ0Lp()(q22)kCk=2k(αn)j=k3fjλ0Lp(Rn)χjLp()χkLp()(q22)kCk=2k(αn)j=k3fjλ0Lp(Rn)χBkLp()(Rn)χBjLp()(Rn)|Bk|(q22)kCk=j=k32(kj)(αnδ11)|2jαfχj|λ0q1()Lp()q1()(Rn)1(q1)+(q22)k,

where

(q22)k=(q2),2kα|j=k3Tθ(fj)χk|λ0q2()Lp()q2()1,(q2)+,2kα|j=k3Tθ(fj)χk|λ0q2()Lp()q2()>1.

If (q1)+ < 1, then by Lemma 2.9 and the fact (q1)+ ≤ (q2)(q21)k, we have

k=2kα|j=k3Tθ(fj)χk|λ0q2()Lp()q2()Ck=j=k32(kj)(αnδ11)|2jαfχj|λ0q1()Lp()q1()(Rn)(q22)k(q1)+Cj=|2jαfχj|λ0q1()Lp()q1()(Rn)k=j+32(kj)(αnδ11)qC,

where q=minkZ(q22)k(q1)+.

If (q1)+ ≥ 1, then 1 ≤ (q1)+ ≤ (q2)(q21)k. Thus, for α < 11, by applying Hölder’s inequality and Lemma 2.9, we get

k=2kα|j=k3Tθ(fj)χk|λ0q2()Lp()q2()Ck=j=k32(kj)(αnδ11)(q1)+/2|2jαfχj|λ0q1()Lp()q1()(Rn)(q22)k(q1)+×j=k32(kj)(αnδ11)((q1)+)/2(q22)k((q1)+)Cj=|2jαfχj|λ0q1()Lp()q1()(Rn)k=j+32(kj)(αnδ11)(q1)+/2qC,

where q=minkZ(q22)k(q1)+. This implies that η210.

Finally, we are going to estimate η23. Let xCk, yCj, jk + 3, then we have |xy| ≥ |y| − |x| ≥ 38|y|.

Hence, we have

|Tθ(fj)(x)|C2jnfjL1(Rn).

Thus, By Lemmas 2.1-2.4 and the fact that 2jα|fχj|λ0q1()Lp()q1() ≤ 1, it follows that

k=2kα|j=k+3Tθ(fj)χk|λ0q2()Lp()q2()Ck=2kα|j=k+32jnfjL1(Rn)χk|λ0q2()Lp()q2()Ck=2kαj=k+32jnfjλ0Lp(Rn)χjLp()χkLp()(q23)kCk=2kαj=k+32jnfχjλ0Lp(Rn)χBkLp()(Rn)χBjLp()(Rn)|Bj|(q23)kCk=j=k+32(kj)(α+nδ12)|2jαfχj|λ0q1()Lp()q1()(Rn)1(q1)+(q23)k,

where

(q23)k=(q2),2kα|j=k+3Tθ(fj)χk|λ0q2()Lp()q2()1,(q2)+,2kα|j=j+3Tθ(fj)χk|λ0q2()Lp()q2()>1.

Noticing that (q2) ≥ (q1)+ and α > −12, by a similar argument about η11, we have η130.

Combing the estimates of η11, η12 and η13, we finish the proof of Theorem 1.1. □

Proof of Theorem 1.2

Let b ∈ BMO(ℝn), fK˙p()α,q1()(Rn). As in the proof of Theorem 1.1, we write

f(x)=j=f(x)χj(x)j=fj(x).

Since

2kα|[b,Tθ](f)χk|ηq2()Lp()q2()C2kα|j=k3[b,Tθ](fj)χk|η21q2()Lp()q2()+C2kα|j=k2k+2[b,Tθ](fj)χk|η22q2()Lp()q2()+C2kα|j=k+3[b,Tθ](fj)χk|η23q2()Lp()q2(),

where

η21={2kα|j=k3[b,Tθ](fj)χk|}k=lq2()(Lp()),η22={2kα|j=k2k+2[b,Tθ](fj)χk|}k=lq2()(Lp()),η23={2kα|j=k+3[b,Tθ](fj)χk|}k=lq2()(Lp())andη=η21+η22+η23.

Hence, it is enough to prove η21, η22, η23Cbλ0, where λ0 = fK˙p()α,q1()(Rn).

We first estimate η22. Noticing that [b, Tθ] is bounded on Lp(⋅)(ℝn) (see Lemma 2.11), as the same argument about in η12 in the proof of Theorem 1.1, we immediately get

k=2kα|j=k2k+2[b,Tθ](fj)χk|λ0bq2()Lp()q2()C.

That is to say, η22Cbλ0.

Next, we estimate η21. Let xCk, jk − 3, suppfjCj. By the estimation of Tθ(fj) in the proof of Theorem 1.1, we have

|Tθ(fj)(x)|C|x|nfjL1(Rn).

