Higher Order Commutators of Fractional Integral Operator on the Homogeneous Herz Spaces with Variable Exponent

Abstract

By decomposing functions, we establish estimates for higher order commutators generated by fractional integral with BMO functions or the Lipschitz functions on the homogeneous Herz spaces with variable exponent. These estimates extend some known results in the literatures.

1. Introduction

Let 𝑏 be a locally integrable function, 0<𝛽<𝑛, and 𝑚∈ℕ; the higher order commutators of fractional integral operator 𝐼^𝑚_𝛽,𝑏 are defined by 𝐼^𝑚_𝛽,𝑏𝑓(𝑥)=_ℝ𝑛𝑏(𝑥)−𝑏𝑦^𝑚𝑥−𝑦^𝑛−𝛽𝑓𝑦𝑑𝑦. (1) Obviously, 𝐼⁰_𝛽,𝑏=𝐼_𝛽 and 𝐼¹_𝛽,𝑏=[𝑏,𝐼_𝛽]. The famous Hardy-Littlewood-Sobolev theorem tells us that the fractional integral operator 𝐼_𝛽 is a bounded operator from the usual Lebesgue spaces 𝐿^𝑝1(ℝ^𝑛) to 𝐿^𝑝2(ℝ^𝑛) when 0<𝑝₁<𝑝₂<∞ and 1/𝑝₁−1/𝑝₂=𝛽/𝑛. Also, many generalized results about 𝐼_𝛽 and the commutator [𝑏,𝐼_𝛽] on some function spaces have been studied; see [1–3] for details.

It is well known that the main motivation for studying the spaces with variable exponent arrived in the nonlinear elasticity theory and differential equations with nonstandard growth. Since the fundamental paper [4] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent 𝐿^𝑝(⋅)(ℝ^𝑛) have been extensively investigated. In the recent twenty years, boundedness of some important operators, for example, the Calderón-Zygmund operators, fractional integrals, and commutators, on 𝐿^𝑝(⋅)(ℝ^𝑛) has been obtained; see [5–7]. Recently, Diening [8] extended the (𝐿^𝑝1(ℝ^𝑛),𝐿^𝑝2(ℝ^𝑛)) boundedness of 𝐼_𝛽 to the Lebesgue spaces with variable exponent. Izuki [7] first introduced the Herz spaces with variable exponent 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛), which is a generalized space of the Herz space 𝐾̇_𝑝^𝛼,𝑞(ℝ^𝑛); see [9, 10], and in case of 𝑏∈BMO(ℝ^𝑛), he obtained the boundedness properties of the commutator [𝑏,𝐼_𝛽]. The paper [11] by Lu et al. indicates that the commutator [𝑏,𝐼_𝛽] with 𝑏∈BMO(ℝ^𝑛) and with 𝑏∈Lip_𝛼(ℝ^𝑛)(0<𝛼≤1) has many different properties. In 2012, Zhou [12] studied the boundedness of 𝐼_𝛽 on the Herz spaces with variable exponent and proved that the boundedness properties of the commutator [𝑏,𝐼_𝛽] also hold in case of 𝑏∈Lip_𝛼(ℝ^𝑛)(0<𝛼≤1). The higher order commutators 𝐼^𝑚_𝛽,𝑏 are recently considered by Wang et al. in the paper [13, 14]; they established the BMO and the Lipschitz estimates for 𝐼^𝑚_𝛽,𝑏 on the Lebesgue spaces with variable exponent 𝐿^𝑝(⋅)(ℝ^𝑛). Motivated by [7, 12–14], in this note, we establish the boundedness of the higher order commutators 𝐼^𝑚_𝛽,𝑏 on the Herz spaces with variable exponent.

