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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


Volume 13 (2015)

Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents

Yanqi Yang / Shuangping Tao
  • Corresponding author
  • College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, 730070, China
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Published Online: 2018-04-10 | DOI: https://doi.org/10.1515/math-2018-0036


Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.RnΩ(x,xy)|xy|nf(y)dy

and Dγ(0 ≤ γ ≤ 1) be the fractional differentiation operator. Let T and T be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TDγDγT and (TT)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents HMK˙p(),λα(),q via the convolution operator Tm, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on HMK˙p(),λα(),q(ℝn) is shown to hold for TDγDγT and (TT)Dγ. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1T2.

Keywords: Variable kernel; Fractional differentiation; Sobolev spaces Iγ(BMO); Morrey-Herz-type Hardy space with variable exponents

MSC 2010: 42B20; 42B25; 42B35

1 Introduction and main results

Let Ω(x, z) : ℝn × ℝn ⟶ ℝ be a measurable function and satisfy the following conditions: Ω(x,λz)=Ω(x,z),foranyx,zRnandλ>0,(1.1) Sn1Ω(x,z)dσ(z)=0,foranyxRn,(1.2)

where 𝕊n − 1 denotes the unit sphere in ℝn (n ≥ 2) with normalized Lebesgue measure dσ. Then the singular integral operator with variable kernel Ω(x, z) is defined by Tf(x)=p.v.RnΩ(x,xy)|xy|nf(y)dy.(1.3)

Boundedness properties of the above operator in a variety of functional spaces have been extensively studied. In particular, Calderón and Zygmund proved that T is bounded on the L2(ℝn) in the Mihlin conditions (see [1]). Other references with results of this sort include [2, 3, 4, 5] and the references within. On the other hand, these estimates played an important role in the theory of non-divergent elliptic equations with discontinuous coefficients (see [6, 7]).

Let 0 ≤ γ ≤ 1. For tempered distributions f ∈ 𝒮(ℝn)(n = 1, 2, …), the fractional differentiation operators Dγ of order γ are defined by Dγf^(ξ)=|ξ|γf^(ξ), i.e.,Dγf(x)=(|ξ|γf^(ξ))(x). We will denote by Iγ the Riesz potential operator of order γ that is defined on the space of tempered distributions modulo polynomials by setting Iγf^(ξ)=|ξ|γf^(ξ). It is easy to see that a locally integrable function b belongs to Iγ(BMO)(ℝn) if and only if Dγb ∈ BMO(ℝn). Strichartz (see [8]) showed that, for γ ∈ (0, 1), Iγ(BMO)(ℝn) is a space of functions modulo constants which is properly contained in Lipγ(ℝn).

We make some conventions. In what follows, χE denotes the characteristic function for a μ-measurable set E. We use the symbol AB to denote that there exsists a positive constant C such that ACB. For any index p ∈ (1, ∞), we denote by p′ its conjugate index, that is, 1p+1p=1.

Denote T and T to be the adjoint of T and the pseudo-adjoint of T respectively (see (3.2) and (3.3) below). Let T1 and T2 be the operators defined in (1.1) which are differentiateded by its kernel Ω1(x, y) and Ω2(x, y). Let T1T2, T1T2 denote the product and pseudo-product of T1 and T2, respectively. In 1957, Calderón and Zygmund found that these operators are closely related to the second order linear elliptic equations with variable coefficients and established the following results of the operators T1,T1,T1T2,T1T2 and D on Lp(ℝn)(1 < p < ∞) (see [1]).

Theorem A

([1]). Let 1 < p < ∞, Ω1(x, y), Ω2(x, y) ∈ Cβ(C), β > 1 satisfy (1.1) and (1.2). Then there is a constant C such that

  1. ∥(T1DDT1)fLp ≲ ∥fLp;

  2. (T1T1)DfLpfLp;

  3. ∥(T1T2T1T2)DfLp ≲ ∥fLp.

In 2015, Chen and Zhu proved that Theorem A was also true on Weighted Lebesgue space and Morrey space (see [9]). In 2016, Tao and Yang obtained the boundedness of those operators on the weighted Morrey-Herz spaces (see [5]), Later, the boundedness of those operators on the Lebesgue spaces with variable exponents were obtained [10]. Inspired by the ideas mentioned previously, the aim of this paper is to deal with the boundedness of the singular integrals with variable kernel and fractional differentiations in the setting of the Morrey-Herz-type Hardy Spaces with variable exponents (which will be defined in the next section).

The main theorems are presented in this section. The definitions of the Morrey-Herz spaces with variable exponents, the Morrey-Herz-type Hardy spaces with variable exponents and the preliminary lemmas are presented in Section 2. In Section 3, we will introduce the spherical harmonical expansions and give the boundedness of Tm, j. The proofs of Theorems are given in Section 4.

Theorem 1.1

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6. Assume that T is defined by (1.3) and Ω(x, y), which satisfies (1.1), (1.2), meet a condition max|j|2nDxγ(j/yj)Ω(x,y)L(Rn×Sn1)<,(1.4)

then we have

  1. (TDγDγT)fMK˙p(),λα(),qfHMK˙p(),λα(),q;

  2. (TT)DγfMK˙p(),λα(),qfHMK˙p(),λα(),q.

Theorem 1.2

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6. Suppose that Ω1(x, y) and Ω2(x, y) satisfy (1.1) and (1.2). If Ω2(x, y) satisfies (1.4) and Ω1(x, y) satisfies max|j|2n(j/yj)Ω1(x,y)L(Rn×Sn1)<,(1.5)

then we have (T1T2T1T2)DγfMK˙p(),λα(),qfHMK˙p(),λα(),q.

