1 Introduction

In mathematical analysis, the Hardy operator is considered a significant averaging operator and has been exercised a lot during the recent past. In [1], Hardy defined the classical Hardy operator as follows:

$$ Pg(z)=\frac{1}{z} \int_{0}^{z}g(t)\,dt,\quad z>0,$$
(1)

and established a sharp \((q,q)\) inequality for it. Faris [2] introduced the n-dimensional version of (1), however, the exact value of norm of the n-dimensional Hardy operator on the Lebesgue space was obtained in [3]. Subsequently, in [4], the authors defined the fractional Hardy operator and its adjoint operator as follows:

$$ Hg(z) = \frac{1}{ \vert z \vert ^{n-\beta}} \int_{ \vert t \vert \leq \vert z \vert } g(t) \,dt,\qquad H^{*}g(z)= \int_{ \vert t \vert > \vert z \vert }\frac{g(t)}{ \vert t \vert ^{n-\beta}}\,dt, \quad z\in \mathbb{R}^{n} \setminus\{0\}, $$
(2)

where \(|z| = \sqrt{\sum_{i=1}^{n} z_{i}^{2}}\). Here, we cite some important readings with regards to the study of Hardy-type operators on different function spaces which include [5–10].

The concept of generalizing function spaces started with the work presented in [11]. However, variable exponent Lebesgue spaces \(L^{p(\cdot)}\) were firstly introduced by Kováčik and Rákosník in [12]. After that, the development of variable Lebesgue spaces was started along with the investigation of boundedness of several operators including the maximal operator on \(L^{p(\cdot)}\) [13–16]. Recently, the theory of generalized function spaces showed deep concern in many fields of mathematical analysis like, for example, in the field of image processing [17], in the analysis of electrorheological fluids models [18], and in the theory of partial differential equations with nonstandard growth conditions [19].

Besides, Izuku introduced Herz spaces with variable exponent \(\dot{K}_{q}^{\alpha,p(\cdot)}\) in [20, 21]. Later on, Almeida and Drihemn [22] gave a new definition of Herz spaces by taking the exponent alpha as a variable. However, the Herz space having all the exponents as variables was defined and studied in [23]. Morrey–Herz spaces with variable exponent \(M\dot{K}^{\alpha,\lambda}_{q,p(\cdot)}\) first appeared in [24]. The ensuing paper [25] made some generalization in the definition of Morrey–Herz spaces given in [24] by replacing the exponent α with \(\alpha(\cdot)\). A few important considerations in this regard can be found in [26–29].

Recent advancements in the field of variable exponent function spaces include the development of its weighted theory based on the Muckenhoupt weights [30]. In [31, 32], Cruz-Uribe with different co-authors gave the continuity criteria for Hardy–Littlewood maximal operator M:

$$Mg(z)=\sup_{B:\mathrm{ball}, z\in B}\frac{1}{ \vert B \vert } \int_{B} \bigl\vert g(t) \bigr\vert \,dt, $$

on weighted \(L^{p(\cdot)}(w)\) spaces. Equivalence between the boundedness of M on \(L^{p(\cdot)}(w)\) and the Muckenhoupt condition was proved by Diening and Hästö in [33]. Izuki and Noi defined the weighted Herz spaces with variable exponents in [34]. However, weighted Morrey–Herz spaces with variable exponents were defined and studied in [35, 36].

The aim of this article is to study the continuity criteria for fractional type Hardy operators on weighted variable exponents Morrey–Herz spaces. It is worth mentioning here that our idea is based on Muckenhoupt theory and on Banach function spaces. We thus extend some results presented in [27]. Also, at an intermediate level, we use the boundedness of fractional integral to control the boundedness of the fractional Hardy operators. The fractional integral can be defined as

$$I_{\beta}(g) (z)= \int_{\mathbb{R}^{n}}\frac{g(t)}{ \vert z-t \vert ^{n-\beta}}\,dt. $$

The variable Lebesgue spaces boundedness property of Riesz potential was reported in [37]. On the weighted Herz spaces, the boundedness of fractional integral operator was obtained by Izuki and Noi [34].

The presentation of this paper includes four sections. The next section is full of necessary notations and definitions. In Sect. 3, we furnish key lemmas which are helpful in proving our main results in Sect. 4.

2 Notations and definitions

In the remainder of this article, the letter C will denote a constant whose value may change from line to line. A nonempty set S is considered to be a measurable set in \(\mathbb{R}^{n}\), and \(\chi_{S}\) represents the characteristic function of S, whereas \(|S|\) represents its Lebesgue measure. Let us first define variable exponent Lebesgue spaces based on the fundamental papers and books [12, 15, 16].