For brevity, |𝐸| denotes the Lebesgue measure for a measurable set 𝐸⊂ℝ^𝑛, and 𝑓_𝐸 denotes the mean value of 𝑓 on 𝐸(𝑓_𝐸=(1/|𝐸|)∫_𝐸𝑓(𝑥)𝑑𝑥). The exponent 𝑝^′(⋅) means the conjugate of 𝑝(⋅), that is, 1/𝑝(⋅)+1/𝑝^′(⋅)=1. 𝐶 denotes a positive constant, which may have different values even in the same line. Let us first recall some definitions and notations.

Definition 1. For 0<𝛾≤1, the Lipschitz space Lip_𝛾(ℝ^𝑛) is the space of functions 𝑓 satisfying 𝑓_Lip𝛾=sup_{𝑥,𝑦∈ℝ𝑛,𝑥≠𝑦}𝑓(𝑥)−𝑓𝑦𝑥−𝑦^𝛾<∞. (2)

Definition 2. For 𝑓∈𝐿¹_loc(ℝ^𝑛), the bounded mean oscillation space BMO(ℝ^𝑛) is the space of functions 𝑓 satisfying 𝑓_BMO=sup_𝐵1|𝐵|_𝐵𝑓(𝑥)−𝑓_𝐵𝑑𝑥<∞, (3) where the supremum is taken over all balls 𝐵 in ℝ^𝑛.

Definition 3. Let 𝑝(⋅)∶𝐸→[1,∞) be a measurable function.(1)The Lebesgue space with variable exponent 𝐿^𝑝(⋅)(𝐸) is defined by 𝐿^𝑝(⋅)(𝐸)=𝑓ismeasurable∶_𝐸𝑓(𝑥)𝜆^𝑝(𝑥).𝑑𝑥<∞forsomeconstant𝜆>0 (4)(2)The space with variable exponent 𝐿^𝑝(⋅)_loc(𝐸) is defined by 𝐿^𝑝(⋅)_loc(𝐸)=𝑓∶𝑓∈𝐿^𝑝(⋅)(𝐾)forallcompactsubsets𝐾⊂𝐸. (5)
The Lebesgue space 𝐿^𝑝(⋅)(𝐸) is a Banach space with the Luxemburg norm 𝑓_{𝐿𝑝(⋅)(𝐸)}=inf𝜆>0∶_𝐸𝑓(𝑥)𝜆^𝑝(𝑥)𝑑𝑥≤1. (6)
We denote 𝑝₋𝑝(𝑥),𝑝=essinf∶𝑥∈𝐸₊𝑝(𝑥),𝒫(𝐸)=𝑝(⋅)=esssup∶𝑥∈𝐸∶𝑝₋>1,𝑝₊,ℬ(𝐸)=𝑝(⋅)(⋅)(𝐸),<∞∶𝑝∈𝒫𝑀isboundedon𝐿^𝑝(⋅)(𝐸), (7) where the Hardy-Littlewood maximal operator 𝑀 is defined by 𝑀𝑓(𝑥)=sup_𝑟>0𝑟^−𝑛_{𝐵(𝑥,𝑟)∩𝐸}𝑓𝑦𝑑𝑦, (8) where 𝐵(𝑥,𝑟)={𝑦∈ℝ^𝑛∶|𝑥−𝑦|<𝑟}.

Proposition 4 (see [15]). If 𝑝(⋅)∈𝒫(𝐸) satisfies 𝑝(𝑥)−𝑝𝑦≤−𝐶,≤1log𝑥−𝑦𝑥−𝑦2,𝑝(𝑥)𝑦≤𝐶−𝑝(|𝑥,𝑦≤|𝑥|,log𝑒+|) (9) then one has 𝑝(⋅)∈ℬ(𝐸).

Let 𝐵_𝑘={𝑥∈ℝ^𝑛∶|𝑥|⩽2^𝑘}, 𝑅_𝑘=𝐵_𝑘⧵𝐵_𝑘−1, and 𝜒_𝑘=𝜒_𝑅𝑘 be the characteristic function of the set 𝑅_𝑘 for 𝑘∈ℤ. For 𝑚∈ℕ, we denote 𝜒_𝑚=𝜒_𝑅𝑚 if 𝑚≥1, and 𝜒₀=𝜒_𝐵0.