Theorem 1.3

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6. Suppose that Ωi(x, y)(i = 1, 2) satisfies (1.1), (1.2) and (1.5), then we have

  1. ∥(T1𝓘 − 𝓘T1)f MK˙p(),λα(),qfHMK˙p(),λα(),q;

  2. (T1T2)IfMK˙p(),λα(),qfHMK˙p(),λα(),q;

  3. ∥(T1T2T1T2)𝓘f MK˙p(),λα(),qfHMK˙p(),λα(),q;

Theorem 1.4

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6. Suppose that Ω(x, y) satisfies(1.1), (1.2) and max|j|2nx(j/yj)Ω(x,y)L(Rn×Sn1)<,(1.6)

then we have

  1. ∥(TDDT)f MK˙p(),λα(),qfHMK˙p(),λα(),q;

  2. ∥(TT)Df MK˙p(),λα(),qfHMK˙p(),λα(),q.

Theorem 1.5

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2 λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6. Suppose that Ω1(x, y) and Ω2(x, y) satisfies(1.1), (1.2). If Ω1(x, y) satisfies (1.5) and Ω2(x, y) satisfies (1.6), then we have (T1T2T1T2)DfMK˙p(),λα(),qfHMK˙p(),λα(),q.

2 Definitions and preliminaries

In this section, the Morrey-Herz spaces with variable exponents MK˙p(),λα(),q and the Morrey-Herz-type Hardy spaces with variable exponents HMK˙p(),λα(),q will be introduced. Some preliminary lemmas will be given as well.

Lebesgue spaces with variable exponent Lp(⋅)(ℝn) become one of important function spaces due to the fundamental paper [11] by Kovóčik Rákosník. In the past 20 years, the theory of these spaces have made progress rapidly. On the other hand, 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 (see [12, 13, 14, 15, 16]). In [17], authors proved the extrapolation theorem which leads to 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 [18]. The boundedness of some typical operators has been studied with keen interest (see [18, 19, 20, 21, 22, 23, 24, 25]). Recently, Xu and Yang have introduced the Morrey-Herz-type Hardy spaces with variable exponents and established the boundedness of singular integral operators on these spaces in [26].

Definition 2.1

([21]). Let α(⋅) be a real-valued function onn. If there exist C > 0 such that for any x, y ∈ ℝn, |xy| < 1/2, |α(x)α(y)|Clog(|xy|),

then α(⋅) is said to be local log-Hölder continuous onn.

If there exist C > 0 such that for all x ∈ ℝn, |α(x)α(0)|Clog(e+1/|x|),

then α(⋅) is said to be log-Hölder continuous at origin.

If there exist α ∈ ℝ and a constant C > 0 such that all x ∈ ℝn, |α(x)α|Clog(e+|x|),

then α(⋅) is said to be log-Hölder continuous at infinity.

Definition 2.2

([11]). Let p: ℝn ⟶ [1, ∞) be a measurable function. The Lebesgue space with variable exponent Lp(⋅)(ℝn) is defined by Lp()(Rn)=fismeasurable:Rn|f(x)|ηp(x)<forsomeconstantη>0.

Equipped with the Luxemburg-Nakano norm fLp()(Rn)=infη>0:Rn|f(x)|ηp(x)dx1

we denote p=essinf{p(x):xRn},p+=esssup{p(x):xRn}.

Then 𝓟(ℝn) consists of all p(⋅) satisfying p > 1 and p+ < ∞.

Let M be te Hardy-littlewood maximal operator. We denote 𝓑(ℝn) to be the set of all functions p(⋅) ∈ 𝓟(ℝn) satisfying the condition that M is bounded on Lp(⋅)(ℝn).

We now make some conventions. Throughout this paper, let k ∈ ℤ, Bk = {x ∈ ℝn : |x| ≤ 2k}, Ck = BkBk − 1, L ∈ ℤ and χk = χCk.

Definition 2.3

([26]). Let 0 < q ≤ ∞, p(⋅) ∈ 𝓟(ℝn), and 0 ≤ λ < ∞. Let α(⋅) be a bounded real-valued measurable function onn. The homogeneous Morrey-Herz space MK˙p(),λα(),q is defined by MK˙p(),λα(),q(Rn)={fLlocp()(Rn{0}):fMK˙p(),λα(),q(Rn)<},

where fMK˙p(),λα(),q(Rn)=supLZ2Lλk=L2α()kfχkLp()q1/q,

with the corresponding modification for q = ∞.

Next let us recall the definition of Morrey-Herz-type Hardy spaces with variable exponents HMK˙p(),λα(),q, which was firstly introduced by Xu and Yang in [26]. To do this, we need some natations. 𝓢(ℝn) denotes the Schwartz spaces of all rapidly decreasing infinitely differentiable functions on ℝn, and 𝓢(ℝn) denotes the dual space of 𝓢(ℝn). Let GNf be the grand maximal function of f defined by GNf(x)=supϕAN|ϕ(f)(x)|,xRn,

where AN={ϕS(Rn):sup|ν|,|β|N,xRn|xνDβϕ(x)|1} and N > n+1 and ϕ is the nontangential maximal operator defined by ϕ(f)(x)=sup|yx|<t|ϕtf(y)|, where ϕt(x)=tnϕ(xt) for any x ∈ ℝn and t > 0.

The grand maximal GN was firstly introduced by Fefferman and Stein in [27] to study classical Hardy spaces. We refer the reader to [28, 29, 30] for details on the classical Hardy spaces. The variable exponent case is shown in [22] by Nakai and Sawano.

Definition 2.4

([26]). Let α(⋅) ∈ L(ℝn), 0 < q ≤ ∞, p(⋅) ∈ 𝓟(ℝn), 0 ≤ λ < ∞, and N > n+1. The homogeneous Morrey-Herz-type Hardy space with variable exponents HMK˙p(),λα(),q is defined by HMK˙p(),λα(),q=fS(Rn):fHMK˙p(),λα(),q=GNfHMK˙p(),λα(),q<.

Obviously, if α(⋅) = α and λ = 0, these spaces were considered by Wang and Liu in [23]. If p(⋅) and α(⋅) are constant and λ = 0, these are the classical Herz type Hardy spaces (see [31]).