Then, it follows that

|[b,Tθ]fj(x)|=|Tθ[(b(x)b)fj](x)|C|x|n(b()b)fjL1(Rn).

Thus, using Lemma 2.4 and Aoki-Rolewicz’s theorem, we obtain

k=2kα|j=k3[b,Tθ](fj)χk|λ0bq2()Lp()q2()Ck=2kα|j=k3|x|n(b()b)fjL1(Rn)χk|λ0bq2()Lp()q2()Ck=2kα|j=k3|x|n(b()b)fjL1(Rn)χk|λ0bLp()(q22)kCk=2k(αn)j=k3|(bbj)fj|λ0bL1(Rn)χkLp()(q22)kCk=2k(αn)j=k3|fj|λ0Lp(Rn)1b(bbj)χBjLp()χkLp()(q22)k,

where bj = bCj = 1|Cj| Cjb(y)dy and

(q22)k=(q2),2kα|j=k3[b,Tθ](fj)χk|λ0bq2()Lp()q2()1,(q2)+,2kα|j=k3[b,Tθ]χk|λ0bq2()Lp()q2()>1.

Applying the Hölder’s inequality and Lemma 2.5 with the case m = 1, we know that

k=2kα|j=k3[b,Tθ](fj)χk|λ0bq2()Lp()q2()Ck=2k(αn)j=k3|fj|λ0Lp(Rn)1b(bbj)χBjLp()χkLp()(q22)kCk=2k(αn)j=k3|fj|λ0Lp(Rn)χjLp()(kj)χBkLp()(q22)kCk=2k(αn)j=k3(kj)|Bk|χBjLp()χBkLp()|fj|λ0Lp(Rn)(q22)k.

Furthermore, by the same argument as η11 in the proof of Theorem 1.1, we have

k=2kα|j=k3[b,Tθ](fj)χk|λ0bq2()Lp()q2()Ck=j=k3(kj)2(kj)(αnδ11)|fχj|λ0q1()Lp(Rn)1(q1)+(q22)k.Cj=|2jαfχj|λ0q1()Lp()q1()k=j+3(kj)2(kj)(αnδ11)q,(p)+1,j=|2jαfχj|λ0q1()Lp()q1()k=j+3(kj)2(kj)(αnδ11)(q1)+/2q,(p)+>1.

C,

where q=minkZ(q22)k(q1)+. Therefore, η21Cbλ0.

Finally, we estimate η23. Let xCk, yCj, jk + 3, supp fjCj. By the estimation of the Tθ(fj) in the proof of Theorem 1.1, we have

|Tθ(fj)(x)|C2jnfjL1(Rn).

From this, it follows

|[b,Tθ]fj(x)|=|Tθ[(b(x)b)fj](x)|C2jn(b()b)fjL1(Rn).

Thus, when α > −12, as in the similar way to estimate η21 before, we obtain

k=2kα|j=k+3[b,Tθ](fj)χk|λ0bq2()Lp()q2()Ck=2kα|j=k+32jn(b()b)fjL1(Rn)χk|λ0bLp()(q23)kCk=2kαj=k+32jn|(b()b)fj|λ0bL1(Rn)χkLp()(q23)kCk=2kαj=k32jn|fj|λ0Lp(Rn)1b(bbk)χBjLp()χkLp()(q23)kCk=2kαj=k+32jn(kj)|Bj|χBkLp()χBjLp()|fj|λ0Lp(Rn)(q23)kCk=j=k+3(jk)2(kj)(α+nδ12)|fχj|λ0q1()Lp()(Rn)1(q1)+(q23)kC,

where

(q23)k=(q2),2kα|j=k+3[b,Tθ](fj)χk|λ0bq2()Lp()q2()1,(q2)+,2kα|j=k+3[b,Tθ]χk|λ0bq2()Lp()q2()>1.

Hence,

η23Cbλ0.

Combing the estimates of η21, η22 and η23, we finish the proof of Theorem 1.2. □

Proof of Theorem 1.3

Let b ∈ Lipβ(ℝn), 0 < β < 1, fK˙p1()α,q1()(Rn). Write

f(x)=j=f(x)χj(x)j=fj(x).

We use similar notations as in the proof of Theorem 1.2. Let

η31={2kα|j=k3[b,Tθ](fj)χk|}k=lq2()(Lp2()),η32={2kα|j=k2k+2[b,Tθ](fj)χk|}k=lq2()(Lp2()),η33=2kα|j=k+3[b,Tθ](fj)χk|k=lq2()(Lp2())andη=η31+η32+η33.

Similarly, we have

[b,Tθ](f)K˙p2()α,q2()(Rn)Cη=C(η31+η32+η33).

We are now going to estimate η31, η32, η33. For simplicity, we also denote λ0 = fK˙p1()α,q1()(Rn).

