Function spaces with variable exponent are being concerned with strong interest not only in harmonic analysis but also in applied mathematics. In the past 27 years, the theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kováčik and Rákosník^[1] in 1991. In Refs. [2-6], the authors proved the boundedness of some integral operators on variable L^p spaces, respectively. Lebesgue and Sobolev spaces with integrability exponent have been widely studied (see Refs. [3, 5] and the references therein). Many applications of these spaces were given, for example, in the modeling of electrorheological fluids, in the study of image processing, and in differential equations with nonstandard growth.

On the other hand, a class of function spaces called Herz-type spaces on ℝⁿ has attracted considerable attention in recent years because the interesting norm includes explicitly both local and global informations of the function. In 2011, Izuki^[7] obtained Herz spaces with variable exponent. In 2012, Wang and Liu^[8] introduced a certain Herz-type Hardy spaces with variable exponent. Subsequently, Wang^[9-12] proved the continuity of some operators on Herz-type Hardy spaces with variable exponent.

Suppose that S^n-1 denotes the unit sphere in ℝⁿ(n≥2) equipped with normalized Lebesgue measure. Let Ω∈Lip_β(S^n-1) for 0 < β≤1 be homogeneous function of degree zero and

$ \int_{{S^{n - 1}}} \Omega \left( {{x^\prime }} \right){\rm{d}}\sigma \left( {{x^\prime }} \right) = 0, $

where x′=x/|x| for any x≠0. In 1958, Stein^[13] introduced the Marcinkiewicz integral related to the Littlewood-Paley g function on ℝⁿ as

$ {\mu _\Omega }(f)(x) = {\left( {\int_0^\infty {{{\left| {{F_{\Omega ,t}}(f)(x)} \right|}^2}} \frac{{{\rm{d}}t}}{{{t^3}}}} \right)^{1/2}}, $

where

$ {F_{\Omega ,t}}(f)(x) = \int_{\left| {x - y} \right| \le t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}f(y){\rm{d}}y} . $

It is shown that μ_Ω is of weak type (1, 1) and of type (p, p) for 1 < p≤2.

For 0 < γ≤1, the Lipschitz space, Lipγ(ℝⁿ), is defined as

$ \begin{array}{l} \text{Lip}{_\gamma }\left( {{\mathbb{R}^n}} \right) = \hfill \\ \left\{ {f:{{\left\| f \right\|}_{{\text{Li}}{{\text{p}}_\gamma }}} = \mathop {\sup }\limits_{x,y \in {\mathbb{R}^n};x \ne y} \frac{{\left| {f(x) - f(y)} \right|}}{{{{\left| {x - y} \right|}^\gamma }}} < \infty } \right\}. \hfill \\ \end{array} $

Let b∈Lip_γ(ℝⁿ). The commutator generated by the Marcinkiewicz integral μ_Ω and b is defined by

$ \begin{array}{*{20}{c}} {\left[ {b,{\mu _\Omega }} \right](f)(x) = \left( {\int_0^\infty | \int_{|x - y| \le t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}} } \right.}\\ {{{\left. {{{\left. {[b(x) - b(y)]f(y){\rm{d}}y} \right|}^2}\frac{{{\rm{d}}t}}{{{t^3}}}} \right)}^{1/2}}.} \end{array} $

Motivated by Refs. [14-15], we will investigate the continuity of the commutators [b, μ_Ω] generated by Marcinkiewicz integral operators with rough kernels μ_Ω and Lipschitz functions b on the Herz-type spaces with variable exponent, where Ω∈L^s(S^n-1) for s≥1.

Throughout this paper, we denote the Lebesgue measure and the characteristic function of a measurable set A⊂ℝⁿ by A and χ_A, respectively. S(ℝⁿ) denotes the space of Schwartz functions, and S′ (ℝⁿ)denotes the dual space of S(ℝⁿ). The notation f≈g means that there exist constants C₁, C₂>0 such that C₁g≤f≤C₂g. Let B_k={x∈ℝⁿ:|x|≤2^k} and A_k=B_k\B_k-1 for k∈${\mathbb{Z}}$. ${\mathbb{Z}}_ +$ and ${\mathbb{N}}$ denote the sets of all positive and non-negative integers, respectively. χ_k=χ_Ak for k∈${\mathbb{Z}}$, ${{\tilde \chi }_k} = {\chi _k}$ if k∈${\mathbb{Z}}$₊, and ${{\tilde \chi }_0} = {\chi _{{B_0}}}$. In addition, δ₂ denotes the same as in Lemma 1.3.