Definition 2.5

([26]). Let p(⋅) ∈ 𝓟(ℝn) and α(⋅) ∈ L(ℝn) be log-Hölder continuous both at the origin and infinity, and nonnegative integer s ≥ [αr2], here αr = α(0), if r < 1, and αr = α, if r ≥ 1, 2αr < ∞ and δ2 as in Lemma 2.6.

  1. A function a onn is called a central (α(⋅), p(⋅))-atom, if it satisfies (1) supp aB(0, r); (2)aLp(⋅) ≤ |B(0, r)|αr/n; (3)na(x)xβdx = 0, |β| ≤ s.

  2. A function a onn is called a central (α(⋅), p(⋅))-atom of restricted type, if it satisfies (1) supp aB(0, r), r ≥ 1; (2)aLp(⋅) ≤ |B(0, r)|αr/n; (3)na(x)xβdx = 0, |β| ≤ s.

Lemma 2.6

([18]). If p(⋅) ∈ 𝓑(ℝn), then there exist constants δ1, δ2 > 0, such that for all balls B ⊂ ℝn and all measurable subsets SB, χBLp()(Rn)χSLp()(Rn)|B||S|,χSLp()(Rn)χBLp()(Rn)|S||B|δ1,χSLp()(Rn)χBLp()(Rn)|S||B|δ2.

Lemma 2.7

([21, Theorem 13]). Let 0 < q < ∞, p(⋅) ∈ 𝓑(ℝn), 0 ≤ λ < ∞, and α(⋅) ∈ L be log-Hölder continuous both at the origin and infinity, 2λ ≤ α(⋅), 2α(0), α < ∞ with δ2 as in Lemma 2.5, then fHMK˙p(),λα(),q if and only if f=k=λkak in the sense of 𝓢(ℝn), where each ak is a central (α(⋅), p(⋅))-atom with support contained in Bk and supLZ2Lλk=L|λk|q<. Moreover, fHMK˙p(),λα(),qinfsupLZ2Lλk=L|λk|q1/q,

where the infimum is taken over all above decompositions of f.

Lemma 2.8

([11]). Let p(⋅) ∈ 𝓟(ℝn). If fLp(⋅) and gLp(⋅), then fg is integrable onn and Rn|f(x)g(x)|dxrpfLp()gLp(),

where rp = 1+1/p − 1/p+.

Lemma 2.9

([26]). Let α(⋅) be a bounded and log-Hölder continuous both at the origin and infinity such that nδ2α(0), α < 2+δ with some δ > max{α(0) − 2, α2} and δ2 as in Lemma 2.6. Suppose T is a Calderón-Zygmund operator associated to a standerd kernel K. If p(⋅) ∈ 𝓑(ℝn), 0 < λ < ∞ and 0 ≤ q < ∞, then we have TfMK˙p(),qα(),λfHMK˙p(),qα(),λ.

Lemma 2.10

([26]). Let p(⋅) ∈ 𝓟(ℝn), q ∈ (0, ∞], and λ ∈ (0, ∞]. If α()L(Rn)P0log(Rn)Plog(Rn), then fMK˙p(),qα(),λmax{supL0,LZ2Lλ(k=L2kqα(0)fχkLp()q)1/q,supL>0,LZ[2Lλ(k=12kqα(0)fχkLp()q)1/q+2Lλ(k=0L2kqαfχkLp()q)1/q]}.

3 Spherical harmonics and boundedness of Tm, l

In this section, we will recall the spherical harmonical expansion and give the boundedness of Tm, l, which are very vital in our proofs of Theorems.

We let 𝓗m denote the space of spherical harmonics homogeneous polynomials of degree m. Let dim𝓗m = dm and {Ym,j}j=1dm be an orthonormal system of 𝓗m. It is showed that {Ym,j}j=1dm, m = 0, 1, …, is a complete orthonormal system in L2(Sn − 1) (see [32]). We can expand the kernel Ω(x, z) in spherical harmonics as Ω(x,z)=m0j=1dmam,j(x)Ym,j(z),

where am,j(x)=Sn1Ω(x,z)Ym,j(z)¯dσ(z).

If ∫Sn − 1Ω(x, z)dσ(z) = 0, then a0, j = 0 for any x ∈ ℝn. Let Tm,jf(x)=(Ym,j||nf)(x).(3.1)

Then T, defined in (1.3), can be written as Tf(x)=m1j=1dmam,j(x)Tm,jf(x).

Let T and T be the adjoint of T and the pseudo-adjoint of T respectively, defined by Tf(x)=m=1j=1dm(1)mTm,j(a¯m,jf)(x)(3.2)

and Tf(x)=m=1j=1dm(1)ma¯m,j(x)Tm,jf(x).(3.3)

Lemma 3.1

Let α(⋅) ∈ L be log-Hölder continuous both at the origin and infinity, 2λ ≤ α(⋅), 2α(0), α < 2+δ with some δ > max{α(0) − 2, α2} and δ2 as in Lemma 2.6. The Tm, j defined by (3.1) is bounded from HMK˙p(),qα(),λtoMK˙p(),qα(),λ, i.e. Tm,jfMK˙p(),qα(),λfHMK˙p(),qα(),λ.


Suppose fMK˙p(),λα(),q. By Lemma 2.7, f=j=λjbj converges in 𝓢(ℝn), where each bj is a central (α(⋅), q(⋅)) – atom with support contained in Bj and fHMK˙p(),λα(),qinfsupLZ2Lλj=L|λj|q1q.

For simplicity, we denote Φ=LZ2Lλj=L|λj|q. By Lemma 2.10, we have Tm,lfMK˙p(),λα(),qqmax{supL0,LZ2Lλqk=L2kqα(0)(Tm,lf)χkLp()q,supL>0,LZ2Lλq(k=12kqα(0)(Tm,lf)χkLp()q+k=0L2kqα()(Tm,lf)χkLp()q)}=:max{I,II+III},

where I=supL0,LZ2Lλqk=L2kqα(0)(Tm,lf)χkLp()q,II=supL>0,LZk=12kqα(0)(Tm,lf)χkLp()q,III=supL>0,LZ2Lλqk=0L2kqα()(Tm,lf)χkLp()q.

