王洪彬^1,2
1. 中国科学院大学数学科学学院, 北京 100049;
2. 山东理工大学数学与统计学院, 山东 淄博 255049
摘要: 引入一类齐型空间上的变指标Herz空间，并建立该空间的块分解.利用此分解得到一类次线性算子在上述变指标Herz空间中的一些有界性.
关键词: Herz空间变指标齐型空间块分解有界性
The theory of function spaces with variable exponent has been extensively studied by researchers since the work of Ková? and Rákosník^[1] appeared in 1991(See Refs.[2-4] and references therein). Inspired by Refs.[5-7], we introduce the Herz space with variable exponent on spaces of homogeneous type and obtain the block decomposition for them. Meanwhile, we obtain some boundedness for a class of sublinear operators on the Herz space with variable exponent on spaces of homogeneous type, using this decomposition.
Firstly we give some notations and basic definitions on variable Lebesgue spaces on spaces of homogeneous type. Let X=(X, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss^[8]. This is a topological space X endowed with a Borel measure μ and a quasi-metric (or quasi-distance) d. The latter is a mapping $d:X\times X\to {{\mathbb{R}}^{+}}=\left\{ t\in \mathbb{R}:t\ge 0 \right\}$ satisfying
(ⅰ)d(x, y)=d(y, x),
(ⅱ)d(x, y)>0 if and only if x≠y,
(ⅲ) there exists a constant K such that d(x, y)≤K[d(x, z)+d(z, y)] for all x, y, z in X.
We postulate that μ(B(x, r))>0 whenever r>0, where B(x, r)={y∈X:d(x, y) < r} denotes the open ball centered at x with a radius r. Our basic assumption relating the measure and the quasi-distance is the existence of a constant A such that

$\mu \left( {B\left( {x, 2r} \right)} \right) \le A\mu \left( {B\left( {x, r} \right)} \right).$

(1)

As known (See Lemma 14.6 in Ref.[9]), from (1) there follows the property

$\frac{{\mu \left( {B\left( {x, R} \right)} \right)}}{{\mu \left( {B\left( {y, r} \right)} \right)}} \le A{\left( {\frac{R}{r}} \right)^N}, N = {\log _2}A, $

(2)

for all the balls B(x, R) and B(y, r) with 0 < r≤R and y∈B(x, r). From (2) we have

$\mu \left( {B\left( {x, r} \right)} \right) \ge C{r^N}, x \in \Omega, 0 < r \le l, $

(3)

for any l < +∞ and any open set Ω?X on which $\mathop {\inf }\limits_{x \in \Omega } {\mkern 1mu} \mu \left( {B\left( {x,l} \right)} \right) > 0$. Condition (3) is also known as the lower Ahlfors regularity condition.
In addition, the space of homogeneous type (X, d, μ) is assumed in Ref.[5] to satisfy the conditions
(ⅰ) μ({x})=0, μ(X)=∞,
(ⅱ) there exist constants a≥2 and A₀>1 such that

$\mu \left( {B\left( {x, ar} \right)} \right) \ge {A_0}\mu \left( {B\left( {x, r} \right)} \right)$

(4)

holds for all x∈X and 0 < r < ∞.
Given a μ-measurable function p:X→[1, ∞), L^p(·)(X) denotes the set of μ-measurable functions f on X such that for some λ>0,

$\int_X {{{\left( {\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)}^{p\left( x \right)}}{\rm{d}}\mu \left( x \right) < \infty .} $

This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm

$\begin{array}{l}{\left\| f \right\|_{{L^{p\left( \cdot \right)}}\left( X \right)}} = \\\inf \left\{ {\lambda > 0:\int_X {{{\left( {\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)}^{p\left( x \right)}}{\rm{d}}\mu \left( x \right) \le 1} } \right\}.\end{array}$

These spaces are referred to as variable Lebesgue spaces on spaces of homogeneous type.
The space $ L_{\text{loc}}^{p\left( \cdot \right)}\left( X \right)$ is defined by $L_{\text{loc}}^{p\left(\cdot \right)}\left(X \right)$:={f:f∈L^p(·)(E) for all compact subsets E ? X}. Define P(X) to be set of p(·):X→[1, ∞) such that

