1. Introduction
Suppose \(
\mathbb{S}^{n-1}, (n\geq 2)\)
denote the unit sphere in \(
\mathbb{R}^{n}\)
equipped with the normalized Lebesgue measure \(
\mathrm{d}\sigma=\mathrm{d}(\sigma^{\prime})\)
. Let \(
\Omega\)
be homogeneous function of degree zero and satisfies
\begin{equation}
\label{1.1}
\int_{\mathbb{S}^{n-1}}\Omega(x’)\mathrm{d}\sigma(x’), \text{where} x’=x/|x|(x\neq 0).
\end{equation}
(1)
The Calderón-Zygmund singular integral operator \(
T_{\Omega}\)
is defined as
\begin{equation}
\label{1.2}
T_{\Omega}h(x)=\mathrm{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}h(y)\mathrm{d}y.
\end{equation}
(2)
Now, we recall the definitions of the corresponding commutators of the Calderón-Zygmund singular integral operator. Suppose that \(
b\in \mathrm{BMO}(\mathbb{R}^{n}),\)
the commutators \(
[b,T_{\Omega}]\)
generated by \(
b \text{and} T\)
is defined as
\begin{equation}
\label{1.3}
[b,T_{\Omega}]h(x)=\mathrm{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}[b(x)-b(y)]h(y)\mathrm{d}y.
\end{equation}
(3)
These operators were firstly introduced by Calderón and Zygmund in [
1,
2], in which they proved that these operator are bounded on \(
L^{p}\)
, where \(
0< p< 1\)
. Coifman
et al. [
3] showed that if \(
\Omega\in\dot{\Lambda}_{\gamma}(\mathbb{S}^{n-1})\)
where \(
\gamma\in
(0,1)\)
and \(
b\in\mathrm{BMO}(\mathbb{R}^{n})\)
, then \(
[b,T_{\Omega}]\)
is bounded on \(
L^{p}\)
. In 2011, Lu Ding and Yan [
4] proved that \(
T_{\Omega}\)
and the commutator \(
[b,T_{\Omega}]\)
are bounded on weighted \(
(L^{p}(\mathbb{R}^{n}),L^{q}(\mathbb{R}^{n}))\)
.
In last 30 years, the function spaces with variable exponent have attracted researchers since the paper [
5] appeared in 1991, see, for example [
6,
7,
8,
9,
10] and their references. Recently, Jingshi Xu and Xiaodi Yang [
11] studied the Herz-Morrey-Hardy spaces with variable exponent and their applications.
Motivated by [
11,
12,
13], our main purpose of this paper is to study some boundedness for commutators of Calderón\(-\)Zygmund operators on Herz-Morrey-Hardy space with two variable exponents. The main tools are properties of variable exponent, BMO function and Lipschitz function.
Definition 1.
Let \(
\Omega\subset\mathbb{R}^{n}\)
be a subset of \(
\mathbb{R}^{n}\)
with the Lebesgue measure \(
>0.\)
For a measurable function \(
p(\cdot):\Omega\rightarrow[1,\infty)\)
, the variable Lebesgue space is defined as
$$
L^{p(\cdot)}(\Omega):=\left\{h \text{is measurable on} \Omega:\rho_{p}(h) < \infty\right\},$$
where $$
\rho_{p}(h):=\int_{\Omega}\left(\frac{|h(x)|}{\mu}\right)^{p(x)}dx 0.$$
The set \(
L^{p(\cdot)}(\Omega)\)
is a quasi Banach space with following Luxemburg-Nakano norm
$$
\|h\|_{L^{p(\cdot)}}:=\inf\left\{\mu>0 : \rho_{p}(\mu^{-1} h)\leq1\right\}.$$
The space \(
L_{\mathrm{loc}}^{p(\cdot)}(\Omega)\)
is defined as
$$
L_{\mathrm{loc}}^{p(\cdot)}(\Omega):=\left\{h: h\chi_{k}\in L^{p(\cdot)}(\mathbb{R}^{n}) \text{for any compact subset} K\subset\Omega\right\}.$$
Suppose \(
\mathcal{P}(\Omega)\)
represents the set of all function \(
p:\Omega\rightarrow[1,\infty)\)
. Assume that \(
p_{-}=\text{ess}\inf_{x\in\Omega} p(x)\)
and \(
p_{+}=\text{ess}\sup_{x\in\Omega} p(x)\)
. Set \(
p_{-}>1 \text{,} p_{+}< \infty\)
and \(
p(\cdot), p'(\cdot)\)
are conjugate exponent function defined by \(
1/p(\cdot)+1/p'(\cdot)=1.\)
Let \(
\mathcal{B}(\Omega)\)
be the set of \(
p(\cdot)\in\mathcal{P}(\Omega)\)
satisfying that the maximal function is bounded on \(
L^{p(\cdot)}\)
.
Definition 2.
(see[11]. Let \(
p(\cdot)\in \mathcal{P}(\mathbb{R}^{n}), 0< q< \infty, 0\leq \lambda n+1\)
. The homogeneous Herz-Morrey-Hardy spaces \(
HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})\)
and nonhomogeneous Herz-Morrey-Hardy spaces
\(
HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})\)
are defined as
$$
HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})=\left\{h\in\mathcal{S^{\prime}}(\mathbb{R}^{n}):
\|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})}
:=\|G_{N}h\|_{M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})}< \infty\right\},$$
$$
HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})=\left\{h\in\mathcal{S^{\prime}}(\mathbb{R}^{n}):\|h\|_{HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})}
:=\|G_{N}h\|_{M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})}< \infty\right\}.$$
respectively.
2. Preliminaries and Lemmas
Proposition 3.