To complete our proof, we only need to show that I, II, IIImnq/2Φ. First, we estimate I: I=supL0,LZ2Lλqk=L2kqα(0)(Tm,lf)χkLp()qsupL0,LZ2Lλqk=L2kqα(0)j=k|λj|[Tm,lbj]χkLp()q+supL0,LZ2Lλqk=L2kqα(0)j=k1|λj|[Tm,lbj]χkLp()q=:I1+I2.

By the result that Tm, l is bounded on Lp(⋅)(ℝn) (see [10]), we have [(Tm,lbl)χk]Lp()mn/2bjLp()mn/2|Bj|αj/n=mn/22αjj.

Therefore, when we get 0 < q ≤ 1, we get I1supL0,LZ2Lλqk=L2kqα(0)j=k|λj|mn/22αjjqmnq/2supL0,LZ2Lλq×k=L2kqα(0)j=k1|λj|q2α(0)jq+j=0|λj|q2αjqmnq/2supL0,LZ2Lλqk=Lj=k1|λj|q2α(0)(kj)q+mnq/2supL0,LZ2Lλqk=L2α(0)kqj=0|λj|q2αjqmnq/2supL0,LZ2Lλqj=1|λj|qk=j2α(0)(kj)q+mnq/2supL0,LZ2jλq|λj|q2(λα)jq2Lλqk=L2α(0)kqmnq/2supL0,LZ2Lλqj=L|λj|q+mnq/2supL0,LZ2Lλqj=L1|λj|qk=j2α(0)(kj)q+mnq/2ΦsupL0,LZj=02(λα)jqk=L2α(0)kqLλqmnq/2Φ+mnq/2supL0,LZj=L12jλq|λj|q2(jL)λqk=j2α(0)(kj)q+mnq/2Φmnq/2Φ+mnq/2ΦsupL0,LZj=L12(jL)λqk=j2α(0)(kj)qmnq/2Φ.

As 1 < q < ∞, we can obtain I1mnq/2supL0,LZ2Lλqk=L2kqα(0)j=k|λj|2αjjqmnq/2supL0,LZ2Lλqk=Lj=k1|λj|2α(0)(kj)q+mnq/2supL0,LZ2Lλqk=L2α(0)kqj=0|λj|2α(0)jqmnq/2supL0,LZ2Lλqk=Lj=k1|λj|q2α(0)(kj)q/2×j=k12α(0)(kj)q/2q/q+mnq/2supL0,LZ2Lλqk=L2α(0)kqj=0|λj|q2αjq/2×j=02αjq/2q/qmnq/2supL0,LZ2Lλqk=Lj=k1|λj|q2α(0)(kj)q/2+mnq/2supL0,LZ2Lλqk=L2α(0)kqj=0|λj|q2αjqmnq/2supL0,LZ2Lλqj=1|λj|qk=j2α(0)(kj)q/2+mnq/2supL0,LZj=02jλq|λj|q2(λα/2)jq2Lλqk=L2α(0)kqmnq/2supL0,LZ2Lλqj=L|λj|q+mnq/2supL0,LZ2Lλqj=L1|λj|qk=j2α(0)(kj)q/2+mnq/2ΦsupL0,LZj=02(λα/2)jqk=L2α(0)kqLλqmnq/22Lλqj=L|λj|q+mnq/22LλqsupL0,LZj=L1|λj|qk=j2α(0)(kj)q/2+mnq/2ΦsupL0,LZj=02(λα/2)jqk=L2α(0)kqLλqmnq/2Φ+mnq/2ΦsupL0,LZj=L12jλq|λj|q2(jL)λqk=j2α(0)(kj)q/2+mnq/2Φmnq/2Φ+mnq/2ΦsupL0,LZj=L12(jL)λqk=j2α(0)(kj)q/2mnq/2Φ.

Hence, we have I1mnq/2Φ.

Secondly, we estimate I2. A simple computation shows that there exists a constant δ > 0 such that Tm, l satisfies the following size condition |Tm,lf|mn/2(diam(suppf))δ|x|(n+δ)f1,whendist(x,suppf)|x|2,

and with the help of Lemma 2.8, we get |Tm,lbj(x)|mn/2|x|(n+δ)2jδBj|bj(y)|dymn/22k(n+δ)2jδbjLp()χBjLp()mn/22j(δαj)k(δ+n)χBjLp().

So by Lemma 2.6 and 2.8, we have (Tm,lbj)χkLp()mn/22j(δαj)k(δ+n)χBjLp()χBkLp()mn/22j(δαj)kδ2kn|Bk|χBkLp()1χBjLp()mn/22j(δαj)kδχBjLp()χBkLp()mn/22(δ+nδ2)(jk)jαj.

Therefore, when 0 < q ≤ 1, by 2α(0) < δ+n δ2 we get I2=supL0,LZ2Lλqk=L2kqα(0)j=k1|λj|(Tm,lbj)χkLp()qmnq/2supL0,LZ2Lλqk=L2kqα(0)j=k1|λj|q2[(δ+nδ2)(jk)jα(0)]qmnq/2supL0,LZ2Lλqj=L|λj|qk=j+112(jk)(δ+nδ2α(0))qmnq/2Φ.

When 1 < q < ∞, let 1/q+1/q = 1. Since 2α(0) ≤ δ+2, by Hölder′s inequality, we have I2mnq/2supL0,LZ2Lλqk=L2α(0)kqj=k1|λj|2(δ+nδ2)(jk)jα(0)qmnq/2supL0,LZk=L2α(0)kqj=k1|λj|q2(jk)(δ+nδ2α(0))q/2×j=k12(jk)(δ+nδ2α(0))q/2q/qmnq/2supL0,LZ2Lλqj=L2α(0)kqj=k1|λj|q2(jk)(δ+nδ2α(0))q/2=mnq/2supL0,LZ2Lλqj=L|λj|qk=j+112(jk)(δ+nδ2α(0))q/2mnq/2Φ.

Hence, we have Imnq/2Φ.