$\begin{array}{l}{p^- } = {\rm{ess}}\;{\rm{inf}}\left\{ {p\left( x \right):x \in X} \right\} > 1, \\{p^ + } = {\rm{ess}}\;{\rm{sup}}\left\{ {p\left( x \right):x \in X} \right\} < \infty .\end{array}$

Denote p'(x)=p(x)/(p(x))-1).
Next we recall some basic properties of the spaces L^p(·)(X). The H?lder inequality is valid in the form

$\int_X {\left| {f\left( x \right)g\left( x \right)} \right|{\rm{d}}\mu \left( x \right)} \le {r_{\rm{p}}}{\left\| f \right\|_{{L^{p\left( \cdot \right)}}}}{\left\| g \right\|_{{L^{p'\left( \cdot \right)}}}}, $

(5)

where

${r_p} = 1 + 1/{p^- }- 1/{p^ + }.$

The standard local logarithmic condition on X is usually introduced in the form

$\begin{array}{*{20}{c}}{\left| {p\left( x \right) - p\left( y \right)} \right| \le \frac{{{A_p}}}{{ - \ln d\left( {x, y} \right)}}, }\\{d\left( {x, y} \right) \le 1/2, x, y \in X, }\end{array}$

(6)

where A_p > 0 is independent of x and y. Condition (6) is known as Dini-Lipschitz condition or log-H?lder continuity condition.
1 Main results and their proofsIn this section, firstly we give the definition of the Herz space with variable exponent on spaces of homogeneous type (X, d, μ). Let x₀ ∈ X, B_k={x∈X:d(x₀, x) < a^k}, and R_k=B_k\B_k-1 for $ k\in \mathbb{Z}$. Denote ${{\mathbb{Z}}_{+}}$ and $\mathbb{N}$ as the sets of all positive and non-negative integers, χ_k=χ_Rk for $ k\in \mathbb{Z}$ if ${{\mathbb{Z}}_{+}}, {{{\tilde{\chi }}}_{k}}={{\chi }_{k}}$, and ${{{\tilde{\chi }}}_{0}}={{\chi }_{{{B}_{0}}}} $, where χ_Rk is the characteristic function of R_k.
Definition 1.1??Let 0 < α < ∞, 0 < p < ∞, and q(·)∈P(X). The homogeneous Herz space with variable exponent $ \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)$ on spaces of homogeneous type (X, d, μ) is defined by

$\dot K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right) = \left\{ {f \in L_{{\rm{loc}}}^{q\left( \cdot \right)}\left( {X\backslash \left\{ {{x_0}} \right\}} \right):{{\left\| f \right\|}_{\dot K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}} < \infty } \right\}, $

where

${{\left\| f \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}={{\left\{ \sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}\left\| f{{\chi }_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p}} \right\}}^{1/p}}.$

The non-homogeneous Herz space with variable exponent $ {K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)$ on spaces of homogeneous type (X, d, μ) is defined by

$K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)=\left\{ f\in L_{\text{loc}}^{q\left( \cdot \right)}\left( X \right):{{\left\| f \right\|}_{K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}<\infty \right\}, $

where

${{\left\| f \right\|}_{K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}={{\left\{ \sum\limits_{k=0}^{\infty }{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}\left\| f{{{\tilde{\chi }}}_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p}} \right\}}^{1/p}}.$

Remark 1.1??When $ X={{\mathbb{R}}^{n}}, d\left( x, y \right)=|x-y|={{\left( \sum\limits_{j=1}^{n}{{{\left( {{x}_{j}}-{{y}_{j}} \right)}^{2}}} \right)}^{1/2}}, {{x}_{0}}=0$ and μ equals Lebesgue measure, we have