Definition 5 (see [7]). For 𝛼∈ℝ,0<𝑞≤∞ and 𝑝(⋅)∈𝒫(ℝ^𝑛).(1)The homogeneous Herz spaces 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) are defined by 𝐾̇_𝑝(⋅)^𝛼,𝑞ℝ^𝑛=𝑓∈𝐿^𝑝(⋅)_locℝ^𝑛⧵{0}∶𝑓_̇𝐾^𝛼,𝑞_{𝑝(⋅)(ℝ𝑛)}<∞, (10) where 𝑓_̇𝐾^𝛼,𝑞_{𝑝(⋅)(ℝ𝑛)}=2^𝛼𝑘𝑓𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}^∞_{𝑘=−∞ℓ𝑞(ℤ)}. (11)(2)The nonhomogeneous Herz spaces 𝐾^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) are defined by 𝐾_𝑝(⋅)^𝛼,𝑞ℝ^𝑛=𝑓∈𝐿^𝑝(⋅)_locℝ^𝑛∶𝑓_𝐾^𝛼,𝑞_{𝑝(⋅)(ℝ𝑛)}<∞, (12) where 𝑓_𝐾^𝛼,𝑞_{𝑝(⋅)(ℝ𝑛)}=2^𝛼𝑚𝑓𝜒_{𝑚𝐿𝑝(⋅)(ℝ𝑛)}^∞_{𝑚=0ℓ𝑞(ℕ)}. (13)

In this note, we obtain the following results.

Theorem 6. Suppose that 𝑏∈Lip_𝛽1(ℝ^𝑛)(0<𝛽₁<1), 𝑝₂(⋅)∈𝒫(ℝ^𝑛) satisfies conditions (9) in Proposition 4. If 0<𝑟<min{1/(𝑝₁)₊,1/(𝑝^′₂)₊}, 0<𝛽+𝑚𝛽₁<𝑛𝑟, 0<𝛼<𝑛𝑟−𝛽−𝑚𝛽₁, 0<𝑞₁≤𝑞₂<∞, and 1/𝑝₁(𝑥)−1/𝑝₂(𝑥)=(𝛽+𝑚𝛽₁)/𝑛, then the higher order commutators 𝐼^𝑚_𝛽,𝑏 are bounded from 𝐾̇^𝛼,𝑞1_𝑝1(⋅)(ℝ^𝑛) to 𝐾̇^𝛼,𝑞2_𝑝2(⋅)(ℝ^𝑛).

Theorem 7. Suppose that 𝑏∈BMO(ℝ^𝑛), 𝑝₂(⋅)∈𝒫(ℝ^𝑛) satisfies conditions (9) in Proposition 4. If 0<𝑟<min{1/(𝑝₁)₊,1/(𝑝^′₂)₊}, 0<𝛽<𝑛𝑟, 0<𝛼<𝑛𝑟−𝛽, 0<𝑞₁≤𝑞₂<∞, and 1/𝑝₁(𝑥)−1/𝑝₂(𝑥)=𝛽/𝑛, then the higher order commutators 𝐼^𝑚_𝛽,𝑏 are bounded from 𝐾̇^𝛼,𝑞1_𝑝1(⋅)(ℝ^𝑛) to 𝐾̇^𝛼,𝑞2_𝑝2(⋅)(ℝ^𝑛).

Remark A. The previous main results generalize the (𝐿^𝑝(⋅)(ℝ^𝑛),𝐿^𝑞(⋅)(ℝ^𝑛)) boundedness of the higher order commutators 𝐼^𝑚_𝛽,𝑏 in [13] to the case of the Herz spaces with variable exponent. If 𝑚=1, our conclusions coincide with the corresponding results in [7, 12]. Moreover, the same boundedness also holds for the nonhomogeneous case.