1 Preliminaries

Firstly we give some notations and basic definitions on variable Lebesgue spaces.

Given an open set E⊂ℝⁿ and a measurable function p(·):E→[1, ∞). p′(·) is the conjugate exponent defined by p′(·)=p(·)/(p(·)-1).

The set P(E) consists of all p(·):E→[1, ∞) satisfying

$ {p^ - } = \text{essinf} \{ p(x):x \in E\} > 1, $

$ {p^ + } = \text{ esssup }\{ p(x):x \in E\} < \infty . $

By L^p(·)(E) we denote the space of all measurable functions f on E such that for some λ>0,

$ \int_E {{{\left( {\frac{{\left| {f(x)} \right|}}{\lambda }} \right)}^{p(x)}}} {\text{d}}x < \infty . $

This is a Banach function space with respect to the Luxemburg-Nakano norm

$ {\left\| f \right\|_{{L^{p( \cdot )}}(E)}} = \inf \left\{ {\lambda > 0:\int_E {{{\left( {\frac{{\left| {f(x)} \right|}}{\lambda }} \right)}^{p(x)}}} {\text{d}}x \leqslant 1} \right\}. $

The space L_loc^p(·)(Ω) is defined by L_loc^p(·)(Ω):={f:f∈L^p(·)(E) for all compact subsets E⊂Ω}.

The set B(ℝⁿ) consists of p(·)∈P(ℝⁿ) satisfying the condition that the Hardy-Littlewood maximal operator M is bounded on L^p(·)(ℝⁿ).

In variable L^p spaces there are some important lemmas^{[1, 3, 7]}, which are given as follows.

Lemma 1.1  If p(·)∈P(ℝⁿ) and satisfies

$ \left| {p(x) - p(y)} \right| \leqslant \frac{C}{{ - \log (|x - y|)}}, $

$ \left| {x - y} \right| \leqslant 1/2, $

(1)

and

$ \left| {p\left( x \right) - p\left( y \right)} \right| \leqslant \frac{C}{{\log (|x| + e)}},\left| y \right| \geqslant \left| x \right|, $

(2)

then p(·)∈B(ℝⁿ), that is, the Hardy-Littlewood maximal operator M being bounded on L^p(·)(ℝⁿ).

Lemma 1.2  Suppose q(·)∈B(ℝⁿ). Then there exists a constant C>0 such that for all balls B in ℝⁿ,

$ \frac{1}{{\left| B \right|}}{\left\| {{\chi _B}} \right\|_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}{\left\| {{\chi _B}} \right\|_{{L^{q'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}} \leqslant C. $

Lemma 1.3  Let q(·)∈B(ℝⁿ). Then there exists a positive constant C such that for all balls B in ℝⁿ and all measurable subsets S⊂B,

$ \frac{{{{\left\| {{\chi _S}} \right\|}_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}}{{{{\left\| {{\chi _B}} \right\|}_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}} \leqslant C{\left( {\frac{{\left| S \right|}}{{\left| B \right|}}} \right)^{{\delta _1}}} $

and

$ \frac{{{{\left\| {{\chi _S}} \right\|}_{{L^{q'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}}{{{{\left\| {{\chi _B}} \right\|}_{{L^{q'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}} \leqslant C{\left( {\frac{{\left| S \right|}}{{\left| B \right|}}} \right)^{{\delta _2}}}, $

where δ₁ and δ₂ are constants with 0 < δ₁, δ₂ < 1.

Lemma 1.4  (Generalized Hölder inequality)  Let p(·)∈Ρ(ℝⁿ). If f∈L^p(·)(ℝⁿ)and g∈L^p′(·)(ℝⁿ), then fg is integrable on ℝⁿ and

$ \int_{{\mathbb{R}^n}} {\left| {f(x)g(x)} \right|{\text{d}}x} \leqslant {r_p}{\left\| f \right\|_{{L^{p( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}{\left\| g \right\|_{{L^{p'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}, $

where r_p=1+1/p^--1/p⁺.

Next we recall some definitions and one lemma for the Herz-type spaces with variable exponent given in Refs. [7-8].