Thirdly, we estimate II. Consider IIk=12kqα(0)j=k|λj|(Tm,lbj)χkLp()q+k=12kqα(0)j=k1|λj|(Tm,lbj)χkLp()q=:II1+II2.

When 0 < q ≤ 1, we get II1mnq/2k=12kqα(0)j=k|λj|2αjjqmnq/2k=12kqα(0)j=k1|λj|q2α(0)jq+j=0|λj|q2αjqmnq/2k=1j=k1|λj|q2α(0)(kj)q+mnq/2k=12α(0)kqj=0|λj|q2αjqmnq/2j=1|λj|qk=j2α(0)(kj)q+mnq/2j=0|λj|q2αjqk=12α(0)kqmnq/2j=1|λj|q+mnq/2j=02jλq|λj|q2αjqk=12α(0)kqmnq/2Φ+mnq/2Φi=j|λi|qj=02(λα)jqk=j2α(0)kqmnq/2Φ.

As 1 < q < ∞, we obtain II1=k=12kqα(0)j=k|λj|(Tm,lbj)χkLp()qmnq/2k=12kqα(0)j=k|λj|2αjjqmnq/2k=1j=k1|λj|2α(0)(kj)q+k=12kqα(0)j=0|λj|2αjqmnq/2k=1j=k1|λj|q2α(0)(kj)q/2j=k12α(0)(kj)q/2q/q+mnq/22α(0)kqj=0|λj|q2αjq/2j=02αjq/2q/qmnq/2k=1|λj|qk=j2α(0)(kj)q/2+mnq/2j=0|λj|q2αjq/2k=12α(0)kqmnq/2k=1|λj|q+mnq/2j=02(λα/2)jq2jλqi=j|λi|qk=12α(0)kqmnq/2Φ+mnq/2Φj=02(λα/2)jqk=12α(0)kqmnq/2Φ.

For II2, as 0 < q ≤ 1, noting that 2α(0) < δ+2, we have II2=k=12kqα(0)j=k1|λj|(Tm,lbj)χkLp()qmnq/2k=12kqα(0)j=k1|λj|q2[(δ+nδ2)(jk)jα(0)]q=mnq/2k=1|λj|qk=j+112(jk)(δ+nδ2α(0))qmnq/2k=1|λj|qmnq/2Φ.

As 1 < q < ∞ and 2α(0) < δ+2, by Hölder′s inequality, one has II2mnq/2k=12kqα(0)j=k1|λj|2(δ+nδ2)(jk)jα(0)qmnq/2j=k1|λj|q2(jk)(δ+nδ2α(0))q/2×j=k12(jk)(δ+nδ2α(0))q/2q/qmnq/2j=12α(0)kqj=k1|λj|q2(jk)(δ+nδ2α(0))q/2=mnq/2j=1|λj|qk=j+112(jk)(δ+nδ2α(0))q/2mnq/2Φ.

So, we have IImnq/2Φ.

Finally, we estimate III. Write IIIsupL>0,LZ2Lλqk=0L2kqαj=k|λj|(Tm,lbj)χkLp()q+supL>0,LZ2Lλqk=0L2kqαj=k1|λj|(Tm,lbj)χkLp()q=:III1+III2.

When 0 < q ≤ 1, by the boundedness of Tm, l in Lp(⋅)(ℝn) (see [10]), we obtain III1supL>0,LZ2Lλqk=0L2kqαj=k|λj|q(Tm,lbj)χkLp()qmnq/2supL>0,LZ2Lλqk=0L2kqαj=k|λj|q2αjjqmnq/2supL>0,LZ2Lλqk=0L2kqαj=k|λj|q2αjq=mnq/2supL>0,LZ2Lλqj=0L|λj|qk=0j2(kj)αq+mnq/2supL>0,LZ2Lλqj=L|λj|qk=0L2(kj)αqmnq/2supL>0,LZ2Lλqj=0L|λj|q+mnq/2supL0,LZj=L2(jλqLλq)2jλqi=j|λi|qk=0L2(kj)αqmnq/2Φ+mnq/2ΦsupL>0,LZj=L2(Lj)αq2(jL)λqmnq/2Φ+mnq/2ΦsupL>0,LZj=L2(jL)q(λα)mnq/2Φ.

As 1 < q < ∞, by the boundedness of Tm, l in Lp(⋅) (see [10]) and Hölder′s inequality, we have III1supL>0,LZ2Lλqj=k|λj|q(Tm,lbj)χkLp()q/2×j=k(Tm,lbj)χkLp()q/2q/qmnq/2supL>0,LZk=0L2Lλqj=k|λj|qbjLp()q/2×j=kbjLp()q/2q/qmnq/2supL>0,LZk=0L2Lλqj=k|λj|q|Bj|αjq/(2n)×j=k|Bj|αjq/(2n)q/qmnq/2supL>0,LZk=0L2Lλqj=k|λj|q|Bj|αjq/(2n)=mnq/2supL>0,LZ2Lλqj=0L|λj|qk=0L2(kj)αq/2+mnq/2supL>0,LZ2Lλqj=L|λj|qk=0L2(kj)αq/2mnq/2supL>0,LZ2Lλqj=0L|λj|q+mnq/2supL0,LZ2Lλqj=L2(jλqLλq)2jλqi=j|λi|qk=0L2(kj)αq/2mnq/2Φ+mnq/2ΦsupL0,LZj=L2(jL)λq2(Lj)αq/2mnq/2Φ+mnq/2ΦsupL0,LZj=L2(jL)q(λα/2)mnq/2Φ.

For III2, as 0 < q ≤ 1 and 2α(0), α < δ+2, we get III2supL>0,LZ2Lλqk=0L2kqαj=1|λj|q2[(δ+nδ2)(jk)jαj]qmnq/2supL>0,LZ2Lλqk=0L2kqαj=1|λj|q2[(δ+nδ2)(jk)jα(0)]q+mnq/2supL>0,LZ2Lλqk=0L2kqαj=0k1|λj|q2[(δ+nδ2)(jk)jα]qmnq/2supL>0,LZ2Lλqk=0L2[α(δ+nδ2)]kq×j=1|λj|q2(δ+nδ2α(0))jq+mnq/2supL>0,LZ2Lλqk=0L|λj|qk=j+12(jk)(δ+nδ2α)qmnq/2supL>0,LZ2Lλqj=1|λj|q+mnq/2supL>0,LZ2Lλqj=0L1|λj|qmnq/2Φ.