$\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)=\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( {{\mathbb{R}}^{n}} \right)$

and

$K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)=K_{q\left( \cdot \right)}^{\alpha, p}\left( {{\mathbb{R}}^{n}} \right)$

where $\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( {{\mathbb{R}}^{n}} \right) $ and $\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( {{\mathbb{R}}^{n}} \right) $ were introduced by Izuki in Ref.[10].
Next, we consider the decomposition of $\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)$. We begin with the notation of central block.
Definition 1.2??Let 0 < α < ∞ and q(·)∈P(X).
(ⅰ) A function a(x) on X is said to be a central (α, q(·))-block if
(a) supp a?B_k.
(b) ‖a‖_L^q(·)(X)≤μ(B_k)^-α
(ⅱ) A fuction a(x) on X is said to be a central α, q(·)-block of restricted type if
(a) supp a?B_k for some 1≤d(x₀, x) < a^k.
(b)‖a‖_L^q(·)(X)≤μ(B_k)^-α.
The decomposition theorem (See below) shows that the central blocks are the "building block" of the Herz space with variable exponent on spaces of homogeneous type.
Theorem 1.1??Let 0 < α < ∞, 0 < p < ∞, and q(·)∈P(X). The following two statements are equivalent:
(ⅰ) $ f\in \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right);$
(ⅱ) f can be represented by

$f\left( x \right)=\sum\limits_{k\in \mathbb{Z}}{{{\lambda }_{k}}{{b}_{k}}\left( x \right)}, $

(7)

where each b_k is a central (α, q(·))-block with support contained in B_k and $ \sum\limits_{k}{|{{\lambda }_{k}}{{|}^{p}}<\infty }$.
Proof??We first prove that (ⅰ) implies (ⅱ). For every $ f\in \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)$, write

$\begin{align} & f\left( x \right)=\sum\limits_{k\in \mathbb{Z}}{f\left( x \right){{\chi }_{k}}\left( x \right)} \\ & \ \ \ \ \ \ \ \ =\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha }}}{{\left\| f{{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}\times \\ & \ \ \ \ \ \ \ \ \frac{f\left( x \right){{\chi }_{k}}\left( x \right)}{\mu {{\left( {{B}_{k}} \right)}^{\alpha }}{{\left\| f{{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}} \\ & \ \ \ \ \ \ \ \ =\sum\limits_{k\in \mathbb{Z}}{{{\lambda }_{k}}{{b}_{k}}\left( x \right)}, \\ \end{align}$

where λ_k=μ(B_k)^α‖fχ_k‖_L^q(·)(X) and ${{b}_{k}}\left( x \right)=\frac{f\left( x \right){{\chi }_{k}}\left( x \right)}{\mu {{\left( {{B}_{k}} \right)}^{\alpha }}{{\left\| f{{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}} $.
It is obvious that supp b_k?B_k and ‖b_k‖_L^q(·)(X)=μ(B_k)^-α. Thus, each b_k is a central (α, q(·))-block with the support B_k and

$\begin{align} & \sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}}=\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p }}\left\| f{{\chi }_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p}} \\ & \ \ \ \ \ \ \ \ \ \ \ =\left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p}<\infty . \\ \end{align}$

Now we prove that (ⅱ) implies (ⅰ). Let $ f\left( x \right)=\sum\limits_{k\in \mathbb{Z}}{{{\lambda }_{k}}{{b}_{k}}}\left( x \right)$ be a decomposition of f which satisfies hypothesis (ⅱ) of Theorem 1.1. For each $ j\in \mathbb{Z}$, by Minkowski inequality, we have

${{\left\| f{{\chi }_{j}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}\le \sum\limits_{k\ge j}{\left| {{\lambda }_{k}} \right|{{\left\| {{b}_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}}.$

(8)

Now we consider two cases for the index p. If 0 < p≤1, from (8) it follows that

$\begin{align} & \left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p} \\ & =\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}\left\| f{{\chi }_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p}} \\ & \le \sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j\ge k}{{{\left| {{\lambda }_{k}} \right|}^{p}}\left\| {{b}_{j}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p}} \right) \\ & \le \sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j\ge k}{{{\left| {{\lambda }_{k}} \right|}^{p}}\mu {{\left( {{B}_{j}} \right)}^{-\alpha p}}} \right) \\ & =\sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}}\sum\limits_{j\ge k}{{{\left( \frac{\mu \left( {{B}_{k}} \right)}{\mu \left( {{B}_{j}} \right)} \right)}^{\alpha p}}} \\ & \le \sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}}\sum\limits_{j\ge k}{A_{0}^{\left( k-j \right)\alpha p}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}}. \\ \end{align}$