2. Proof of Theorems 6 and 7

To prove our main results, we need the following lemmas.

Lemma 8 (see [4]). Let 𝑝(⋅)∈𝒫(ℝ^𝑛); if 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛) and 𝑔∈𝐿^𝑝′(⋅)(ℝ^𝑛), then _ℝ𝑛𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤𝑟_𝑝𝑓_{𝐿𝑝(⋅)(ℝ𝑛)}𝑔_{𝐿𝑝′(⋅)(ℝ𝑛)}, (14)
where 𝑟_𝑝=1+1/𝑝₋−1/𝑝₊.

Lemma 9 (see [7]). Let 𝑝(⋅)∈ℬ(ℝ^𝑛); then for all balls 𝐵 in ℝ^𝑛, 1|𝐵|𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝′(⋅)(ℝ𝑛)}≤𝐶. (15)

Lemma 10 (see [7]). Let 𝑝^′₂(⋅)∈ℬ(ℝ^𝑛); then for all balls 𝐵 in ℝ^𝑛 and all measurable subsets 𝑆⊂𝐵, one can take a constant 0<𝑟<1/(𝑝^′₂)₊, so that 𝜒_{𝑆𝐿𝑝′2(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝′2(⋅)(ℝ𝑛)}|𝑆|≤𝐶|𝐵|^𝑟. (16)

Lemma 11 (see [8]). Suppose that 𝑝₁(⋅)∈𝒫(ℝ^𝑛) satisfies conditions (9) in Proposition 4, 0<𝛽<𝑛/(𝑝₁)₊ and 1/𝑝₁(𝑥)−1/𝑝₂(𝑥)=𝛽/𝑛; then 𝐼_𝛽𝑓_{𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶𝑓_{𝐿𝑝1(⋅)(ℝ𝑛)}. (17)

Lemma 12 (see [13]). Suppose that 𝑝₁(⋅), 𝑝₂(⋅)∈𝒫(ℝ^𝑛).(1)Let 0<𝛽<𝑛/(𝑝₁)₊, 𝑏∈BMO(ℝ^𝑛). If 𝑝₂(⋅) satisfies conditions (9) in Proposition 4 and 1/𝑝₁(𝑥)−1/𝑝₂(𝑥)=𝛽/𝑛, then 𝐼^𝑚_𝛽,𝑏𝑓_{𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶‖𝑏‖^𝑚_BMO𝑓_{𝐿𝑝1(⋅)(ℝ𝑛)}. (18)(2)Let 0<𝛽+𝑚𝛽₁<𝑛/(𝑝₁)₊, 𝑏∈𝐿𝑖𝑝_𝛽1(ℝ^𝑛)(0<𝛽₁<1). If 𝑝₂(⋅) satisfies conditions (9) in Proposition 4 and 1/𝑝₁(𝑥)−1/𝑝₂(𝑥)=(𝛽+𝑚𝛽₁)/𝑛, then 𝐼^𝑚_𝛽,𝑏𝑓_{𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶‖𝑏‖^𝑚_{𝐿𝑖𝑝𝛽1}𝑓_{𝐿𝑝1(⋅)(ℝ𝑛)}. (19)

Lemma 13 (see [16]). Let 𝑏∈BMO(ℝ^𝑛), 𝑘>𝑗(𝑘,𝑗∈ℕ); one has(1)𝐶⁻¹||𝑏||^𝑚_BMO≤sup_{𝐵⊂ℝ𝑛}(1/||𝜒_𝐵||_𝐿^𝑝(⋅)_(ℝ^𝑛₎)||(𝑏−𝑏_𝐵)^𝑚𝜒_𝐵||_𝐿^𝑝(⋅)_(ℝ^𝑛₎≤𝐶||𝑏||^𝑚_BMO;(2)||(𝑏−𝑏_𝐵𝑗)^𝑚𝜒_𝐵𝑘||_𝐿^𝑝(⋅)_(ℝ^𝑛₎≤𝐶(𝑘−𝑗)^𝑚||𝑏||^𝑚_BMO  ×||𝜒_𝐵𝑘||_𝐿^𝑝(⋅)_(ℝ^𝑛₎.