Definition 1.1  Let α∈ℝ, 0 < p≤∞, and q(·)∈P(ℝⁿ). The homogeneous Herz space $\dot K_{q\left( \cdot \right)}^{\alpha , p}$(ℝⁿ) consists of all f∈L_loc^q(·)(ℝⁿ\{0}) such that

$ {\left\| f \right\|_{\dot K_{q( \cdot )}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} = {\left\{ {\sum\limits_{k = - \infty }^\infty {{2^{k\alpha p}}} \left\| {f{\chi _k}} \right\|_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}^p} \right\}^{1/p}} < \infty . $

The non-homogeneous Herz space K_q(·)^{α, p}(ℝⁿ) is defined as the set of all f∈L_loc^q(·)(ℝⁿ) such that

$ {\left\| f \right\|_{K_{q( \cdot )}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} = {\left\{ {\sum\limits_{k = 0}^\infty {{2^{k\alpha p}}} \left\| {f{{\tilde \chi }_k}} \right\|_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}^p} \right\}^{1/p}} < \infty . $

Let G_N(f)(x) be the grand maximal function of f(x) defined by

$ {G_N}(f)(x) = \mathop {\sup }\limits_{\phi \in {A_N}} \left| {\phi _\nabla ^*(f)(x)} \right|, $

where A_N={ϕ∈S(ℝⁿ):$\mathop {\sup }\limits_{\left| \alpha \right|, \left| \beta \right| \le N} \left| {{x^\alpha }{D^\beta }\phi \left( x \right)} \right|$ and $\phi _\nabla ^*(f)(x) = \mathop {{\rm{sup}}}\limits_{|y - x| < t} \left| {{\phi _t}*f(y)} \right|$ with ${\phi _t}\left( x \right) = {t^{ - n}}\phi \left( {x/t} \right)$.

Definition 1.2  Let α∈ℝ, 0 < p < ∞, q(·)∈P(ℝⁿ) and N>n+1.

(ⅰ) The homogeneous Herz-type Hardy space $H\dot K_{q\left( \cdot \right)}^{\alpha , p}$(ℝⁿ) consists of all f∈S′(ℝⁿ) such that G_N(f)(x)∈$\dot K_{q\left( \cdot \right)}^{\alpha , p}$(ℝⁿ) and ${{\left\| f \right\|}_{H\dot{K}_{q\left( \cdot \right)}^{\alpha , p}\left( {{\mathbb{R}}^{n}} \right)}}$=${{\left\| {{G}_{N}}\left( f \right) \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha , p}\left( {{\mathbb{R}}^{n}} \right)}}$.

(ⅱ) The non-homogeneous Herz-type Hardy space HK_q(·)^{α, p}(ℝⁿ) is defined as the set of all f∈S′(ℝⁿ) such that G_N(f)(x)∈K_q(·)^{α, p}(ℝⁿ) and ‖f‖_{HKq(·)^{α, p}}(ℝⁿ)=‖G_N(f)‖_{Kq(·)^{α, p}}(ℝⁿ).

For x∈ℝ we denote by [x] the largest integer less than or equal to x.

Definition 1.3  Let nδ₂≤α < ∞, q(·)∈P(ℝⁿ), b∈L_loc¹(ℝⁿ), and integer s≥[α-nδ₂].

(ⅰ) A function a(x) on ℝⁿ is said to be a central (α, q(·))-atom, if it satisfies

1) supp a⊂B(0, r)={x∈ℝⁿ:|x| < r},

2) ‖a‖_{L^q(·)(ℝⁿ)}≤|B(0, r)|^-α/n,

3) ∫_ℝⁿa(x)x^βdx=∫_ℝⁿa(x)b(x)x^βdx=0, |β|≤s.

(ⅱ) A function a(x) on ℝⁿ is said to be a central (α, q(·))-atom of the restricted type, if it satisfies the above conditions 2) and 3) and

$ {\left. 1 \right)^\prime }{\text{supp}}a \subset B\left( {0,r} \right),r \geqslant 1. $

If r=2^k for some k∈${\mathbb{Z}}$ in Definition 1.3, then the corresponding central (α, q(·))-atom is called a dyadic central (α, q(·))-atom.

Lemma 1.5  Let nδ₂≤α < ∞, 0 < p < ∞ and q (·)∈B(ℝⁿ).Then ${{\left\| f \right\|}_{H\dot{K}_{q\left( \cdot \right)}^{\alpha , p}}}\left( {{\mathbb{R}}^{n}} \right)$ (or HK_q(·)^{α, p}(ℝⁿ)) if and only if