As 1 < q < ∞ and 2α(0), αδ+2, by Hölder′s inequality, we have III2mnq/2supL>0,LZ2Lλqk=0L2kqαj=k1|λj|2[(δ+nδ2)(jk)jαj]qmnq/2supL>0,LZ2Lλqk=0L2kqαj=1|λj|2[(δ+nδ2)(jk)jα(0)]q+mnq/2supL0,LZ2Lλqk=0L2kqαj=0k1|λj|2[(δ+nδ2)(jk)jα]qmnq/2supL>0,LZ2Lλqk=0L2[α(δ+nδ2)]kq×j=12(δ+nδ2α(0))jq+mnq/2supL>0,LZ2Lλqk=0Lj=0k1|λj|2(jk)(δ+nδ2α)qmnq/2supL>0,LZ2Lλqj=1|λj|q2(δ+nδ2α(0))jq/2j=12(δ+nδ2α(0))jq/2q/q+mnq/2supL>0,LZ2Lλqk=0Lj=0k1|λj|q2(jk)(δ+nδ2α)q/2j=0k12(jk)(δ+nδ2α)q/2q/qmnq/2supL>0,LZ2Lλqj=1|λj|q2(δ+nδ2α(0))jq/2+mnq/2supL>0,LZ2Lλqk=0Lj=0k1|λj|q2(jk)(δ+nδ2α)q/2mnq/2supL>0,LZ2Lλqj=1|λj|q+mnq/2supL>0,LZ2Lλqj=0L1|λj|qk=j+1L2(jk)(δ+nδ2α)q/2mnq/2supL>0,LZ2Lλqj=1|λj|q+mnq/2supL>0,LZ2Lλqj=0L1|λj|qmnq/2Φ.

Thus, we have IIImnq/2 Φ.

Combining the estimates I, II, III, we complete the proof of Lemma 3.1. □

Lemma 3.2

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, 0 ≤ λ < ∞ and t(x) be a homogeneous of degreen − 1 and locally integrable in |x| > 0. Let bLip(ℝn) and K is defined by Kf(x)=limε0|xy|>εt(x)(b(x)b(y))f(y)dy.

Suppose that t(x) ∈ 𝓒1(Sn − 1), ∫Sn − 1t(x)xjdσ(x) = 0( j = 1, …, n), and α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6, then we have KfMK˙p(),λα(),q(tL(Sn1)+tL(Sn1))bLfHMK˙p(),λα(),q.


Let k(x, y) = t(xy)(b(x) − b(y)). For all x, x0, y ∈ ℝn with |xx0| ≤ 1/2 |yx|, then k satisfies the following inequalities |k(x,y)k(x0,y)|sL(Sn1)bL|xx0||yx|n1(3.4)

and |k(x,y)|tL(Sn1)bL|yx|n.(3.5)

This, together with the boundedness of K on L2(ℝn) (see [33]), tells us K is a generalized Calderón-Zygmund operator (see [34]). Thus, by applying Lemma 2.9, we see that K is bounded from HMK˙p(),λα(),q to MK˙p(),λα(),q with bound (∥∇ sL(Sn − 1)+∥sL(Sn − 1))∥∇ bL, i.e. KfMK˙p(),λα(),q(sL(Sn1)+sL(Sn1))bLfHMK˙p(),λα(),q.

Therefore, the proof of Lemma 3.2 is finished. □

Lemma 3.3

Let p(⋅) ∈ 𝓑(ℝn), 0 < q < ∞, 0 ≤ λ < ∞, bLip(ℝn), and T be a singular operator which is defined by Tf(x)=limε0|xy|>εK(xy)f(y)dy,

where K(x) ∈ 𝓒3(Sn − 1) satisfies ∫Sn − 1K(x)dσ(x) = 0 and Kx) = λnK(x) for x ∈ ℝn∖{0}. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), 2α(0), α2+δ with some δ > max{α(0) − δ2, α2} and δ2 as in Lemma 2.6, then, for fC0(Rn), [b,T]fxjMK˙p(),λα(),qmax|β|2βKL(Sn1)bLfHMK˙p(),λα(),q.


With an argument similar to that used in the proof of Lemma 5.2 in [9], together with Lemma 2.9 and Lemma 3.1, it is not difficult to obtain Lemma 3.3. Thus, we omit the details here. □

4 Proofs of of Theorems 1.1-1.5

Proof Theorem 1.1

Let Ω(x,y)=m1j=1dmam,j(x)Ym,j(y).

From [8], for any x, we can write the coefficients am, j as am,j(x)=(1)nmn(m+n2)nSn1Lyn(Ω(x,y))Ym,j(y)dσ(y),m1,(4.1)

where L(F) = |x|2Δ F(x).

We will firstly prove the conclusion (1). Write (TDγDγT)f=m=1j=1dm(am,jTm,jDγDγam,jTm,j)f=m=1j=1dm(am,jDγTm,jDγam,jTm,j)f=m=1j=1dm[am,j,Dγ]Tm,jf.

By condition (4.1), it follows that Dγam,j(x)=(1)nmn(m+n2)nSn1DxγLyn(Ω(x,y))Ym,j(y)dσ(y),m1.

Further, by applying the condition (1.4), we have Dγam,jLm2n.(4.2)

Moreover, [b, Dγ] is a generalized Calderónf-Zygmund operator (see [35]), which is defined by [b,Dγ]f(x)=C(γ)Rn(b(x)b(y))|xy|n+γf(y)dy.