Proof of Theorem 6. Let 𝑓∈𝐾̇^𝛼,𝑞1_𝑝1(⋅)(ℝ^𝑛); we can write 𝑓(𝑥)=^∞_𝑗=−∞𝑓(𝑥)𝜒_𝑗(𝑥)=^∞_𝑗=−∞𝑓_𝑗(𝑥). (20)
For 0<𝑞₁/𝑞₂≤1, applying the inequality ^∞_𝑖=1𝑎_𝑖^𝑞1/𝑞2≤^∞_𝑖=1𝑎^𝑞1/𝑞2_𝑖𝑎_𝑖>0,𝑖=1,2…, (21) we obtain 𝐼^𝑚_𝛽,𝑏𝑓^𝑞1_{̇𝐾2𝑝2(⋅)𝛼,𝑞(ℝ𝑛)}=𝐶^∞_𝑘=−∞2^{𝛼𝑞2𝑘}𝐼^𝑚_𝛽,𝑏𝑓𝜒_𝑘^𝑞2_{𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1/𝑞2≤𝐶^∞_𝑘=−∞2^{𝛼𝑞1𝑘𝑘−2}_𝑗=−∞𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1+𝐶^∞_𝑘=−∞2^{𝛼𝑞1𝑘∞}_{𝑗=𝑘−1}𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1=𝑈₁+𝑈₂. (22)
We first estimate 𝑈₁. Noting that if 𝑥∈𝑅_𝑘,𝑦∈𝑅_𝑗, and 𝑗≤𝑘−2, then |𝑥−𝑦|∼|𝑥|∼2^𝑘, we get 𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_𝑘≤_𝑅𝑗𝑏(𝑥)−𝑏𝑦^𝑚𝑥−𝑦^𝑛−𝛽𝑓_𝑗𝑦𝑑𝑦⋅𝜒_𝑘(𝑥)≤𝐶2^{𝑘(𝛽−𝑛)}_𝑅𝑗𝑏(𝑥)−𝑏𝑦^𝑚𝑓_𝑗𝑦𝑑𝑦⋅𝜒_𝑘(𝑥)≤𝐶2^{𝑘(𝛽+𝑚𝛽1−𝑛)}‖𝑏‖^𝑚_{Lip𝛽1𝑅𝑗}𝑓_𝑗𝑦𝑑𝑦⋅𝜒_𝑘(𝑥). (23)
By Hölder’s inequality, Lemmas 9 and 10, we have 𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^{𝑘(𝛽+𝑚𝛽1−𝑛)}‖𝑏‖^𝑚_Lip𝛽1𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×𝜒_{𝐵𝑘𝐿𝑝2(⋅)(ℝ𝑛)}𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}≤𝐶2^{𝑘(𝛽+𝑚𝛽1)}‖𝑏‖^𝑚_Lip𝛽1𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}𝜒_𝐵𝑘𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}≤𝐶2^{𝑘(𝛽+𝑚𝛽1)}‖𝑏‖^𝑚_Lip𝛽1𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}𝜒_𝐵𝑗𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)(ℝ𝑛)}𝜒_{𝐵𝑘𝐿𝑝′2(⋅)(ℝ𝑛)}≤𝐶2^{𝑘(𝛽+𝑚𝛽1)}2^{𝑛𝑟(𝑗−𝑘)}‖𝑏‖^𝑚_Lip𝛽1𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}𝜒_𝐵𝑗𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}. (24)
Note that 𝐼_{𝛽+𝑚𝛽1}𝜒_𝐵𝑗(𝑥)≥𝐼_{𝛽+𝑚𝛽1}𝜒_𝐵𝑗(𝑥)⋅𝜒_𝐵𝑗(𝑥)=_𝐵𝑗𝑑𝑦𝑥−𝑦^{𝑛−𝛽−𝑚𝛽1}⋅𝜒_𝐵𝑗(𝑥)≥𝐶2^{𝑗(𝛽+𝑚𝛽1)}⋅𝜒_𝐵𝑗(𝑥). (25) By Lemmas 8 and 11, we obtain 𝜒_𝐵𝑗𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}≤𝐶2^−𝑛𝑗𝜒_{𝐵𝑗𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^−𝑛𝑗2^{−𝑗(𝛽+𝑚𝛽1)}𝐼_{𝛽+𝑚𝛽1}𝜒_{𝐵𝑗𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^{−𝑗(𝛽+𝑚𝛽1)}2^−𝑛𝑗𝜒_{𝐵𝑗𝐿𝑝1(⋅)(ℝ𝑛)}≤𝐶2^{−𝑗(𝛽+𝑚𝛽1)}𝜒_𝐵𝑗𝐿⁻¹_{𝑝′1(⋅)(ℝ𝑛)}. (26)