We first estimate η32. Noticing that [b, Tθ] is bounded Lp1(⋅)(ℝn) to Lp2(⋅)(ℝn) (Lemma 2.12), as the same argument about in η12 in the proof of Theorem 1.1, we immediately get

k=2kα|j=k2k+2[b,Tθ](fj)χk|λ0bLipβ(Rn)q2()Lp2()q2()C.

Thus, η32CbLipβ(ℝn)λ0.

Now let us turn to estimate η31. Let xCk, jk − 3, suppfjCj. By the estimation of η21 in the proof of Theorem 1.2 and the generalized Hölder’s inequality, we have

k=2kα|j=k3[b,Tθ](fj)χk|λ0bLipβ(Rn)q2()Lp2()q2()Ck=2k(αn)j=k3|(bbj)fj|λ0bLipβ(Rn)Lp1(Rn)χkLp2()(q22)kCk=2k(αn)j=k3fjλ0Lp1(Rn)(bbj)χBkLp2()bLipβ(Rn)(q22)kCk=2k(αn)j=k3|fj|λ0Lp1(Rn)1bLipβ(Rn)(bbj)χBjLp1()χkLp2()(q22)kCk=2k(αn)j=k3|fj|λ0Lp1(Rn)χBkLp2()(bbj)χBjLp1()bLipβ(Rn)(q22)k,

where

(q22)k=(q2),2kα|j=k3[b,Tθ](fj)χk|λ0bLipβ(Rn)q2()Lp1()q2()1,(q2)+,2kα|j=k3[b,Tθ]χk|λ0bLipβ(Rn)q2()Lp1()q2()>1.

As 1p1(x)1p2(x)=βn, we use the following fact (see [38])

C1|B|β/nχBLp2()(Rn)χBLp1()(Rn)C2|B|β/nχBLp2()(Rn).

Together with Lemma 2.2-2.4 and Lemma 2.7, we obtain

k=2kα|j=k3[b,Tθ](fj)χk|λ0bLipβ(Rn)q2()Lp2()q2()Ck=2k(αn)j=k3|fj|λ0Lp1(Rn)|Bj|β/n|Bk|β/nχBjLp1()χkLp1()(q22)kk=2k(αn)j=k3|fj|λ0Lp1(Rn)χBjLp1()χkLp1()(q22)kCk=2k(αn)j=k3|Bk|χBjLp1()χBkLp1()|fj|λ0Lp1(Rn)(q22)kCk=j=k32(kj)(αnδ11)|fj|λ0q1()Lp1(Rn)1(q1)+(q22)k.

Furthermore, by the same argument as η11 in the proof of Theorem 1.1, we get

k=2kα|j=k3[b,Tθ](fj)χk|λ0bLipβ(Rn)q2()Lp1()q2()C.

This implies that

η31CbLipβ(Rn)λ0.

Finally, we estimate η33. Let xCk, yCj, jk + 3, suppfjCj. By the estimation of the η23 in the proof of Theorem 1.2 and the generalized Hölder’s inequality, we have

k=2kα|j=k+3[b,Tθ](fj)χk|λ0q2()Lp2()q2()Ck=2kαj=k+32jn|(bbk)fj|λ0bLipβ(Rn)L1(Rn)χkLp2()(q23)kCk=2k(αn)j=k+32jn|fj|λ0L1(Rn)(bbk)χBkLp2()bLipβ(Rn)(q23)k,

where

(q23)k=(q2),2kα|j=k+3[b,Tθ](fj)χk|λ0q2()Lp1()q2()1,(q2)+,2kα|j=j+3[b,Tθ](fj)χk|λ0q2()Lp1()q2()>1.

Observe that 1/p1(x) − 1/p2(x) = β/n, which implies 1/p2(x)1/p1(x)=β/n. Hence, when α > −22, as argued about η31 before, one has

k=2kα|j=k+3[b,Tθ](fj)χk|λ0q2()Lp2()q2()Ck=2k(αn)j=k+32jn|Bk|χBkLp2()χBjLp2()|fj|λ0Lp1(Rn)(q23)kCk=j=k+32(kj)(α+nδ22)|fχj|λ0q1()Lp1()(Rn)1(q1)+(q23)kC.

Therefore,

η33CbLipβ(Rn)λ0.

Summing up the estimates of η31, η32, and η33, it follows that

[b,Tθ](f)K˙p2()α,q2()(Rn)CbLipβ(Rn)fK˙p1()α,q1()(Rn).

This finishes the proof of Theorem 1.3. □

Acknowledgement

The authors would like to express their deep thanks to the referee for his/her very careful reading and many valuable comments and suggestions. This work is supported by National Natural Science Foundation of China (Grant No. 11561062).

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About the article

Received: 2018-07-04

Accepted: 2018-11-17

Published Online: 2018-12-31


Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1607–1620, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0133.

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© 2018 Yang and Tao, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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