$ f = \sum\limits_{k = - \infty }^\infty {{\lambda _k}} {a_k}\left( {{\text{or}}\sum\limits_{k = 0}^\infty {{\lambda _k}} {a_k}} \right), $

in the sense of S′(ℝⁿ), where each a_k is a central (α, q(·))-atom(or central (α, q(·))-atom of restricted type) with support contained in B_k and $\sum\limits_{k=-\infty }^{\infty }{{{\left| {{\lambda }_{k}} \right|}^{p}}}<\infty \left( \text{ or }\sum\limits_{k=0}^{\infty }{{{\left| {{\lambda }_{k}} \right|}^{p}}}<\infty \right)$. Moreover,

$ {\left\| f \right\|_{H\dot K_{q\left( \cdot \right)}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} \approx \inf {\left( {\sum\limits_{k = - \infty }^\infty {{{\left| {{\lambda _k}} \right|}^p}} } \right)^{1/p}} $

$ \left( {{\text{or}}{{\left\| f \right\|}_{HK_{q\left( \cdot \right)}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} \approx \inf {{\left( {\sum\limits_{k = 0}^\infty {{{\left| {{\lambda _k}} \right|}^p}} } \right)}^{1/p}}} \right), $

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

2 Main results and their proofs

Let b∈Lip_γ(ℝⁿ). It is easy to know that |[b, μ_Ω]|≤C‖b‖_LipγT_{Ω, γ}, where

$ {{\bar T}_{\Omega ,\gamma }}f(x) = \int_{{\mathbb{R}^n}} {\frac{{\left| {\Omega (x - y)} \right|}}{{{{\left| {x - y} \right|}^{n - \gamma }}}}} \left| {f(y)} \right|{\text{d}}y. $

In Ref. [16], the authors proved that T_{Ω, γ} is bounded from L^q₁(·)(ℝⁿ) to L^q₂(·)(ℝⁿ) for 1/q₁(x)-1/q₂(x)=γ/n and q₁(·)∈P(ℝⁿ) satisfying conditions(1) and (2) in Lemma 1.1 with q₁⁺ < n/γ. So we can get the following theorem.

Theorem 2.1  Suppose that b∈Lip_γ(ℝⁿ) with 0 < γ≤1. If q₁(·)∈P(ℝⁿ) satisfies conditions (1) and (2) in Lemma 1.1 with q₁⁺ < n/γ, 1/q₁(x)-1/q₂(x)=γ/n, Ω∈L^s(S^n-1)(s>q₂⁺) with 1≤s′ < q₁^-. Then [b, μ_Ω] is bounded from L^q₁(·)(ℝⁿ) to L^q₂(·)(ℝⁿ).

Next, we will give the continuity about the commutator [b, μ_Ω] on Herz-type Hardy spaces with variable exponent. Before stating our result, let us recall the definition of the L^s-Dini condition. We say that Ω satisfies the L^s-Dini condition if Ω(x′)∈L^s(S^n-1) with s≥1 is homogeneous of degree zero on ℝⁿ, and

$ \int_0^1 {\frac{{{\omega _s}(\delta )}}{\delta }} {\text{d}}\delta < \infty , $

where ω_s(δ) denotes the integral modulus of continuity of order s of Ω defined by

$ {\omega _s}(\delta ) = \mathop {\sup }\limits_{\left\| \rho \right\| \le \delta } {\left( {\int_{{S^{n - 1}}} {{{\left| {\Omega \left( {\rho x'} \right) - \Omega \left( {x'} \right)} \right|}^s}} {\rm{d}}\sigma \left( {x'} \right)} \right)^{1/s}} $

and ρ is a rotation on S^n-1 and $\left\| \rho \right\| = \mathop {\sup }\limits_{x' \in {S^{n - 1}}} \left| {\rho x' - x'} \right|$.

Theorem 2.2 Suppose that b∈Lip_γ(ℝⁿ) with 0 < γ≤1. If q₁(·)∈P(ℝⁿ) satisfies conditions (1) and (2) in Lemma 1.1 with q₁⁺ < n/γ, 1/q₁(x)-1/q₂(x)=γ/n, Ω∈L^s(S^n-1)(s>q₂⁺) with 1≤s′ < q₁^- and satisfies

$ \int_0^1 {\frac{{{\omega _s}(\delta )}}{{{\delta ^{1 + \gamma }}}}} {\text{d}}\delta < \infty . $

Let 0 < p1≤p2 < ∞ and nδ2≤α < nδ2+γ(or 0 < max(nδ2, α2)≤α1 < nδ2+γ). Then [b, μΩ] maps $H\dot{K}_{{{q}_{1}}\left( \cdot \right)}^{\alpha , {{p}_{1}}}\left( {{\mathbb{R}}^{n}} \right)$(or HKα1, p1q1(·)(ℝn)) continuously into $\dot{K}_{{{q}_{2}}\left( \cdot \right)}^{\alpha , {{p}_{2}}}\left( {{\mathbb{R}}^{n}} \right)$(or ${K}_{{{q}_{2}}\left( \cdot \right)}^{{\alpha}_{2} , {{p}_{2}}}\left( {{\mathbb{R}}^{n}} \right)$)).