If 1 < p < ∞, by (8) and the H?lder inequality we have

$\begin{align} & {{\left\| f{{\chi }_{j}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}} \\ & \le \sum\limits_{k\ge j}{\left| {{\lambda }_{k}} \right|\left\| {{b}_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{1/2}\left\| {{b}_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{1/2}} \\ & \le {{\left( \sum\limits_{k\ge j}{{{\left| {{\lambda }_{k}} \right|}^{p}}\left\| {{b}_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p/2}} \right)}^{1/p}}\times \\ & \ \ \ {{\left( \sum\limits_{k\ge j}{\left\| {{b}_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{{p}'/2}} \right)}^{1/{p}'}} \\ & \le {{\left( \sum\limits_{k\ge j}{{{\left| {{\lambda }_{k}} \right|}^{p}}\mu {{\left( {{B}_{k}} \right)}^{-\alpha p/2}}} \right)}^{1/p}}\times {{\left( \sum\limits_{k\ge j}{\mu {{\left( {{B}_{k}} \right)}^{-\alpha {p}'/2}}} \right)}^{1/{p}'}}. \\ \end{align}$

Therefore,

$\begin{align} & \left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p} \\ & \le C\sum\limits_{j\in \mathbb{Z}}{\mu {{\left( {{B}_{j}} \right)}^{\alpha p}}}\left( \sum\limits_{k\ge j}{{{\left| {{\lambda }_{k}} \right|}^{p}}\mu {{\left( {{B}_{k}} \right)}^{-\alpha p/2}}} \right)\times \\ & \ \ \ {{\left( \sum\limits_{k\ge j}{\mu {{\left( {{B}_{k}} \right)}^{-\alpha {p}'/2}}} \right)}^{p/{p}'}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}}\sum\limits_{j\le k}{A_{0}^{\alpha \left( j-k \right)p/2}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}}. \\ \end{align}$

This leads to $f\in \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right) $ and then completes the proof of Theorem 1.1.
Remark 1.2??From the proof of Theorem 1.1, it is easy to see that if $f\in \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right) $ and $ f\left( x \right)=\sum\limits_{k\in \mathbb{Z}}{{{\lambda }_{k}}{{b}_{k}}}\left( x \right)$ is a central (α, q(·))-block decomposition, then

${{\left\| f \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}\approx {{\left( \sum\limits_{k\in \mathbb{Z}}{{{\left| {{\lambda }_{k}} \right|}^{p}}} \right)}^{1/p}}.$

By an argument similar to the proof of Theorem 1.1, we can obtain the decomposition characterizations of the non-homogeneous Herz space with variable exponent of X as follows.
Theorem 1.2??Let 0 < α < ∞, 0 < p < ∞, and q(·)∈P(X). The following two statements are equivalent:
(ⅰ) $f\in K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right); $
(ⅱ) f can be represented by

$f\left( x \right)=\sum\limits_{k=0}^{\infty }{{{\lambda }_{k}}{{b}_{k}}\left( x \right)}, $

(9)

where each b_k is a central (α, q(·))-block of restricted type with support contained in B_k and $\sum\limits_{k\ge 0}{|{{\lambda }_{k}}{{|}^{p}}<\infty } $.
Moreover, the norms ${{\left\| f \right\|}_{K_{q\left( \cdot \right)}^{\alpha, p}}}\left( X \right) $ and inf ${{\left( \sum\limits_{k\ge 0}{|{{\lambda }_{k}}{{|}^{p}}} \right)}^{1/p}} $ are equivalent, where the infimum is taken over all decompositions of f as in (9).
As applications of the decomposition theorems, let us come to investigate the boundedness for some sublinear operators on the Herz space with variable exponent on X.
Theorem 1.3??Let X be bounded, q(·)∈P(X) satisties condition (6), 0 < p < ∞, and $ 0<\alpha <1-\frac{1}{{{q}^{-}}}$. If a sublinear operator T satisfies