Combining (24) and (26), we have the estimate 𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟)}×‖𝑏‖^𝑚_Lip𝛽1𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}. (27) Thus, 𝑈₁≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1×^∞_𝑘=−∞^𝑘−2_𝑗=−∞2^𝛼𝑗𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}2^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟+𝛼)𝑞1}. (28)
If 1<𝑞₁<∞, noting that 𝛽+𝑚𝛽₁−𝑛𝑟+𝛼<0, by Hölder’s inequality, we have 𝑈₁≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1×^∞_𝑘=−∞^𝑘−2_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟+𝛼)𝑞1/2}×^𝑘−2_𝑗=−∞2^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟+𝛼)𝑞′1/2𝑞1/𝑞′1}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^∞_𝑘=𝑗+22^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟+𝛼)𝑞1/2}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (29)
If 0<𝑞₁≤1, by inequality (21), we have 𝑈₁≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1×^∞_𝑘=−∞^𝑘−2_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟+𝛼)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^∞_𝑘=𝑗+22^{(𝑘−𝑗)(𝛽+𝑚𝛽1−𝑛𝑟+𝛼)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (30)
Next, we estimate 𝑈₂. By Lemma 12(2), we obtain 𝑈₂≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞_𝑘=−∞^∞_{𝑗=𝑘−1}2^𝛼𝑗𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}2^{𝛼(𝑘−𝑗)𝑞1}. (31)
If 1<𝑞₁<∞, by Hölder’s inequality, we have 𝑈₂≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞_𝑘=−∞^∞_{𝑗=𝑘−1}2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{𝛼(𝑘−𝑗)𝑞1/2}×^∞_{𝑗=𝑘−1}2^{𝛼(𝑘−𝑗)𝑞′1/2𝑞1/𝑞′1}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^𝑗+1_𝑘=−∞2^{𝛼(𝑘−𝑗)𝑞1/2}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (32)
If 0<𝑞₁≤1, by inequality (21), we have 𝑈₂≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞∞_{𝑘=−∞𝑗=𝑘−1}2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{𝛼(𝑘−𝑗)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^𝑗+1_𝑘=−∞2^{𝛼(𝑘−𝑗)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_Lip𝛽1𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (33)
Combining the estimates for 𝑈₁ and 𝑈₂, the proof of Theorem 6 is completed.