Definition 2.1

Let \(p(\cdot):\mathbb{R}^{n}\rightarrow[1,\infty)\) be a measurable function. The Lebesgue space with variable exponent \(L^{p{(\cdot)}}(\mathbb{R}^{n})\) is the set of all measurable functions f such that

$$F_{p}(f)= \int_{\mathbb{R}^{n}} \bigl\vert f(x) \bigr\vert ^{p(x)}\,dx< \infty. $$

The space \(L^{p{(\cdot)}}(\mathbb{R}^{n})\) turns out to be Banach function space with respect to the norm

$$\Vert f \Vert _{L^{p(\cdot)}}=\inf \biggl\{ \sigma>0:F_{p} \biggl(\frac{f}{\sigma}\biggr)= \int_{\mathbb{R}^{n}} \biggl(\frac{ \vert f(x) \vert }{\sigma} \biggr)^{p(x)}\,dx \leq1 \biggr\} . $$

Definition 2.2

We denote by \(\mathcal{P}(\mathbb{R}^{n})\) the set of all measurable functions \(p(\cdot):\mathbb{R}^{n}\rightarrow(1,\infty)\) such that

$$1< p_{-}\leq p(x)\leq p_{+}< \infty, $$

where

$$p_{-}:=\operatorname*{essinf}_{x\in\mathbb{R}^{n}}p(x), \qquad p_{+}:= \operatorname*{esssup}_{x\in\mathbb{R}^{n}}p(x). $$

Definition 2.3

Suppose that \(p(\cdot)\) is a real-valued function on \(\mathbb{R}^{n}\). We say that

  1. (i)

    \(\mathcal{C}^{\log}_{\mathrm{loc}}(\mathbb{R}^{n})\) is the set of all local log-Hölder continuous functions \(p(\cdot)\) satisfying

    $$\bigl\vert p(x)-p(y) \bigr\vert \le\frac{-C}{\log( \vert x-y \vert )},\quad \vert x-y \vert < \frac{1}{2}, x,y\in \mathbb{R}^{n}. $$
  2. (ii)

    \(\mathcal{C}^{\log}_{0}(\mathbb{R}^{n})\) is the set of all local log-Hölder continuous functions \(p(\cdot)\) satisfying at origin

    $$\bigl\vert p(x)-p(0) \bigr\vert \lesssim\frac{C}{\log( \vert e+\frac{1}{ \vert x \vert } \vert )},\quad x\in \mathbb{R}^{n}. $$
  3. (iii)

    \(\mathcal{C}^{\log}_{\infty}(\mathbb{R}^{n})\) is a set of all log-Hölder continuous functions satisfying at infinity

    $$\bigl\vert p(x)-p_{\infty} \bigr\vert \leq\frac{C_{\infty}}{\log(e+ \vert x \vert )},\quad x \in \mathbb{R}^{n}. $$
  4. (iv)

    \(\mathcal{C}^{\log}(\mathbb{R}^{n})=\mathcal{C}^{\log }_{\infty} \cap\mathcal{C}^{\log}_{\mathrm{loc}}\) denotes the set of all global log-Hölder continuous functions \(p(\cdot)\).

It was proved in [38] that if \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\cap\mathcal{C}^{\log}( \mathbb{R}^{n})\), then the Hardy–Littlewood maximal operator M is bounded on \(L^{p(\cdot)}(\mathbb {R}^{n})\).

Suppose that \(w({x})\) is a weight function on \(\mathbb{R}^{n}\), which is a nonnegative and locally integrable function on \(\mathbb{R}^{n}\). Let \(L^{p(\cdot)}(w)\) be the space of all complex-valued functions f on \(\mathbb{R}^{n}\) such that \(fw^{\frac{1}{p(\cdot)}}\in L^{p(\cdot)}(\mathbb{R}^{n})\). The space \(L^{p(\cdot)}(w)\) is a Banach function space equipped with the norm

$$\Vert f \Vert _{L^{p(\cdot)}(w)}= \bigl\Vert fw^{\frac{1}{p(\cdot)}} \bigr\Vert _{L^{p(\cdot)}}. $$

Benjamin Muckenhoupt introduced the theory of \(A_{p}\) \((1< p<\infty)\) weights on \(\mathbb{R}^{n}\) in [30]. Recently, in [34, 39] Izuki and Noi generalized the Muckenhoupt \(A_{p}\) class by taking p as variable.

Definition 2.4

Let \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\). A weight w is an \(A_{p(\cdot)}\) weight if

$$\sup_{B}\frac{1}{ \vert B \vert } \bigl\Vert w^{1/p(\cdot)}\chi_{B} \bigr\Vert _{L^{p(\cdot)}} \bigl\Vert w^{-1/p( \cdot)}\chi_{B} \bigr\Vert _{L^{p^{\prime}(\cdot)}}< \infty. $$

In [40], authors proved that \(w\in A_{p(\cdot)}\) if and only if M is bounded on the space \(L^{p(\cdot)}\).