Thus, we see that [b, Dγ]f(x) is bounded from HMK˙p(),λα(),q to MK˙p(),λα(),q by applying Lemma 2.9. Namely [b,Dγ]fMK˙p(),λα(),qDγbBMOfHMK˙p(),λα(),q.(4.3)

Then by dmmn − 2 (see [7]), (4.2), (4.3) and Lemma 3.1, we have (TDγDγT)fMK˙p(),λα(),qm=1j=1dm[am,j,Dγ]Tm,jfMK˙p(),λα(),qm=1j=1dmDγam,jBMOTm,jfMK˙p(),λα(),qm=1j=1dmmn2Dγam,jLfHMK˙p(),λα(),qm=1mn2mn2m2nfHMK˙p(),λα(),qfHMK˙p(),λα(),q.

Now let us turn to estimate (2). By applying the definition of T and T we can deduce that (TT)Dγf=m=1j=1dm(1)m[a¯m,j,Tm,j]Dγf.(4.4)

To estimate MK˙p(),λα(),q norm of (TT)Dγ, we first consider [b, Tm.j]Dγ for any fixed bIγ(BMO). Noting that b(x) − b(y) = (b(x) − b(z)) − (b(y) − b(z)), for any x, y, z ∈ ℝn, then we have [b,Tm,j]Dγf=[b,DγTm,j]fTm,j[b,Dγ]f.

Thus, we get by (4.3) and Lemma 3.1 Tm,j[b,Dγ]fMK˙p(),λα(),qmn2DγbBMOfHMK˙p(),λα(),q.(4.5)

Further, we estimate the MK˙p(),λα(),q norm of [b, DγTm, j]f. From the fact that [b, DγTm, j] is a generalized Calderón-Zygmund operator with kernel (see [9]) |km,j(x,y)|mn21+γDγbBMO1|xy|n,

then we get by Lemma 2.9 [b,DγTm,j]fMK˙p(),λα(),qmn2+γDγbBMOfHMK˙p(),λα(),q.(4.6)

Then, combining (4.5) with (4.6), we have [b,Tm,j]DγfMK˙p(),λα(),qmn2+γDγbBMOfHMK˙p(),λα(),q+mn2DγbBMOfHMK˙p(),λα(),qmn2+γDγbBMOfHMK˙p(),λα(),q.(4.7)

By condition (4.2), (4.4) and (4.7), we get (TT)DγfMK˙p(),λα(),qm=1j=1dm[a¯m,j,Tm,j]DγfMK˙p(),λα(),qm=1j=1dmmn2+γDγa¯m,jBMOfHMK˙p(),λα(),qm=1j=1dmmn2+γDγa¯m,jLfHMK˙p(),λα(),qm=1mn2mn2+γm2nfHMK˙p(),λα(),qfHMK˙p(),λα(),q.

Thus we finish the proof of Theorem 1.1. □

Proof of Theorem 1.2

Let T1f(x)=RnΩ1(x,xy)|xy|nf(y)dyandT2f(x)=RnΩ2(x,xy)|xy|nf(y)dy.

Write Ω1(x,y)=m1j=1dmam,j(x)Ym,j(y)andΩ2(x,y)=λ1μ=1dλbλ,μ(x)Yλ,μ(y),

where am,j(x)=Sn1Ω1(x,z)Ym,j(z)¯dσ(z)andbλ,μ(x)=Sn1Ω2(x,z)Yλ,μ(z)¯dσ(z).

For any x ∈ ℝn, with a similar argument used in the proof of Theorem 1.1 in terms of (1.4) and (1.5), we can obtain that am,jLm2n.(4.8) Dγbλ,μLm2n.(4.9)

Let Tm,jf(x)=Ym,j||nf(x)andTλ,μf(x)=Yλ,μ||nf(x).

Since Ω1(x, y) and Ω2(x, y) satisfy (1.2), then we get T1f(x)=m1j=1dmam,j(x)Tm,jf(x)andT2f(x)=λ1μ=1dλbλ,μ(x)Tλ,μf(x).

Write (see [9]) (T1T2)f(x)=m=1j=1dmλ=1μ=1dλam,j(x)bλ,μ(x)(Tm,jTλ,μf)(x),(T1T2)f(x)=m=1j=1dmλ=1μ=1dλam,jTm,j(bλ,μTλ,μf)(x).

Then (T1T2T1T2)Dγf=m=1j=1dmλ=1μ=1dλam,j(bλ,μ(x)Tm,jTm,jbλ,μ(x))Tλ,μDγf=m=1j=1dmλ=1μ=1dλam,j(bλ,μ(x)Tm,jTm,jbλ,μ(x))DγTλ,μf=m=1j=1dmλ=1μ=1dλam,j[bλ,μ,Tm,j]DγTλ,μf.

Therefore, together with (4.7), (4.8), (4.9) and Lemma 3.1, we obtain (T1T2T1T2)DγfMK˙p(),λα(),qm=1j=1dmλ=1μ=1dλam,jL[bλ,μ,Tm,j]DγTλ,μfMK˙p(),λα(),qm=1j=1dmλ=1μ=1dλam,jLDγbλ,μBMOmn2+γTλ,μfMK˙p(),λα(),qm=1j=1dmλ=1μ=1dλam,jLDγbλ,μLmn2+γλn2fHMK˙p(),λα(),qm=1mn2m2nmn2+γλ=1λn2λ2nλn2fHMK˙p(),λα(),qfHMK˙p(),λα(),q.

This finishes the proof of Theorem 1.2. □

Proof of Theorem 1.3

We estimate that term exactly as we did for the corresponding boundedness in Theorem 1.1 in the above arguments. Without loss of generality, we only have to prove (2) and (3) of Theorem 1.3. By using the fact that Ω1(x, y) and Ω2(x, y) satisfy (1.5), we have shown that am,jLm2n,bλ,μLλ2n.(4.10)

Firstly, let′s prove (2). As in the proof of Theorem 1.1, we can get (T1T1)If=m=1j=1dm(1)m[a¯m,j,Tm,j]If.