(see[14]. Given a function \(
p(\cdot): \mathbb{R}^{n} \rightarrow [ 1 , \infty).\)
If \(
p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)
satisfies
\begin{equation}
\label{2.1}
| p(x) – p(y)|\leq \frac{ -C}{Log( |x – y|)}; | x – y| \leq 1/ 2 ,
\end{equation}
(4)
and
\begin{equation}
\label{2.2}
| p(x) – p(y)|\leq \frac{ C}{Log( e +|x|)}; |y|\geq|x|,
\end{equation}
(5)
then, \(
p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\)
.
Lemma 4. (see[5]). (Generalized Hölder’s Inequality) Given \(
p(\cdot), p_{1}(\cdot), p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\)
.
- For every \(
h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}) \text{and} g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\)
, we have
$$
\int_{\mathbb{R}^{n}}|h(x) g(x)| dx \leq C\|h\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})},$$
where \(
C_{p}=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}\)
.
- For every \(
h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}),\)
and \(
g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\)
, when \(
\frac{1}{p(\cdot)}=\frac{1}{p_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}\)
, we have
$$
\|h(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{p_{2}}(\mathbb{R}^{n})}\|h(x)\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})},$$
where \(
C_{p_{1},p_{2}}=[1+\frac{1}{p_{1-}}-\frac{1}{p_{1+}}]^{\frac{1}{p_{-}}}\)
.
Lemma 5. (see[15, 16]). Given \(
p(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\)
. If there exist positive constants \(
C,\)
\(
\delta_{1}\)
and \(
\delta_{2}\)
such that \(
\delta_{1}, \delta_{2}< 1\)
, then for all balls \(
B\subset\mathbb{R}^{n}\)
and all measurable subset \(
R\subset B,\)
we have
$$
\frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{|R|}{|B|},
\frac{\|\chi_{R}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{2}}, \frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{
\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{1}}.$$
Lemma 6. (see[17]). If \(
p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}),\)
then there exists a constant \(
C > 0\)
such that for any ball \(
B\)
in \(
\mathbb{R}^{n}\)
, we have
$$
\frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C.
$$
Now, the BMO function and BMO norm are defined as
\begin{align*}
\mathrm{BMO}(\mathbb{R}^{n})&:=\left\{b\in L^{1}_{loc}(\mathbb{R}^{n}):\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}< 0\right\},\\
\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}&:=\sup\limits_{Q:\text{cube}}|Q|^{-1}\int_{Q}|b(x)-b_{Q}|\text{d}x.
\end{align*}
respectively.
Lemma 7. (see[18]). Given \(
p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}), b\in \mathrm{BMO}(\mathbb{R}^{n})\)
. If \(
i,j\in\mathbb{Z}\)
with \(
i< j\)
, then we have
- \(
C^{-1}\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}
\leq\sup\limits_{B}\frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\|(b-b_{B})\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
\leq C\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}.\)
- \(
\|(b-b_{B_{i}})\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C( j-i)\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}.\)
Lemma 8. (see[18]). Suppose that \(
p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}), q\in[0,\infty)\)
and \(
\lambda\in[0,\infty)\)
. If \(
\alpha(\cdot)\)
is log-Hölder’s continuous both at origin and at infinity, then
\begin{eqnarray*}
&&\qquad\|h\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q}(\mathbb{R}^{n})}\approx max \left\{{\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda }
\left(\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q} \left\|h\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}},
\right.\\
&&\left.
\sup\limits_{L>0,L\in \mathbb{Z}} \left[2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{-1}
2^{k\alpha(0)q}\left\|h\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q} + 2^{-L\lambda }\left( \sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|h\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}.
\end{eqnarray*}
Lemma 9. (see[4]). Let \(
\Omega\)
satisfies \(
L^{r}\)
-Dini condition \(
r\in[1,\infty)\)
. If there exist constants \(
C>0\)
and \(
R>0\)
such that \(
|y|< R/2\)
, then for every \(
x\in\mathbb{R}^{n}\)
, we have
$$
\left(\int_{R< |x|< 2R}\left|\frac{\Omega(x-y)}{|x-y|^{n}}-\frac{\Omega(x)}{|x|^{n}}\right|^{r}dx\right)^{\frac{1}{r}}
\leq CR^{(\frac{n}{r}-n)}\left\{\frac{|y|}{R}+\int_{|y|/2R< \delta< |y|/R}\frac{w_{r}(\delta)}{\delta}d\delta\right\}.$$
Lemma 10. (see[11]). Suppose that \(
p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}), q\in[0,\infty)\)
and \(
\lambda\in[0,\infty)\)
. Let \(
\alpha(\cdot)\)
is log-Hölder’s continuous both at origin and at infinity. If \(
2\lambda\leq \alpha(\cdot), n\delta_{2}\leq \alpha(0), \alpha< \infty \text{and} \delta_{2}\)
as defined in Lemma 5. Then \(
h\in HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n}) \left(\text{or} HM{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})\right)\)
if and only if \(
h=\sum\limits_{k=-\infty}^{\infty}\lambda_{k}g_{k} \left(\text{or} \sum\limits_{k=0}^{\infty}\lambda_{k}g_{k}\right)\)
, in the sense of \(
\mathcal{S}'(\mathbb{R}^{n})\)
, where each \(
g_{k}\)
be a central \(
(\alpha(\cdot),p(\cdot))\)
-atom (or central \(
(\alpha(\cdot), p(\cdot))\)
-atom of restricted type) with support contained in \(
B_{k}\)
and \(
\sup\limits_{L\in\mathbb{Z}} 2^{-L\lambda}\sum\limits_{k=-\infty}^{L}|\lambda_{k}|^{q} < \infty\)
or \(
\left(\sup\limits_{L\in\mathbb{Z}} 2^{-L\lambda}\sum\limits_{k=0}^{L}|\lambda_{k}|^{q}< \infty\right).\)
Also,
\(
\|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}
\left(\sum\limits_{k=-\infty}^{L}|\lambda_{k}|^{q}\right)^{1/q}\left(\text{or}
\|h\|_{HM{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}
\left(\sum\limits_{k=0}^{L}|\lambda_{k}|^{q}\right)^{1/q}\right),\)
where infimum is taken over all above decomposition of \(h\)
.