$\begin{matrix} \left| Tf\left( x \right) \right|\le C{{\left\| f \right\|}_{{{L}^{1}}\left( X \right)}}/\mu \left( B\left( {{x}_{0}}, \text{d}\left( {{x}_{0}}, x \right) \right) \right), \\ \text{if}\ \text{dist}\left( x, \text{supp}\ f \right)>\frac{\text{d}\left( {{x}_{0}}, x \right)}{2K}, \\\end{matrix}$

(10)

for any integrable function f with a compact support and T is bounded on L^q(·)(X), then T is bounded on $ \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)$ and $K_{q\left( \cdot \right)}^{\alpha, p}\left( X \right) $, respectively.
To prove Theorem 1.3, we need the follwing auxiliary result.
Lemma 1.1^[11]??Let X be bounded, the measure μ satisfies condition (3), and p(·) satisfies condition (6). Then

${{\left\| {{\chi }_{B\left( x, r \right)}} \right\|}_{p\left( \cdot \right)}}\le C{{\left[\mu \left( B\left( x, r \right) \right) \right]}^{\frac{1}{p\left( x \right)}}}$

with C>0 independent of x∈X and r > 0.
Proof of Theorem 1.3??It suffices to prove that T is bounded on $\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right) $. Suppose of $ f\in \dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)$. By Theorem 1.1, $f\left( x \right)=\sum\limits_{j\in \mathbb{Z}}{{{\lambda }_{j}}{{b}_{j}}}\left( x \right) $, where each b_j is a central (α, q(·))-block with support contained in B_j and

${{\left\| f \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}\approx {{\left( \sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}} \right)}^{1/p}}.$

Therefore, we get

$\begin{align} & \left\| Tf \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p} \\ & =C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left\| \left( Tf \right){{\chi }_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p} \\ & \le C\left[\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\times \right. \\ & \ \ \ {{\left( \sum\limits_{j=-\infty }^{k-2}{\left| {{\lambda }_{j}} \right|{{\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}} \right)}^{p}}+ \\ & \ \ \ \ \sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\times \\ & \left. \ \ \ \ {{\left( \sum\limits_{j=k-1}^{\infty }{\left| {{\lambda }_{j}} \right|{{\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}} \right)}^{p}} \right] \\ & =:C\left( {{I}_{1}}+{{I}_{2}} \right). \\ \end{align}$

Let us first estimate I₁. By (5) and (10) we get

$\begin{align} & \left| T{{b}_{j}}\left( x \right) \right| \\ & \le C\mu {{\left( B\left( {{x}_{0}}, d\left( {{x}_{0}}, x \right) \right) \right)}^{-1}}\int_{{{B}_{j}}}{\left| {{b}_{j}}\left( y \right) \right|\text{d}\mu \left( y \right)} \\ & \le C\mu {{\left( {{B}_{k}} \right)}^{-1}}{{\left\| {{b}_{j}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}{{\left\| {{\chi }_{{{B}_{j}}}} \right\|}_{{{L}^{{q}'\left( \cdot \right)}}\left( X \right)}} \\ & \le C\mu {{\left( {{B}_{k}} \right)}^{-1}}\mu {{\left( {{B}_{j}} \right)}^{-\alpha }}{{\left\| {{\chi }_{{{B}_{j}}}} \right\|}_{{{L}^{{q}'\left( \cdot \right)}}\left( X \right)}}. \\ \end{align}$

So by Lemma 1.1 we have

$\begin{align} & {{\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}} \\ & \le C\mu {{\left( {{B}_{k}} \right)}^{-1}}\mu {{\left( {{B}_{j}} \right)}^{-\alpha }}{{\left\| {{\chi }_{{{B}_{j}}}} \right\|}_{{{L}^{{q}'\left( \cdot \right)}}\left( X \right)}}\times \\ & \ \ \ {{\left\| {{\chi }_{{{B}_{k}}}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}} \\ & \le C\mu {{\left( {{B}_{k}} \right)}^{-1}}\mu {{\left( {{B}_{j}} \right)}^{-\alpha }}\mu {{\left( {{B}_{j}} \right)}^{1-\frac{1}{q\left( x \right)}}}\mu {{\left( {{B}_{k}} \right)}^{\frac{1}{q\left( x \right)}}} \\ & =C\mu {{\left( {{B}_{k}} \right)}^{-1+\frac{1}{q\left( x \right)}}}\mu {{\left( {{B}_{j}} \right)}^{-\alpha +1-\frac{1}{q\left( x \right)}}}. \\ \end{align}$