In the proof of Theorem 2.2, we need the following lemmas in Refs. [3, 5, 17-18].

Lemma 2.1  Given E and p(·)∈P(E), let f:E×E→ℝ be a measurable function (with respect to product measure) such that for almost every y∈E, f(·, y)∈L^p(·)(E). Then

$ {\left\| {\int_E f ( \cdot ,y){\text{d}}y} \right\|_{{L^{p( \cdot )}}(E)}} \leqslant C\int_E {{{\left\| {f\left( { \cdot ,y} \right)} \right\|}_{{L^p}( \cdot )(E)}}{\text{d}}y} . $

Lemma 2.2  Define a variable exponent $\tilde{q}(\cdot )$ by $\frac{1}{p(x)}=\frac{1}{\tilde{q}(x)}+\frac{1}{q}$(x∈ℝⁿ). Then we have

$ {\left\| {fg} \right\|_{{L^p}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \leqslant C{\left\| f \right\|_{{L^{\tilde q( \cdot )}}}}_{\left( {{\mathbb{R}^n}} \right)}{\left\| g \right\|_{{L^q}\left( {{\mathbb{R}^n}} \right)}} $

for all measurable functions f and g.

Lemma 2.3  Let p(·)∈P(ℝⁿ) satisfies conditions (1) and (2) in Lemma 1.1. Then

$ {\left\| {{\chi _Q}} \right\|_{{L^{p( \cdot )}}\left( {{\mathbb{R}^n}} \right)}} \approx \left\{ {\begin{array}{*{20}{c}} {{{\left| Q \right|}^{\frac{1}{{p\left( x \right)}}}}}&{{\text{if}}\left| Q \right| \leqslant {2^n}\;{\text{and}}\;x \in Q,} \\ {{{\left| Q \right|}^{\frac{1}{{p\left( \infty \right)}}}}}&{{\text{if}}\left| Q \right| \geqslant 1} \end{array}} \right. $

for every cube (or ball) Q⊂ℝⁿ, where p(∞)=$\underset{x\to \infty }{\mathop{\lim }}\, p\left( x \right)$.

Lemma 2.4  Suppose that Ω satisfies the L^s-Dini condition (1≤s < ∞). Then for any R>0 and x∈ℝⁿ, when |y| < R/2, there is a constant C>0 such that

$ \begin{array}{*{20}{c}} {{{\left( {\int_{R < \left| x \right| < 2R} {{{\left| {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^n}}} - \frac{{\Omega (x)}}{{{{\left| x \right|}^n}}}} \right|}^s}} {\text{d}}x} \right)}^{1/s}} \leqslant } \\ {C{R^{\frac{n}{s} - n}}\left\{ {\frac{{\left| y \right|}}{R} + \int_{\left| y \right|/2R < \delta < \left| y \right|/R} {\frac{{{\omega _s}(\delta )}}{\delta }{\text{d}}\delta } } \right\}.} \end{array} $

Proof of Theorem 2.2  We only prove homogeneous case. In Ref. [19], the authors proved K_q(·)^{α₁, p₂}(ℝⁿ)⊂K_q(·)^{α₂, p₂}(ℝⁿ) for 0 < α₂≤α₁. So the non-homogeneous case can be proved in the same way. Let $f\in H\dot{K}_{{{q}_{1}}\left( \cdot \right)}^{\alpha , {{p}_{1}}}\left( {{\mathbb{R}}^{n}} \right)$ and b∈Lip_γ(ℝⁿ). By Lemma 1.5 we get $f=\sum\limits_{j=-\infty }^{\infty }{{{\lambda }_{j}}}{{a}_{j}}$, where ${{\left\| f \right\|}_{H\dot{K}_{{{q}_{1}}\left( \cdot \right)}^{\alpha , {{p}_{1}}}\left( {{\mathbb{R}}^{n}} \right)}}$≈$\inf {{\left( \sum\limits_{j=-\infty }^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{{{p}_{1}}}}} \right)}^{1/{{p}_{1}}}}$ (the infimum is taken over above decompositions of f), and a_j is a dyadic central (α, q₁(·))-atom with the support B_j. We have