Lemma 11(see [19]). Let \(p(\cdot)\in \mathcal{P}(\Omega)\)
and \(
h:\Omega\times \Omega\rightarrow \mathbb{R}\)
is a measurable function (with respect to product measure) such that, for almost every \(
y\in \Omega, h(\cdot,y)\in L^{p(\cdot)}(\Omega)\)
. Then
$$
\left\|\int_{\Omega}h(\cdot,y)dy\right\|_{L^{p(\cdot)}(\Omega)}\leq C \int_{\Omega}\left\|h(\cdot,y)\right\|_{L^{p(\cdot)}(\Omega)}dy.$$
Lemma 12. (see[19]). Suppose \(
p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)
satisfies conditions (4) and (5) of Proposition (3), then for any ball (or cube) \(
Q\subset\mathbb{R}^{n}\)
, we have
$$
{\|\chi_{Q}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\approx\left\{\begin{array}{ll}
|Q|^{\frac{1}{p(x)}}, \text{if} |Q|\leq 2^{n};\\
|Q|^{\frac{1}{p(\infty)}}, \text{if} |Q|\geq 1,
\end{array}\right.$$
where \(
p(\infty)=\lim\limits_{x\rightarrow\infty}p(x).\)
3. Main Results
In this section, we formulate and prove the main results of this paper.
Theorem 13.
Let \(
p(\cdot)\in \mathcal{B}(\mathbb{R}^{n}) \text{and} \Omega\in L^{r}(\mathbb{S}^{n-1})(r>p^{+})\)
satisfies
\begin{equation}
\int^{1}_{0}\frac{w_{r}(\delta)}{\delta^{1+\beta}}d\delta< \infty, 0< \beta\leq1.
\end{equation}
(6)
Suppose that \(
0< q< \infty, 0\leq\lambda< \infty \text{and} \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\)
satisfies conditions (4) and (5) of Proposition 3. If \(
2\lambda\leq\alpha(\cdot), n\delta_{2}\leq \alpha(0),\alpha_{\infty}< \beta+n\delta_{2},\)
then \(
T_{\Omega}\)
is bounded from \(
HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
or \(
\left(HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right)\)
to \(
M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
or \(
\left(MK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right).\)
Proof.
It suffices to prove for \(
HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
. Assume that \(
h\in HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
, then by Lemma 10,
\(
h=\sum\limits_{j=-\infty}^{\infty}\lambda_{j}g_{j}\)
converges in \(
\mathcal{S}'(\mathbb{R}^{n}),\)
where \(
\|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}
(\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q})^{1/q},\)
and \(
g_{j}\)
is a dyadic central \(
(\alpha(\cdot),p(\cdot))\)
-atom with support contained in \(
B_{j}\)
. For simplicity, we take \(
\Upsilon=\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}.\)
By virtue of Lemma 8, we have
\begin{eqnarray*}
&&\|T_{\Omega}(h)\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q}(\mathbb{R}^{n})}\approx max \left\{{\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda }
\left(\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q} \left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}},
\right.\\
&&\left.
\sup\limits_{L>0,L\in \mathbb{Z}} \left[2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{-1}
2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q} + 2^{-L\lambda }\left( \sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}\\
&&\approx max \left\{E,F+G\right\}.
\end{eqnarray*}
Let
\begin{eqnarray*}
E&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\
F&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\
G&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}.
\end{eqnarray*}
To finish our proof, we only need to show that there exists a constant \(
C>0\)
, such that \(
E, F \text{,} G\leq C\Upsilon.\)
First we prove that \(
E\leq C\Upsilon\)
.
\begin{eqnarray*}
E&=&\sup\limits_{L\leq0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&& +\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}:=E_{1}+E_{2}.
\end{eqnarray*}
By the \(
\left(L^{p(\cdot)}(\mathbb{R}^{n}),L^{p(\cdot)}(\mathbb{R}^{n})\right)\)
-boundedness of the \(
T_{\Omega}\)
(see[13]), we get
$$
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq |B_{j}|^{-\alpha_{j}/n}=2^{-j\alpha_{j}}.$$
Therefore, when \(
0< q\leq1,\)
we obtain
\begin{eqnarray*}
E_{1}&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
2^{-j\alpha_{j}}\right)^{q}\\
&\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left[\left(\sum\limits_{j=k}^{-1}|\lambda_{j}|
2^{-j\alpha(0)}+\sum\limits_{j=0}^{\infty}|\lambda_{j}|2^{-j\alpha_{\infty}}\right)^{q}\right]\\
&\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q}
2^{(k-j)\alpha(0)q}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\left(
\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)}\sum\limits_{j=0}^{\infty}|\lambda_{j}|2^{-j\alpha_{\infty}}
\right)^{q}\right]\\
&\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=-\infty}^{j}
2^{(k-j)\alpha(0)q}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}}
\sum\limits_{j=0}^{\infty}2^{-j\lambda q} |\lambda_{j}|^{q}2^{(\lambda-\alpha_{\infty})jq}2^{-L\lambda q}\sum\limits_{k=\infty}^{L}2^{k\alpha(0)q}\right]\\
&\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}
+\sup\limits_{L\leq0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=L}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=-\infty}^{j}2^{(k-j)\alpha(0)q}\right.\\
&& +\left.\Upsilon\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=0}^{\infty}2^{(\lambda-\alpha_{\infty})jq}\sum\limits_{k=\infty}^{L}2^{[k\alpha(0)-L\lambda]q}\right]\\
&\leq& C \left[\Upsilon+\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=L}^{-1}2^{-j\lambda q}|\lambda_{j}|^{q}2^{(j-L)\lambda q}\sum\limits_{k=-\infty}^{j}2^{(k-j)\alpha(0)q}+\Upsilon\right]\\
&\leq& C \left[\Upsilon+\Upsilon\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=L}^{-1}2^{(j-L)\lambda q}\sum\limits_{k=-\infty}^{j}2^{(k-j)\alpha(0)q}+\Upsilon\right]\leq C \Upsilon.