We showed that [b, Tm, j] is a special Calderón-Zygmund operator, so it is a bounded operator from HMK˙p(),λα(),q to MK˙p(),λα(),q by applying Lemma 2.9. Thus we have [b,Tm,j]fMK˙p(),λα(),qmn2bLfHMK˙p(),λα(),q.(4.11)

Then by (4.10), we get (T1T1)IfMK˙p(),λα(),qm=1mn2m3n/2fHMK˙p(),λα(),qfHMK˙p(),λα(),q.

Thus the conclusion (2) is proved. We now estimate (3). Write (T1T2T1T2)If=m=1j=1dmλ=1μ=1dλ[bλ,μ,Tm,j]Tλ,μIf.

Therefore, by (4.10), (4.11) and Lemma 3.1, we get (T1T2T1T2)IfMK˙p(),λα(),qm=1j=1dmλ=1μ=1dλam,jLbλ,μLmn2Tλ,μIf]MK˙p(),λα(),q.m=1mn2m2nmn/2λ=1λn2λ2nλn/2fHMK˙p(),λα(),qfHMK˙p(),λα(),q.

Thus the conclusion (3) is also proved. Hence the proof of Theorem 1.3 is finished. □

Proof of Theorem 1.4

In the first place, we will prove the conclusion (1). Write D=k=1nRkxk, where 𝓡k denotes the Riesz transform. As in the proof of Theorem 1.1, we have (TDDT)f(x)=m=1j=1dm[am,j,D]Tm,jf(x)=m=1j=1dmk=1nRk[am,j,xk]Tm,jf(x)+m=1j=1dmk=1n[am,j,Rk]xk(Tm,jf)(x)=:J1+J2.

We have by the Leibniz′s rules that J1=m=1j=1dmk=1nRk(xk(am,j)Tm,jf).

Thus we deduce from (4.1) that am,jxk(x)=(1)nmn(m+n2)nSn1xkLyn(Ω(x,y))Ym,j(y)dσ(y),m1.

From this and (1.6), we get for k = 1, …, n, am,jxkLm2n.(4.12)

By using the fact that RkgWMK˙p,1α,λ(Rn)gMK˙p,1α,λ(Rn),dmmn2 and Lemma 3.2, then we have J1MK˙p(),λα(),qm=1j=1dmk=1nRk(xk(am,j)Tm,jf)MK˙p(),λα(),qm=1j=1dmm2nmn/2fHMK˙p(),λα(),qm=1mn2m2nmn/2fHMK˙p(),λα(),qfHMK˙p(),λα(),q.

By Lemma 3.3 and (4.12), a trivial computation shows that for I2, J2WMK˙p,1α,λ(Rn)m=1j=1dmk=1nam,jLTm,jfWMK˙p,1α,λ(Rn)m=1j=1dmm2nmn/2fMK˙p,1α,λ(Rn)m=1mn2m2nmn/2fMK˙p,1α,λ(Rn)fMK˙p,1α,λ(Rn).

Combining the estimates above, we arrive at the desired boundedness (TDDT)fMK˙p(),λα(),qfHMK˙p(),λα(),q.

We posterior prove the conclusion (2). Write D=k=1nRkxk, we have (TT)Df(x)=m=1j=1dm(1)m[a¯m,j,Tm,j]Df(x)=k=1nm=1j=1dm(1)m[a¯m,j,Tm,j]xk(Rkf)(x).(4.13)

We now turn to estimate the MK˙p(),λα(),q norm of [a¯m,j,Tm,j]xk(Rkf). Applying (4.12), Lemma 3.3 and the fact that for any multi-index β and x ∈ ℝn∖{0}, m = 1, 2, …. (see [1]), |β(|x|m)Ym,j|C(n)|x|m|β|m|β|+(n2)/2.(4.14)

Hence, we get [a¯m,j,Tm,j]xk(Rkf)MK˙p(),λα(),qa¯m,jLmax|β|2βYm,jL(Sn1)RkfMK˙p(),λα(),qm2nmn/2+1fHMK˙p(),λα(),qm3n/2+1fHMK˙p(),λα(),q.(4.15)

Combining the estimates of (4.13) with (4.15), we have (TT)DfMK˙p(),λα(),qm=1mn2m3n/2+1fHMK˙p(),λα(),qfHMK˙p(),λα(),q.

Consequently, the proof of Theorem 1.4 is completed. □

Proof of Theorem 1.5

Similarly to the proof of Theorem 1.2, we easily see that (T1T2T1T2)Df=m=1d=1dmλ=1μ=1dλam,j[bλ,μ,Tm,j]DTλ,μf,

where am, j and bλ, μ are the same as in the proof of Theorem 1.2. By (1.5) and (1.6), we have am,jLm2n.(4.16) bλ,μLλ2n.(4.17)

Write D=k=1nxkRk, it then follows that (T1T2T1T2)DfMK˙p(),λα(),qm=1j=1dmλ=1μ=1dλam,jL[bλ,μ,Tm,j](k=1nxkRkTλ,μf)MK˙p(),λα(),qk=1nm=1j=1dmλ=1μ=1dλam,jL[bλ,μ,Tm,j](xkRkTλ,μf)MK˙p(),λα(),q.

The above estimate, via Lemma 3.1, leads to (T1T2T1T2)DfMK˙p(),λα(),qm=1j=1dmλ=1μ=1dλam,jLbλ,μLmax|β|2βYm,jL(Sn1)Tλ,μRkfMK˙p(),λα(),q.

We thus obtain from (4.14), (4.16), (4.17) and Lemma 3.1 that (T1T2T1T2)DfMK˙p(),λα(),qm=1mn/21m2nmn/2+1λ=1λn/21λ2nλn/2fHMK˙p(),λα(),qfHMK˙p(),λα(),q.

Consequently, the proof of Theorem 1.5 is finished. □


This work is supported by National Natural Science Foundation of China (Grant No. 11561062;11661061).


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About the article

Received: 2017-12-17

Accepted: 2018-02-09

Published Online: 2018-04-10

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 326–345, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0036.

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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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Yanqi Yang and Shuangping Tao
Open Mathematics, 2018, Volume 16, Number 1, Page 1607

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