(11)

Therefore, when 0 < p≤1, by $ 0<\alpha <1-\frac{1}{{{q}^{-}}}$, we get

$\begin{align} & {{I}_{1}}=\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\times \\ & {{\left( \sum\limits_{j=-\infty }^{k-2}{\left| {{\lambda }_{j}} \right|{{\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}} \right)}^{p}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j=-\infty }^{k-2}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\times \right. \\ & \left. \mu {{\left( {{B}_{k}} \right)}^{\left[-1+\frac{1}{q\left( x \right)} \right]p}}\mu {{\left( {{B}_{j}} \right)}^{\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]p}} \right) \\ & \le C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\sum\limits_{k\ge j+2}{{{\left( \frac{\mu \left( {{B}_{j}} \right)}{\mu \left( {{B}_{k}} \right)} \right)}^{\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]p}}} \\ & \le C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\sum\limits_{k\ge j+2}{A_{0}^{\left( j-k \right)\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]p}} \\ & \le C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\le C\left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p}. \\ \end{align}$

(12)

When 1 < p < ∞, take 1/p+1/p'=1. Since $ 0<\alpha <1-\frac{1}{{{q}^{-}}}$, by (11) and the H?lder inequality, we have

$\begin{align} & {{I}_{1}}\le C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j=-\infty }^{k-2}{\left| {{\lambda }_{j}} \right|\mu {{\left( {{B}_{k}} \right)}^{-1+\frac{1}{q\left( x \right)}}}}\times \right. \\ & \ \ \ \ {{\left. \mu {{\left( {{B}_{j}} \right)}^{-\alpha +1-\frac{1}{q\left( x \right)}}} \right)}^{p}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\left( \sum\limits_{j=-\infty }^{k-2}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\times {{\left( \frac{\mu \left( {{B}_{j}} \right)}{\mu \left( {{B}_{k}} \right)} \right)}^{\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]p/2}} \right)}\times \\ & \ \ \ \ \ {{\left( \sum\limits_{j=-\infty }^{k-2}{{{\left( \frac{\mu \left( {{B}_{j}} \right)}{\mu \left( {{B}_{k}} \right)} \right)}^{\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]{p}'/2}}} \right)}^{p/{p}'}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\left( \sum\limits_{j=-\infty }^{k-2}{{{\left| {{\lambda }_{j}} \right|}^{p}}A_{0}^{\left( j-k \right)\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]p/2}} \right)} \\ & =C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\sum\limits_{k\ge j+2}{A_{0}^{\left( j-k \right)\left[-\alpha +1-\frac{1}{q\left( x \right)} \right]p/2}} \\ & \le C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\le C\left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p}. \\ \end{align}$

(13)

Let us now estimate I₂. Similarly, we consider two cases for p. When 0 < p≤1, by L^q(·)(X) boundedness of T, we have

$\begin{align} & {{I}_{2}}=\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\times \\ & \ \ \ \ {{\left( \sum\limits_{j=k-1}^{\infty }{\left| {{\lambda }_{j}} \right|{{\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|}_{{{L}^{q\left( \cdot \right)}}\left( X \right)}}} \right)}^{p}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j=k-1}^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{p}}\left\| {{b}_{j}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p}} \right) \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j=k-1}^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{p}}\mu {{\left( {{B}_{j}} \right)}^{-\alpha p}}} \right) \\ & =C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\sum\limits_{k\le j+1}{{{\left( \frac{\mu \left( {{B}_{k}} \right)}{\mu \left( {{B}_{j}} \right)} \right)}^{\alpha p}}} \\ & \le C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\le C\left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p}. \\ \end{align}$

(14)

When 1 < p < ∞. take 1/p+1/p'=1. By L^q(·)(X) boundedness of T and the H?lder inequality. we have