\end{eqnarray*}
when \(
1< q1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\)
. Since \(
r>p^{+}\)
, so by Lemma 4 and Lemma 11, we get
\begin{eqnarray}\label{3.2}
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}&&\leq
\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|\chi_{k}
\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
|g_{j}(y)|dy\nonumber\\
&&\leq \int_{B_{j}}\left\|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{k}\|_{L^{\widetilde{p}(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|dy.
\end{eqnarray}
(8)
Using Lemma 9, we get
\begin{eqnarray}
\left\|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right\|_{L^{r}(\mathbb{R}^{n})}
&&\leq C 2^{(k-1)(\frac{n}{r}-n)}
\left\{\frac{|y|}{2^{k-1}}+\int^{|y|/2^{k-1}}_{|y|/2^{k}}\frac{w_{r}(\delta)}{\delta}d\delta\right\}\nonumber\\
&&\leq C 2^{(k-1)(\frac{n}{r}-n)}
\left(2^{j-k}+2^{(j-k)\beta}\int^{1}_{0}\frac{w_{r}(\delta)}{\delta^{1+\beta}}d\delta\right)\nonumber\\
&&\leq C 2^{(k-1)(\frac{n}{r}-n)}2^{(j-k)\beta}.
\end{eqnarray}
(9)
By Lemma 12, we obtain
\begin{align}
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
&\leq \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}\left\|\chi_{k}
\right\|_{L^{\tilde{p}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|g_{j}(y)|dy\nonumber\\
&\leq C \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}\left(\|\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|B_{k}|^{\frac{1}{r}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.
\end{align}
(10)
From (8) and Lemmas 4-6, we get
\begin{align}\label{3.4}
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
&\leq C 2^{-nk+(j-k)\beta}\left(\|\chi_{B_{k}}\|^{-1}_{L^{p'(\cdot)}(\mathbb{R}^{n})}|B_{k}|\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&\leq C 2^{(j-k)\beta}\left(\frac{\|\chi_{B_{j}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B_{k}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&\leq C 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{j}}.
\end{align}
(11)
So, when \(
0< q\leq1,\)
we obtain
\begin{align}
E_{2}&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&\leq C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|)
2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\nonumber\\
&\leq C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&\leq C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\left(\sum\limits_{k=j+1}^{-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&\leq C \Upsilon.
\end{align}
(12)
When \(
0< q< \infty,\)
and \(
1/q+1/q^{\prime}=1,\)
by \(
n\delta_{2}\leq \alpha(0) < \beta +n\delta_{2}\)
and Hölder inequality, we have
\begin{eqnarray}
E_{2}&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|^{q}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\right)\nonumber\\
&& \times\left(\sum\limits_{j=-\infty}^{k-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\
&\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\nonumber\\
&\leq& C \Upsilon.
\end{eqnarray}
(13)
Next we prove that \(
F\leq C\Upsilon.\)
\begin{eqnarray}
F&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}
\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\nonumber\leq \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&& + \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&:=&F_{1}+F_{2}.
\end{eqnarray}
(14)
Since \(
0< q\leq1\)
, we get
\begin{eqnarray}
F_{1}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\leq C \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
2^{-j\alpha_{j}}\right)^{q}\nonumber\\
&\leq& C \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left[\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q}
2^{-j\alpha(0)q}+\sum\limits_{j=0}^{\infty}|\lambda_{j}|^{q}2^{-j\alpha_{\infty}q}\right]\nonumber\\
&\leq& C \left[\sum\limits_{k=-\infty}^{-1}\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q}
2^{(k-j)\alpha(0)q}+\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\sum\limits_{j=0}^{\infty}|\lambda_{j}|^{q}2^{-j\alpha_{\infty}q}\right]\nonumber\\
&\leq& C \left[\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=-\infty}^{j}
2^{(k-j)\alpha(0)q}+\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\sum\limits_{j=0}^{\infty}2^{-j\lambda q}\sum\limits_{l=0}^{j}|\lambda_{l}|^{q}2^{(\lambda-\alpha_{\infty})jq}\right]\nonumber\\
&\leq& C \left[\Upsilon+\Upsilon\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\sum\limits_{j=0}^{\infty}2^{(\lambda-\alpha_{\infty})jq}\right]\nonumber\\
&\leq& C \Upsilon.
\end{eqnarray}
(15)
when \(
0< q< \infty\)
and \(
1/q+1/q^{\prime}=1,\)
we deduce
\begin{eqnarray*}
F_{1}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\leq C \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
2^{-j\alpha_{j}}\right)^{q}\\
&\leq& C \left[\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=k}^{-1}|\lambda_{j}|
2^{(k-j)\alpha(0)}\right)^{q}
+\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=0}^{\infty}
|\lambda_{j}|2^{-j\alpha_{\infty}}\right)^{q}\right]\\
&\leq& C \left[\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q}
2^{(k-j)\alpha(0){\frac{q}{2}}}\right)\times\left(\sum\limits_{j=k}^{-1}
2^{(k-j)\alpha(0){\frac{q^{\prime}}{2}}}\right)^{\frac{q}{q^{\prime}}}\right.\\
&& \left.+\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=0}^{\infty}
|\lambda_{j}|^{q}2^{-j\alpha_{\infty}\frac{q}{2}}\right)
\times\left(\sum\limits_{j=0}^{\infty}2^{-j\alpha_{\infty}\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}
\right]
\end{eqnarray*}
\begin{eqnarray*}
&& \leq C \left[\sum\limits_{k=-\infty}^{-1}\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q}
2^{(k-j)\alpha(0){\frac{q}{2}}}
+ \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\sum\limits_{j=0}^{\infty}
|\lambda_{j}|^{q}2^{-j\alpha_{\infty}\frac{q}{2}}\right]\\
&& \leq C \left[\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=\infty}^{j}
2^{(k-j)\alpha(0){\frac{q}{2}}}+\sum\limits_{j=0}^{\infty}2^{-j\lambda q}\sum\limits_{l=-\infty}^{j}|\lambda_{l}|^{q} 2^{(\lambda-\frac{\alpha_{\infty}}{2})jq} \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\right]\\
&& \leq C \left[\Upsilon\sum\limits_{k=\infty}^{j}
2^{(k-j)\alpha(0){\frac{q}{2}}}+\Upsilon\sum\limits_{j=0}^{\infty} 2^{(\lambda-\frac{\alpha_{\infty}}{2})jq} \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\right]\nonumber\\
&&\leq C \Upsilon.