$ \begin{array}{*{20}{c}} {\left\| {\left[ {b,{\mu _\Omega }} \right](f)} \right\|_{\dot K_{{q_2}\left( \cdot \right)}^{\alpha ,{p_2}}\left( {{\mathbb{R}^n}} \right)}^{{p_1}}} \\ { = {{\left\{ {\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_2}}}} \left\| {\left[ {b,{\mu _\Omega }} \right](f){\chi _k}} \right\|_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}^{{p_2}}} \right\}}^{{p_1}/{p_2}}}} \\ { \leqslant \sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} \left\| {\left[ {b,{\mu _\Omega }} \right](f){\chi _k}} \right\|_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}^{{p_1}} \leqslant C\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} } \\ {{{\left( {\sum\limits_{j = - \infty }^{k - 2} {\left| {{\lambda _j}} \right|} {{\left\| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right){\chi _k}} \right\|}_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right)}^{{p_1}}} + C\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} } \\ {{{\left( {\sum\limits_{j = k - 1}^\infty {\left| {{\lambda _j}} \right|} {{\left\| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right){\chi _k}} \right\|}_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right)}^{{p_1}}}} \\ { = :{U_1} + {U_2}.} \end{array} $

(3)

We first estimate U₁. Note that

$ \begin{array}{*{20}{c}} {\left| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right)(x)} \right| \leqslant } \\ {\left( {\int_0^{|x|} | \int_{|x - y| \leqslant t|} {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}}} [b(x) - b(y)]} \right.} \\ {{{\left. {{{\left. {{a_j}(y){\text{d}}y} \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} \right)}^{1/2}} + } \\ {\left( {\int_{|x|}^\infty | \int_{|x - y| \leqslant t} {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}}} [b(x) - b(y)]} \right.} \\ {{{\left. {{{\left. {{a_j}(y){\text{d}}y} \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} \right)}^{1/2}}} \\ { = :{U_{11}} + {U_{12}}.} \end{array} $

When x∈A_k and |x-y|≤t with t≤|x|, it follows from j≤k-2 that |x-y|~|x|. By mean value theorem we have

$ \left| {\frac{1}{{|x - y{|^2}}} - \frac{1}{{|x{|^2}}}} \right| \leqslant \frac{{|y|}}{{|x - y{|^3}}}. $

(4)

Then by (4), the Minkowski inequality, the generalized Hölder inequality and the vanishing moments of a_j we have

$ \begin{gathered} {U_{11}} \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\left( {\int_{|x - y|}^{|x|} {\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}{\text{d}}y \hfill \\ \;\;\;\;\; \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\left| {\frac{1}{{|x - y{|^2}}} - \frac{1}{{|x{|^2}}}} \right|^{1/2}}{\text{d}}y \hfill \\ \;\;\;\;\; \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|\frac{{|y{|^{1/2}}}}{{|x - y{|^{3/2}}}}{\text{d}}y \hfill \\ \;\;\;\;\; \leqslant C{2^{(j - k)/2}}\int_{{A_j}} {\left| {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^n}}} - \frac{{\Omega (x)}}{{{{\left| x \right|}^n}}}} \right|} \hfill \\ \;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\text{d}}y. \hfill \\ \end{gathered} $

Similarly, we consider U₁₂. Noting that |x-y|~|x|, by the Minkowski inequality and the vanishing moments of a_j we have

$ \begin{gathered} {U_{12}} \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\left( {\int_{|x|}^\infty {\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}{\text{d}}y \hfill \\ \;\;\;\;\;\; \leqslant C\int_{{A_j}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^n}}} - \frac{{\Omega (x)}}{{|x{|^n}}}} \right|} \hfill \\ \;\;\;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\text{d}}y. \hfill \\ \end{gathered} $

So we have

$ \begin{array}{*{20}{c}} {\left| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right)(x)} \right| \leqslant C\int_{{A_j}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^n}}} - \frac{{\Omega (x)}}{{|x{|^n}}}} \right|} } \\ {\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\text{d}}y.} \end{array} $