$\begin{align} & {{I}_{2}}\le C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\times \left( \sum\limits_{j=k-1}^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{p}}\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p/2}} \right)\times \\ & {{\left( \sum\limits_{j=k-1}^{\infty }{\left\| \left( T{{b}_{j}} \right){{\chi }_{k}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{{p}'/2}} \right)}^{p/{p}'}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\mu {{\left( {{B}_{k}} \right)}^{\alpha p}}}\left( \sum\limits_{j=k-1}^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{p}}\left\| {{b}_{j}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{p/2}} \right)\times \\ & {{\left( \sum\limits_{j=k-1}^{\infty }{\left\| {{b}_{j}} \right\|_{{{L}^{q\left( \cdot \right)}}\left( X \right)}^{{p}'/2}} \right)}^{p/{p}'}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\left( \sum\limits_{j=k-1}^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{p}}{{\left( \frac{\mu \left( {{B}_{k}} \right)}{\mu \left( {{B}_{j}} \right)} \right)}^{\alpha p/2}}} \right)}\times \\ & {{\left( \sum\limits_{j=k-1}^{\infty }{{{\left( \frac{\mu \left( {{B}_{k}} \right)}{\mu \left( {{B}_{j}} \right)} \right)}^{\alpha {p}'/2}}} \right)}^{p/{p}'}} \\ & \le C\sum\limits_{k\in \mathbb{Z}}{\left( \sum\limits_{j=k-1}^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{p}}{{\left( \frac{\mu \left( {{B}_{k}} \right)}{\mu \left( {{B}_{j}} \right)} \right)}^{\alpha p/2}}} \right)} \\ & =C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\sum\limits_{k\le j+1}{{{\left( \frac{\mu \left( {{B}_{k}} \right)}{\mu \left( {{B}_{j}} \right)} \right)}^{\alpha p/2}}} \\ & \le C\sum\limits_{j\in \mathbb{Z}}{{{\left| {{\lambda }_{j}} \right|}^{p}}}\le C\left\| f \right\|_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}^{p}. \\ \end{align}$

(15)

Combining inequalities (12)-(15), we have

${{\left\| Tf \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}\le C{{\left\| f \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha, p}\left( X \right)}}.$

Thus, the proof of Theorem 1.3 is completed.
References

[1]	Ková? ik O, Rákosník J. On spaces Lp(x) and Wk, p(x)[J].Czechoslovak Math J, 1991, 41:592–618.
[2]	Cruz-Uribe D, Fiorenza A. Variable Lebesgue spaces:foundations and harmonic analysis (applied and numerical harmonic analysis)[M].Heidelberg: Springer, 2013.
[3]	Diening L, Harjulehto P, H?st? P, et al. Lebesgue and Sobolev spaces with variable exponents[M]. Lecture Notes in Math, vol. 2017. Heidelberg: Springer, 2011.
[4]	Harjulehto P, H?st? P, Pere M. Variable exponent Lebesgue spaces on metric spaces:the Hardy-Littlewood maximal operator[J].Real Anal Exchange, 2004, 30(1):87–104.
[5]	Liu Z, Zeng Y. The Herz spaces on spaces of homogeneous type and their applications[J].Acta Math Sci Ser A Chin Ed, 1999, 19:270–277.
[6]	Lu S, Yang D. The decomposition of weighted Herz space on ${{mathbb{R}}.{n}}$ and its applications[J].Sci China Ser A-Math, 1995, 38:147–158.
[7]	Wang H. The decomposition for the Herz spaces[J].Panamer Math J, 2015, 25:15–28.
[8]	Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis[J].Bull Amer Math Soc, 1977, 83:569–645.DOI:10.1090/S0002-9904-1977-14325-5
[9]	Hajlasz P, Koskela P. Sobolev met Poincare[J].Mem Amer Math Soc, 2000, 145(688):1–106.
[10]	Izuki M. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization[J].Anal Math, 2010, 36:33–50.DOI:10.1007/s10476-010-0102-8
[11]	Almeida A, Hasanov J, Samko S. Maximal and potential operators in variable exponent Morrey spaces[J].Georgian Math J, 2008, 15:195–208.