\end{eqnarray*}
Now we prove that \(
F_{2}\leq C \Upsilon.\)
When \(
0< q\leq1\)
, from (11) and \(
n\delta_{2}\leq \alpha(0) < \beta+n\delta_{2}\)
, we have
\begin{eqnarray*}
F_{2}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\leq C \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\nonumber\\
&&\leq C \sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\leq C \sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\left(\sum\limits_{k=j+1}^{-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&&\leq C \Upsilon.
\end{eqnarray*}
when \(
0< q\leq\infty,\)
and \(
1/q+1/q^{\prime}=1,\)
by \(
n\delta_{2}\leq \alpha(0) < \beta+n\delta_{2}\)
and Hölder inequality, we obtain
\begin{eqnarray}
&F_{2}&=\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&&\leq C \sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|^{q}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\right)\times\left(\sum\limits_{j=-\infty}^{k-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\
&&\leq C \sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\nonumber\\
&&\leq C \Upsilon.
\end{eqnarray}
(16)
Finally we prove that \(
G\leq C\Upsilon.\)
\begin{eqnarray*}
&G&=\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&& +\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&&:=G_{1}+G_{2}.
\end{eqnarray*}
When \(
0< q\leq1,\)
by the boundedness of the commutator \(
[b,T]\)
in \(
L^{p(\cdot)}(\mathbb{R}^{n})\)
, we obtain
\begin{eqnarray}
G_{1}&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q} \leq C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
2^{-j\alpha_{j}}\right)^{q}\nonumber\\
&& \leq C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{L}|\lambda_{j}|^{q}
2^{-j\alpha_{\infty}q}+\sum\limits_{j=L}^{\infty}|\lambda_{j}|^{q}2^{-j\alpha_{\infty}q}\right)\nonumber\\
&& \leq C \left(\sup\limits_{L>0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{j=0}^{L}|\lambda_{j}|^{q}\sum\limits_{k=0}^{j}2^{(k-j)\alpha_{\infty}q}
+\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=L}^{\infty}|\lambda_{j}|^{q}\sum\limits_{k=0}^{L}2^{(k-j)\alpha_{\infty}q}\right)\nonumber\\
&&\leq C \left(\sup\limits_{L>0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{j=0}^{L}|\lambda_{j}|^{q}\sum\limits_{k=0}^{j}2^{(k-j)\alpha_{\infty}q}
\right.\left.+\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{j\lambda q-L\lambda q}2^{-j\lambda q}\sum\limits_{l=-\infty}^{j}|\lambda_{l}|^{q}\sum\limits_{k=0}^{L}2^{(k-j)\alpha_{\infty}q}\right)\nonumber\\
&&\leq C \left(\Upsilon
+\Upsilon\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)\lambda q}2^{(L-j)\alpha_{\infty}q}\right)\leq C \left(\Upsilon
+\Upsilon\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{[(j-L)(\lambda-\alpha_{\infty} )]q}\right)\leq C \Upsilon.
\end{eqnarray}
(17)
When \(0< q< \infty,\)
by using Hölder inequality, we have
\begin{eqnarray*}
&G_{1}&=\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&& \leq C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|^{q}
|B_{j}|^{-j\alpha_{j}\frac{q}{2n}}\right)\times\left(\sum\limits_{j=k}^{\infty}
|B_{j}|^{-j\alpha_{j}\frac{q^{\prime}}{2n}}\right)^{\frac{q}{q^{\prime}}}\\
&& \leq C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|^{q}
2^{(k-j)\alpha_{\infty}\frac{q}{2}}\right)\\
&& \leq C \left[\sup\limits_{L>0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{j=0}^{L}|\lambda_{j}|^{q}\sum\limits_{k=0}^{j}2^{(k-j)\alpha_{\infty}\frac{q}{2}}
+\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=L}^{\infty}|\lambda_{j}|^{q}\sum\limits_{k=0}^{L}2^{(k-j)\alpha_{\infty}\frac{q}{2}}\right]\\
&& \leq C \left[\sup\limits_{L>0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{j=0}^{L}|\lambda_{j}|^{q}
+\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)\lambda q}2^{-j\lambda q}\sum\limits_{l=-\infty}^{j}|\lambda_{l}|^{q}\sum\limits_{k=0}^{L}2^{(k-j)\alpha_{\infty}\frac{q}{2}}\right]\\
&& \leq C \left[\Upsilon
+\Upsilon\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)\lambda q}2^{(L-j)\frac{\alpha_{\infty}}{2}q}\right]\\
&& \leq C \left[\Upsilon
+\Upsilon\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)(\lambda-\frac{\alpha_{\infty}}{2}) q}\right]\nonumber\\
&& \leq C \Upsilon.