Proof of Theorem 7. Let 𝑓∈𝐾̇^𝛼,𝑞1_𝑝1(⋅)(ℝ^𝑛); we can write 𝑓(𝑥)=^∞_𝑗=−∞𝑓(𝑥)𝜒_𝑗(𝑥)=^∞_𝑗=−∞𝑓_𝑗(𝑥). (34)
By inequality (21), we obtain 𝐼^𝑚_𝛽,𝑏𝑓^𝑞1_{̇𝐾2𝑝2(⋅)𝛼,𝑞(ℝ𝑛)}=𝐶^∞_𝑘=−∞2^{𝛼𝑞2𝑘}𝐼^𝑚_𝛽,𝑏𝑓𝜒_𝑘^𝑞2_{𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1/𝑞2≤𝐶^∞_𝑘=−∞2^{𝛼𝑞1𝑘𝑘−2}_𝑗=−∞𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1+𝐶^∞_𝑘=−∞2^{𝛼𝑞1𝑘∞}_{𝑗=𝑘−1}𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1=𝑉₁+𝑉₂. (35)
For 𝑉₁, using Hölder’s inequality and Lemma 8, we have 𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_𝑘≤𝐶2^{𝑘(𝛽−𝑛)}_𝑅𝑗𝑏(𝑥)−𝑏𝑦^𝑚𝑓_𝑗𝑦𝑑𝑦⋅𝜒_𝑘(𝑥)≤𝐶2^{𝑚𝑘(𝛽−𝑛)}_𝑖=0𝐶^𝑖_𝑚𝑏(𝑥)−𝑏_𝐵𝑗^𝑚−𝑖×_𝑅𝑗𝑏_𝐵𝑗−𝑏𝑦^𝑖𝑓_𝑗𝑦𝑑𝑦≤𝐶2^{𝑘(𝛽−𝑛)}𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×^𝑚_𝑖=0𝐶^𝑖_𝑚𝑏(𝑥)−𝑏_𝐵𝑗^𝑚−𝑖𝑏_𝐵𝑗−𝑏^𝑖𝜒_{𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}. (36) By Lemmas 9, 10, and 13, we have 𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^{𝑘(𝛽−𝑛)}𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×^𝑚_𝑖=0𝐶^𝑖_𝑚𝑏(𝑥)−𝑏_𝐵𝑗^𝑚−𝑖𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}×𝑏_𝐵𝑗−𝑏^𝑖𝜒_{𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}≤𝐶2^{𝑘(𝛽−𝑛)}𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×^𝑚_𝑖=0𝐶^𝑖_𝑚𝑘−𝑗^𝑚−𝑖‖𝑏‖^𝑚−𝑖_BMO𝜒_{𝐵𝑘𝐿𝑝2(⋅)(ℝ𝑛)}×‖𝑏‖^𝑖_BMO𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}=𝐶𝑘−𝑗+1^𝑚‖𝑏‖^𝑚_BMO𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×2^{𝑘(𝛽−𝑛)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)(ℝ𝑛)}𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}≤𝐶𝑘−𝑗+1^𝑚‖𝑏‖^𝑚_BMO𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×2^𝑘𝛽𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}𝜒_𝐵𝑘𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}≤𝐶𝑘−𝑗+1^𝑚‖𝑏‖^𝑚_BMO𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×2^𝑘𝛽𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}𝜒_𝐵𝑘𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)(ℝ𝑛)}𝜒_{𝐵𝑘𝐿𝑝′2(⋅)(ℝ𝑛)}≤𝐶𝑘−𝑗+1^𝑚‖𝑏‖^𝑚_BMO𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×2^𝑘𝛽2^{𝑛𝑟(𝑗−𝑘)}𝜒_{𝐵𝑗𝐿𝑝′1(⋅)(ℝ𝑛)}𝜒_𝐵𝑗𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}. (37)
Note that 𝐼_𝛽𝜒_𝐵𝑗(𝑥)≥_𝐵𝑗𝑑𝑦𝑥−𝑦^𝑛−𝛽⋅𝜒_𝐵𝑗(𝑥)≥𝐶2^𝑗𝛽⋅𝜒_𝐵𝑗(𝑥). (38) By Lemmas 8 and 11, we obtain 𝜒_𝐵𝑗𝐿⁻¹_{𝑝′2(⋅)(ℝ𝑛)}≤𝐶2^−𝑛𝑗𝜒_{𝐵𝑗𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^−𝑛𝑗2^−𝑗𝛽𝐼_𝛽(𝜒_𝐵𝑗)_{𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶2^−𝑗𝛽2^−𝑛𝑗𝜒_{𝐵𝑗𝐿𝑝1(⋅)(ℝ𝑛)}≤𝐶2^−𝑗𝛽𝜒_𝐵𝑗𝐿⁻¹_{𝑝′1(⋅)(ℝ𝑛)}. (39)