Using Lemma 2.1 and the Minkowski inequality we have

$ \begin{align} & {{\left\| \left[ b,{{\mu }_{\Omega }} \right]\left( {{a}_{j}} \right){{\chi }_{k}} \right\|}_{L{{q}_{2}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}} \\ & \le \int_{{{B}_{j}}}{\left\| \left| \frac{\Omega (\cdot -y)}{|\cdot -y{{|}^{n}}}-\frac{\Omega (\cdot )}{|\cdot {{|}^{n}}} \right| \right.} \\ & {{\left. (b(\cdot )-b(y)){{\chi }_{k}}(\cdot ) \right\|}_{L{{q}_{2}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}}\left| {{a}_{j}}(y) \right|\text{d}y \\ & \le \int_{{{B}_{j}}}{\left\| \left| \frac{\Omega (\cdot -y)}{|\cdot -y{{|}^{n}}}-\frac{\Omega (\cdot )}{|\cdot {{|}^{n}}} \right| \right.} \\ & b(\cdot )-b(0)|{{\chi }_{k}}(\cdot )|{{|}_{L{{q}_{2}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}}\left| {{a}_{j}}(y) \right|\text{d}y+ \\ & \int_{{{B}_{j}}}{\left| \frac{\Omega (\cdot -y)}{|\cdot -y{{|}^{n}}}-\frac{\Omega (\cdot )}{|\cdot {{|}^{n}}} \right|{{\chi }_{k}}(}\cdot )|{{|}_{{{L}{{{q}_{2}}}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}} \\ & |b(0)-b(y)|\left| {{a}_{j}}(y) \right|\text{d}y \\ & =:{{U}_{13}}+{{U}_{14}}. \\ \end{align} $

For U₁₃, noting s>q₂⁺, we denote ${{{\tilde{q}}}_{2}}(\cdot )>1$ and $\frac{1}{{{q}_{2}}(x)}=\frac{1}{{{{\tilde{q}}}_{2}}(x)}+\frac{1}{s}$ By Lemma 2.2 we have

$ \begin{gathered} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right||b( \cdot ) - b(0)|{\chi _k}( \cdot )} \right\|_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {|b( \cdot ) - b(0)|{\chi _k}( \cdot )} \right\|_{L{{\bar q}_2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{\left\| b \right\|_{Li{p_\gamma }}}{2^{k\gamma }}{\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{\chi _{{B_k}}}} \right\|_{L_2^{\bar q}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

When |B_k|≤2ⁿ and x_k∈B_k, by Lemma 2.3 we have

$ \begin{gathered} {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \approx {\left| {{B_k}} \right|^{\frac{1}{{{{\tilde q}_2}\left( {{x_k}} \right)}}}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \approx {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}\left( \cdot \right)\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}. \hfill \\ \end{gathered} $

When |B_k|≥1 we have

$ \begin{gathered} {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \approx {\left| {{B_k}} \right|^{\frac{1}{{{{\tilde q}_2}\left( \infty \right)}}}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \approx {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}\left( \cdot \right)\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}. \hfill \\ \end{gathered} $

So we obtain

$ {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \approx {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}\left( \cdot \right)\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}. $

Meanwhile, by Lemma 2.4 we have

$ \begin{gathered} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{{{\left| \cdot \right|}^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}\left\{ {\frac{{|y|}}{{{2^k}}} + \int_{|y|/{2^k}}^{|y|/{2^{k - 1}}} {\frac{{{\omega _s}(\delta )}}{\delta }{\text{d}}\delta } } \right\} \hfill \\ \leqslant {2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}\left( {{2^{j - k + 1}} + {2^{(j - k + 1)\gamma }}\int_0^1 {\frac{{{\omega _s}(\delta )}}{\delta }{\text{d}}\delta } } \right) \hfill \\ \leqslant C{2^{\left( {k - 1} \right)\left( {\frac{n}{s} - n} \right)}}{2^{\left( {j - k} \right)\gamma }}. \hfill \\ \end{gathered} $

So by the generalized Hölder inequality we have

$ \begin{gathered} {U_{13}} = \int_{{B_j}} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|} \right.} \hfill \\ \;\;\;\;\;\;\;\;\;\;\left| {b( \cdot ) - b(0)} \right|{\chi _k}( \cdot )\left\| {_{_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right.\left| {{a_j}(y)} \right|{\text{d}}y \hfill \\ \;\;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{k\gamma }}{2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}{2^{(j - k)\gamma }} \hfill \\ \;\;\;\;\;\;\;\;\;\;{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}\int_{{B_j}} {\left| {{a_j}(y)} \right|{\text{d}}y} \hfill \\ \;\;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + \left( {j - k} \right)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;\;{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{\chi _{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

(5)

For U₁₄, similar to the method of U₁₃ we have

$ \begin{gathered} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{\chi _k}( \cdot )} \right\|_{L{{\tilde q}_2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{\chi _{{B_k}}}} \right\|_{L{{\tilde q}_2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}{2^{(j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{2^{ - kn + (j - k)\gamma - k\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