\end{eqnarray*}
For \(
G_{2}\leq C \Upsilon\)
, when \(
0< q< \infty,\)
and \(
1/q+1/q^{\prime}=1,\)
from (7), \(n\delta_{2} \leq\alpha(0), \alpha_{\infty}< \beta +n\delta_{2}\)
and applying Hölder inequality, we have
\begin{eqnarray*}
G_{2}&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
2^{(j-k)(\beta+n\delta_{2})-j\alpha_{j}}\right)^{q}\\
&\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|
2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\\
&& + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}|
2^{(j-k)(\beta+n\delta_{2})-j\alpha_{\infty}}\right)^{q}\\
&\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|
2^{[(\beta+n\delta_{2})-\alpha(0)]j}\right)^{q}\\
&& + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}|
2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]}\right)^{q}\\
&\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|
2^{j\frac{[(\beta+n\delta_{2})-\alpha(0)]}{2}}\right)^{q} \times\left(\sum\limits_{j=-\infty}^{-1}(k-j)^{q^{\prime}}
2^{j[(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\\
&& + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}|
2^{(j-k)\frac{[(\beta+n\delta_{2})-\alpha_{\infty}]}{2}}\right)^{q} \times\left(\sum\limits_{j=0}^{k-1}(k-j)^{q^{\prime}}
2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\\
&\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}
2^{[(\beta+n\delta_{2})-\alpha(0)]j\frac{q}{2}} + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\\
&\leq& C \left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}
\right.\left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\
\end{eqnarray*}
\begin{eqnarray*}
&\leq& C \left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}
\right.\left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{L}
2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\
&\leq& C \left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}
+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\right]\nonumber\\
&\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{L-1}|\lambda_{j}|^{q}\leq C \Upsilon.
\end{eqnarray*}
The proof is completed.
Theorem 14.
Suppose that \( b\in \mathrm{BMO}(\mathbb{R}^{n}), p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\) and let \(\Omega\in L^{r}(\mathbb{S}^{n-1})(r>p^{+})\)
satisfies (7). Let \(
0< q< \infty, 0\leq\lambda< \infty \text{and} \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\)
satisfies conditions (4) and (5) of Proposition 3. If \(
2\lambda\leq\alpha(\cdot), n\delta_{2}\leq \alpha(0),\alpha_{\infty}< \beta+n\delta_{2},\)
then \(
[b,T_{\Omega}]\)
is bounded from \(
HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
or \(
\left(HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right)\)
to \(
M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
or \(
\left(MK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right).\)
Proof.
It suffices to prove for \(
HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
. Set \(
b\in \mathrm{BMO}(\mathbb{R}^{n})\)
and \(
h\in HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\)
. By Lemma 10, \(
h=\sum\limits_{j=-\infty}^{\infty}\lambda_{j}g_{j}\)
converges in \(
\mathcal{S}'(\mathbb{R}^{n}),\)
where \(
\|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}
\left(\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\right)^{1/q},\)
and \(
g_{j}\)
is a dyadic central \(
(\alpha(\cdot),p(\cdot))\)
-atom with support contained in \(
B_{j}\)
. For simplicity, we denote \(
\Upsilon=\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}.\)
By virtue of Lemma 8, we rewrite
$$
\begin{array}{ll}
\|T^{b}_{\Omega}(h)\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q}(\mathbb{R}^{n})} \approx max \left\{{\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda }
\left(\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q} \left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}},
\right.\\
\\
\left.
\sup\limits_{L>0,L\in \mathbb{Z}} \left[2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{-1}
2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q} + 2^{-L\lambda }\left( \sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}\\ \approx max \left\{E’,F’+G’\right\}
\end{array}$$
where
\begin{eqnarray*}
&E’&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}
\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\
&F’&=\sum\limits_{k=-\infty}^{-1}
2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\
&G’&=\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}.
\end{eqnarray*}
To complete the prove, we only need to show that there exists a constant \(
C>0\)
, such that \(
E’,F’ \text{,} G’\leq C\Upsilon.\)
First we show that \(
E’\leq C\Upsilon.\)
\begin{eqnarray*}
&E’&=\sup\limits_{L\leq0,L\in \mathbb{Z}}
2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\\
&&\leq \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T^{b}_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&& +\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T^{b}_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&&:=E’_{1}+E’_{2}.
\end{eqnarray*}
By the \(
\left(L^{p(\cdot)}(\mathbb{R}^{n}),L^{p(\cdot)}(\mathbb{R}^{n})\right)\)
-boundedness of the \(
T^{b}_{\Omega}\)
(see [13]) and following the same way as we estimated \(
E_{1}\)
in Theorem 13, we get
\(
E’_{1}\leq C \|b\|_{\ast}\Upsilon.\)
Now, we estimate \(E’_{2}\)
. For each \(
k\in \mathbb{Z}\)
and \(
x\in A_{k}\)
, by Lemma 7 and Minkowski inequality, we get
\begin{eqnarray*}
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
&\leq&
\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|(b(\cdot)-b(y))\chi_{k}
\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|\mathrm{d}y\\
&\leq&\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right||b(\cdot)-b_{B_{j}}|\chi_{k}
\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|\mathrm{d}y\\
&&+\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|\chi_{k}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|b_{B_{j}}-b(y)||g_{j}(y)|\mathrm{d}y.