Combining (37) and (39), we have the estimate 𝐼^𝑚_𝛽,𝑏𝑓_𝑗𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶𝑘−𝑗+1^𝑚×‖𝑏‖^𝑚_BMO2^{(𝑘−𝑗)(𝛽−𝑛𝑟)}𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}. (40) Thus, 𝑉₁≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑘=−∞^𝑘−2_𝑗=−∞2^𝛼𝑗𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}×𝑘−𝑗+1^𝑚2^{(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞1}. (41)
In case of 1<𝑞₁<∞, noting that 𝛽−𝑛𝑟+𝛼<0, by Hölder’s inequality, we have 𝑉₁≤𝐶‖𝑏‖^𝑚𝑞1_BMO×^∞_𝑘=−∞^𝑘−2_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞1/2}×^𝑘−2_𝑗=−∞𝑘−𝑗+1^𝑚𝑞′2^{(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞′1/2𝑞1/𝑞′1}≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^∞_𝑘=𝑗+22^{(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞1/2}≤𝐶‖𝑏‖^𝑚𝑞1_BMO𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (42)
In case of 0<𝑞₁≤1, by inequality (21), we have 𝑉₁≤𝐶‖𝑏‖^𝑚𝑞1_BMO×^∞_𝑘=−∞^𝑘−2_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}×𝑘−𝑗+1^𝑚𝑞12^{(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}×^∞_𝑘=𝑗+2𝑘−𝑗+1^𝑚𝑞12^{(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_BMO𝑓^𝑞1_{̇𝐾1𝑝1𝛼,𝑞(⋅)(ℝ𝑛)}. (43)
For 𝑉₂, by Lemma 12(1), we obtain 𝑉₂≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑘=−∞^∞_{𝑗=𝑘−1}2^𝛼𝑗𝑓_{𝑗𝐿𝑝1(⋅)(ℝ𝑛)}2^{𝛼(𝑘−𝑗)𝑞1}. (44)
If 1<𝑞₁<∞, by Hölder’s inequality, we have 𝑉₂≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑘=−∞^∞_{𝑗=𝑘−1}2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{𝛼(𝑘−𝑗)𝑞1/2}×^∞_{𝑗=𝑘−1}2^{𝛼(𝑘−𝑗)𝑞′1/2𝑞1/𝑞′1}≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^𝑗+1_𝑘=−∞2^{𝛼(𝑘−𝑗)𝑞1/2}≤𝐶‖𝑏‖^𝑚𝑞1_BMO𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (45)
If 0<𝑞₁≤1, by inequality (21), we have 𝑉₂≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞∞_{𝑘=−∞𝑗=𝑘−1}2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}2^{𝛼(𝑘−𝑗)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_BMO^∞_𝑗=−∞2^{𝛼𝑗𝑞1}𝑓_𝑗^𝑞1_{𝐿𝑝1(⋅)(ℝ𝑛)}^𝑗+1_𝑘=−∞2^{𝛼(𝑘−𝑗)𝑞1}≤𝐶‖𝑏‖^𝑚𝑞1_BMO𝑓^𝑞1_{̇𝐾1𝑝1(⋅)𝛼,𝑞(ℝ𝑛)}. (46)
Combining the estimates for 𝑉₁ and 𝑉₂, consequently, we have proved Theorem 7.

Acknowledgments

The authors thank the referees for their valuable comments to the original version of this note. This paper is supported by the NSF of China (no. 11201003); the Natural Science Foundation of Anhui Higher Education Institutions of China (no. KJ2011A138; no. KJ2013B034).