So by the generalized Hölder inequality we have

$ \begin{gathered} {U_{14}} = \int_{{B_j}} {{{\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|}_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \hfill \\ \;\;\;\;\;\;\;\;\;|b(0) - b(y)|\left| {{a_j}(y)} \right|{\text{d}}y \hfill \\ \;\;\;\;\;\;\;\;\; \leqslant C{2^{ - kn + (j - k)\gamma - k\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\int_{{B_j}} {\left| {b(0) - b(y)} \right|} \left| {{a_j}(y)} \right|{\text{d}}y \hfill \\ \;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + 2(j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;{\left\| {{\chi _{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + 2(j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;{\left\| {{\chi _{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

(6)

By (5), (6), Lemma 1.2 and Lemma 1.3 we have

$ \begin{gathered} {\left\| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right){\chi _k}} \right\|_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + (j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{X_{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{(j - k)\gamma }}{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}\frac{{{{\left\| {{\chi _{{B_j}}}} \right\|}_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}}}{{{{\left\| {{\chi _{{B_k}}}} \right\|}_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}}} \hfill \\ \leqslant C{2^{ - j\alpha + (j - k)\left( {\gamma + n{\delta _2}} \right)}}{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}. \hfill \\ \end{gathered} $

So we have

$ \begin{gathered} {U_1} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} \hfill \\ \;\;\;\;\;\;{\left( {\sum\limits_{j = - \infty }^{k - 2} {\left| {{\lambda _j}} \right|} {2^{ - j\alpha + (j - k)\left( {\gamma + n{\delta _2}} \right)}}} \right)^{{p_1}}} \hfill \\ \;\;\; = C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\sum\limits_{k = - \infty }^\infty {{{\left( {\sum\limits_{j = - \infty }^{k - 2} {\left| {{\lambda _j}} \right|} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right)}}} \right)}^{{p_1}}}} . \hfill \\ \end{gathered} $

When 1 < p₁ < ∞, take 1/p₁+1/p₁′=1. Since γ+nδ₂-α>0, by the Hölder inequality we have

$ \begin{array}{l} {U_1} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\;\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = - \infty }^{k - 2} {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}/2}}} \right)} \hfill \\ \;\;\;\;\;\;{\left( {\sum\limits_{j = - \infty }^{k - 2} {{2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right)p_1^\prime /2}}} } \right)^{{p_1}/p_1^\prime }} \hfill \\ \;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = - \infty }^{k - 2} {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}/2}}} \right)} \hfill \\ \;\;\; = C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} \left( {\sum\limits_{k = j + 2}^\infty {{2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}/2}}} } \right) \hfill \\ \;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{array} $

(7)

When 0 < p₁≤1, we have

$ \begin{array}{l} {U_1} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = - \infty }^{k - 2} {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}}}} \right)} \hfill \\ \;\;\;\;\; = C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} \left( {\sum\limits_{k = j + 1}^\infty {{2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}}}} } \right) \hfill \\ \;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{array} $

(8)

Next we estimate U₂, by the (L^q₁(·)(ℝⁿ), L^q₂(·)(ℝⁿ))-boundedness of the commutator [b, μ_Ω] we have

$ \begin{gathered} {U_2} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\;\;\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} {\left( {\sum\limits_{j = k - 1}^\infty {\left| {{\lambda _j}} \right|} {{\left\| {{a_j}} \right\|}_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right)^{{p_1}}} \hfill \\ \;\;\;\;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{k = - \infty }^\infty {{{\left( {\sum\limits_{j = k - 1}^\infty {\left| {{\lambda _j}} \right|} {2^{(k - j)\alpha }}} \right)}^{{p_1}}}} . \hfill \\ \end{gathered} $

If 0<p₁≤1, then we have

$ \begin{gathered} {U_2} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} \sum\limits_{k = - \infty }^{j + 1} {{2^{(k - j)\alpha {p_1}}}} \hfill \\ \;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{gathered} $

(9)

If 1 < p₁ < ∞, by the Hölder inequality we have

$ \begin{gathered} {U_2} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = k - 1}^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(k - j)\alpha {p_1}/2}}} \right)} \hfill \\ \;\;\;\;\;\;\;{\left( {\sum\limits_{j = k - 1}^\infty {{2^{(k - j)\alpha {p^\prime }1/2}}} } \right)^{{p_1}/{p^\prime }_1}} \hfill \\ \;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{gathered} $

(10)

Thus, by (3), (7)-(10) we complete the proof of Theorem 2.2.