\end{eqnarray*}
Since \(
\tilde{p}(\cdot)>1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\)
. Since \(
r>p^{+}\)
, so by Lemma 4 and Lemma 11, we get
\begin{eqnarray*}
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
&\leq& \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}\left\|b(\cdot)-b_{B_{j}}\chi_{k}
\right\|_{L^{\tilde{p}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|g_{j}(y)|dy\\
&&+\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|\right\|_{L^{r}(\mathbb{R}^{n})}
\|\chi_{B_{k}}\|_{L^{\tilde{p}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|b_{B_{j}}-b(y)||g_{j}(y)|dy
\end{eqnarray*}
From (8) and Lemmas 4-6, we get
\begin{eqnarray}
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}&\leq& C \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}(k-j)\|b\|_{\ast}\left(
\|\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|B_{k}|^{\frac{1}{r}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&&+\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|\right\|_{L^{r}(\mathbb{R}^{n})}
\|(b_{B_{j}}-b)\chi_{B_{j}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\left(
\|\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|B_{k}|^{\frac{1}{r}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&&\leq C \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|\right\|_{L^{r}(\mathbb{R}^{n})}(k-j)\|b\|_{\ast}\left(
\|\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|B_{k}|^{\frac{1}{r}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&& +\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|\right\|_{L^{r}(\mathbb{R}^{n})}\|b\|_{\ast}
\left(\|\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|B_{k}|^{\frac{1}{r}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&&\leq C \|b\|_{\ast} (k-j)2^{-nk+(j-k)\beta}\left(\|\chi_{B_{k}}\|^{-1}_{L^{p'(\cdot)}(\mathbb{R}^{n})}|B_{k}|\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&&\leq C \|b\|_{\ast} (k-j)2^{(j-k)\beta}\left(\frac{\|\chi_{B_{j}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B_{k}}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}}\right)
\|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&&\leq C \|b\|_{\ast} (k-j)2^{-j\alpha_{j}+(j-k)(\beta+n\delta_{2})}.\label{aaa}
\end{eqnarray}
(18)
Therefore, when \(
0< q\leq1\)
, we obtain
\begin{eqnarray*}
&E'_{2}&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
(k-j)2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
(k-j) 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\left(\sum\limits_{k=j+1}^{-1}
(k-j) 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\Upsilon.
\end{eqnarray*}
When \(
0< q\leq\infty,\)
and \(
1/q+1/q^{\prime}=1,\)
by \(
n\delta_{2}\leq \alpha(0) < \beta +n\delta_{2}\)
and Hölder inequality, we have
\begin{eqnarray*}
&E'_{2}&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|^{q}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\right)\nonumber\\
&& \times\left(\sum\limits_{j=-\infty}^{k-1}
(k-j)^{q^{\prime}} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\Upsilon.
\end{eqnarray*}
Now we prove that \(
F'\leq C\Upsilon.\)
\begin{eqnarray*}
F'&=& \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left\|T_{\Omega}^{b}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&& + \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&:=&F'_{1}+F'_{2}.
\end{eqnarray*}
To estimate \(
F'_{1}\)
. By the boundedness of the \(
T^{b}_{\Omega}\)
on \(
L^{p(\cdot)}(\mathbb{R}^{n})\)
(see[
13]) and following the same way as we estimated \(
F_{1}\)
in Theorem 13, we get
\begin{equation}
F’_{1} \leq C \|b\|^{q}_{\ast} \Upsilon.
\end{equation}
(19)
For \(
F_{2}\)
, when \(
0< q\leq1\)
, by inequality (18) and \(
n\delta_{2}\leq \alpha(0)< \varepsilon+n\delta_{2}\)
, we have
\begin{eqnarray}
F’_{2}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
(k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
(k-j) 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\left(\sum\limits_{k=j+1}^{-1}
(k-j) 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\
&&\leq C \|b\|^{q}_{\ast}\Upsilon.
\end{eqnarray}
(20)
when \(0< q\leq\infty,\)
and \(1/q+1/q^{\prime}=1.\)
by \(n\delta_{2}\leq \alpha(0) < \beta+n\delta_{2}\)
and Hölder inequality, we obtain
\begin{eqnarray*}
F’_{2}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(
\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\
&\leq& C \|b\|^{q}_{\ast}\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|^{q}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\right)\nonumber \times\left(\sum\limits_{j=-\infty}^{k-1}
(k-j)^{q^{\prime}} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\
&\leq& C \|b\|^{q}_{\ast}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{-1}
2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\nonumber\\
&\leq& C \|b\|^{q}_{\ast}\Upsilon.
\end{eqnarray*}
Finally, we show that \(
G’\leq C\Upsilon.\)
\begin{eqnarray*}
G’&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}
\left\|T_{\Omega}^{b}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\
&\leq& \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&& +\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q},\\
&:=&G’_{1}+G’_{2}.
\end{eqnarray*}
For \(
G’_{1}\)
, by the boundedness of the commutator \(
T_{\Omega}^{b}\)
in \(
L^{p(\cdot)}(\mathbb{R}^{n})\)
(see [
13]
), and following the same way as we estimated \(
G_{1}\)
in Theorem 13, we get
\begin{equation}
G’_{1}\leq C \|b\|^{q}_{\ast}\Upsilon.
\end{equation}
(21)
Now, we estimate \(G’_{2}\)
. When \(0< q< \infty,\)
and \(1/q+1/q^{\prime}=1,\)
from inequality (18), since \(n\delta_{2} \leq\alpha(0), \alpha_{\infty}< \beta+n\delta_{2}\)
and applying Hölder inequality, we obtain
\begin{eqnarray*}
G’_{2}&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\
&\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|
(k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{j}}\right)^{q}\\
&\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|
(k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\\
&& + C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}|
(k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{\infty}}\right)^{q}\\
&\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|
(k-j) 2^{[(\beta+n\delta_{2})-\alpha(0)]j}\right)^{q}\\
&& + C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}|
(k-j) 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]}\right)^{q}\\
&\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|
2^{j\frac{[(\beta+n\delta_{2})-\alpha(0)]}{2}}\right)^{q}\left(\sum\limits_{j=-\infty}^{-1}(k-j)^{q^{\prime}}
2^{j[(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\\
&& + C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}|
2^{(j-k)\frac{[(\beta+n\delta_{2})-\alpha_{\infty}]}{2}}\right)^{q}
\left(
\sum\limits_{j=0}^{k-1}(k-j)^{q^{\prime}}2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]
\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\
&\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}
2^{[(\beta+n\delta_{2})-\alpha(0)]j\frac{q}{2}}\right)+ C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\nonumber\\
&\leq& C \|b\|_{\ast}\left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\right.\left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\
&\leq& C \|b\|_{\ast}\left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\right. \left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{L}
2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\
&\leq& C \|b\|_{\ast}\left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}
+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\right]\nonumber\\
&\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{L-1}|\lambda_{j}|^{q}\nonumber\\
&\leq& C \|b\|_{\ast}\Upsilon.
\end{eqnarray*}
The proof is completed.
Theorem 15.
Suppose that \(b\in \dot{\Lambda}_{\gamma}(\mathbb{R}^{n})(0< \gamma\leq 1), p_{1}(\cdot), p_{2}(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\)
be such that \(
p_{1}^{+}q^{+}_{2})\)
with \(
1\leq r’ < p_{1}^{-}\)
and satisfies
$$ \int^{1}_{0}\frac{w_{r}(\delta)}{\delta^{1+\gamma}}d\delta < \infty.$$
Let \(
0< q< \infty, 0\leq\lambda< \infty \text{and} \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\)
satisfies conditions (4) and (5) of Proposition 3. If \(
2\lambda\leq\alpha(\cdot), n\delta_{2}\leq \alpha(0),\alpha_{\infty}< \gamma+n\delta_{2},\)
then \(
[b,T_{\Omega}]\)
is bounded from \(
HM\dot{K}^{\alpha(\cdot),q}_{p_{1}(\cdot),\lambda}\)
or \(
\left(HMK^{\alpha(\cdot),q}_{p_{1}(\cdot),\lambda}\right)\)
to \(
M\dot{K}^{\alpha(\cdot),q}_{p_{2}(\cdot),\lambda}\)
or \(
\left(MK^{\alpha(\cdot),q}_{p_{2}(\cdot),\lambda}\right).\)
Proof.
The prove of this Theorem follows almost similarly to that of Theorem 14. Instead of giving all details, we only give the modifications required for the estimation of \(
E”, F” \text{and} G”\)
.
Note that if \(
x\in B_{k} \text{for each} k\in\mathbb{Z}, y\in B_{j}\)
and \(
j\leq k-1\)
. Let \(
\tilde{p}(\cdot)>1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\)
, since \(
r>p^{+}\)
, so by Lemmas 10 and 12, we get
\begin{eqnarray*}
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}&\leq&
\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right|(b(\cdot)-b(y))\chi_{k}\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|dy\\
&\leq& \int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}
\right||b(\cdot)-b_{B_{j}}|\chi_{k}\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|dy\\
&& +\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|\chi_{k}
\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}|b_{B_{j}}-b(y)||g_{j}(y)|dy
\end{eqnarray*}
Since \(
\tilde{p_{2}}(\cdot)>1 \text{and} 1/p_{2}(\cdot)=1/\tilde{p_{2}}(\cdot)+1/r\)
, by \(
r>p^{+}\)
and Lemmas 11 and 12, we deduced
\begin{eqnarray*}
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}&\leq& \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}\left\|(b-b_{B_{j}})\chi_{k}
\right\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|g_{j}(y)|dy\\
&& +\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|\right\|_{L^{r}(\mathbb{R}^{n})}
\|\chi_{B_{k}}\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|b_{B_{j}}-b(y)||g_{j}(y)|dy\\
&\leq& C \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}\left\|(b-b_{B_{j}})\chi_{k}
\right\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}\|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\|\chi_{j}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}\\
&& +\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|
\right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}
\|(b-b_{B_{j}})\chi_{B_{j}}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}
\|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}.
\end{eqnarray*}
By Lemma 9, we have
\begin{eqnarray}
\left\|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right\|_{L^{r}(\mathbb{R}^{n})}
&\leq& C 2^{(k-1)(\frac{n}{r}-n)}
\left\{\frac{|y|}{2^{k-1}}+\int^{|y|/2^{k-1}}_{|y|/2^{k}}\frac{w_{r}(\delta)}{\delta}d\delta\right\}\nonumber\\
&\leq& C 2^{(k-1)(\frac{n}{r}-n)}
\left(2^{j-k}+2^{(j-k)\gamma}\int^{1}_{0}\frac{w_{r}(\delta)}{\delta^{1+\gamma}}d\delta\right)\nonumber\\
&\leq& C 2^{(k-1)(\frac{n}{r}-n)}2^{(j-k)\gamma}.
\end{eqnarray}
(22)
by Lemma 4-6,we have
\begin{eqnarray}
\|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}&\leq & C 2^{(k-1)(\frac{n}{s}-n)+(j-k)\gamma}\|b\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}2^{\gamma k}
\left(\frac{\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}{|B_{k}|^{\frac{1}{r}+\frac{\gamma}{n}}}
\right)\|\chi_{j}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}\|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&& + C 2^{(k-1)(\frac{n}{s}-n)+(j-k)\gamma}\|b\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}2^{\gamma k}
\left(\frac{\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}{|B_{k}|^{\frac{1}{r}+\frac{\gamma}{n}}}\right)
\|\chi_{j}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}\|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&\leq& C \|b\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}2^{-nk+(j-k)\gamma}
\left(\|\chi_{B_{k}}\|^{-1}_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}|B_{k}|\right) \|\chi_{j}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}\|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\nonumber\\
\end{eqnarray}
(23)
\begin{eqnarray}
&\leq& C \|b\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}2^{(j-k)\gamma}\left(
\frac{\|\chi_{j}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B_{k}}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}}\right) \|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\nonumber\\
&\leq& C \|b\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}2^{(j-k)(\gamma+n\delta_{2})-j\alpha_{j}}.
\end{eqnarray}
(24)
From this, following the same calculations as we did for \(
E’, F’ \text{and} G’\)
in Theorem 14, we get
\begin{equation}
E”, F”, G”\leq C \|b\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}\Upsilon.
\end{equation}
(25)
Acknowledgments
This paper is supported by Shendi University.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