Remark 2.5

([34])

Suppose \(p(\cdot),q(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\cap\mathcal {C}^{\log}( \mathbb{R}^{n})\) and \(p(\cdot)\le q(\cdot)\), then we have

$$A_{1}\subset A_{p(\cdot)}\subset A_{q(\cdot)}. $$

Definition 2.6

Suppose \(p_{1}(\cdot),p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and \(\beta\in(0,n)\) such that \(\frac{1}{p_{2}(x)}=\frac{1}{p_{1}(x)}-\frac{\beta}{n}\). A weight w is said to be \(A(p_{1}(\cdot),p_{2}(\cdot))\) weight if

$$\Vert \chi_{B} \Vert _{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})} \Vert \chi_{B} \Vert _{(L^{p_{1}( \cdot)}(w^{p_{1}(\cdot)}))^{\prime}}\le C \vert B \vert ^{1-\frac{\beta}{n}}. $$

Definition 2.7

([34])

Suppose \(p_{1}(\cdot),p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and \(\beta\in(0,n)\) such that \(\frac{1}{p_{2}(x)}=\frac{1}{p_{1}(x)}-\frac{\beta}{n}\). Then \(w\in A_{(p_{1}(\cdot),p_{2}(\cdot))}\) if and only if \(w^{p_{2}(\cdot)}\in A_{1+p_{2}(\cdot)/p_{1}^{\prime}(\cdot)}\).

It is well known that Herz spaces play an important role in harmonic analysis. After they have been introduced in [41], the theory of these spaces had a remarkable development in part due to its usefulness in applications. For instance, they appear in the characterization of multipliers on Hardy spaces [42], in the summability of Fourier transforms [43], and in regularity theory for elliptic equations in divergence form [44]. For a detailed study of Herz-type spaces, we recommend the reader to see the book [45]. Now, we define variable exponent weighted Morrey–Herz space \(M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(w)\). Let \(B_{k}=\{x\in\mathbb{R}^{n}:|x|\leq2^{k}\},~A_{k}=B_{k}\setminus B_{k-1}\), and \(\chi_{k}=\chi_{A_{k}}\) for \(k\in\mathbb{Z}\).

Definition 2.8

Let w be a weight on \(\mathbb{R}^{n}\), \(\lambda\in[0,\infty)\), \(q\in(0,\infty)\), \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\), and \(\alpha(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}\) with \(\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\). The space \(M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(w)\) is the set of all measurable functions f given by

$$M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(w)= \bigl\{ f\in L^{p( \cdot)}_{loc} \bigl(\mathbb{R}^{n}\setminus\{0\},w\bigr): \Vert f \Vert _{M\dot{K}^{ \alpha(\cdot),\lambda}_{q,p(\cdot)}(w)}< \infty \bigr\} , $$

where

$$\Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(w)}=\sup_{k_{0} \in{Z}}2^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha( \cdot)q} \Vert f\chi_{k} \Vert ^{q}_{L^{p(\cdot)}(w)} \Biggr)^{1/q}. $$

Obviously, \(M\dot{K}^{\alpha(\cdot),0}_{q,p(\cdot)}(w)=\dot{K}^{ \alpha(\cdot)}_{q,p(\cdot)}(w)\) is the weighted Herz space with variable exponent (see [22]).

3 Key lemmas

We start this section with some useful lemmas that will be helpful in proving our main results.

Lemma 3.1

([46])

If X is a Banach function space, then

  1. (i)

    The associated space \(X^{\prime}\) is also a Banach function space.

  2. (ii)

    \(\|\cdot\|_{(X^{\prime})^{\prime}}\) and \(\|\cdot\|_{X}\) are equivalent.

  3. (iii)

    If \(g\in X\) and \(f\in X^{\prime}\), then

    $$\int_{\mathbb{R}^{n}} \bigl\vert f(x)g(x) \bigr\vert \,dx\leq \Vert g \Vert _{X} \Vert f \Vert _{X^{\prime}} $$

    is the generalized Hölder inequality.

Lemma 3.2

Suppose that X is a Banach function space, we have that, for all balls B,

$$1\leq\frac{1}{ \vert B \vert } \Vert \chi_{B} \Vert _{X} \Vert \chi_{B} \Vert _{X^{\prime}}. $$

Lemma 3.3

([47])

Consider Banach function space X. Let M be a Hardy–Littlewood maximal operator that is weakly bounded on X, that is,

$$\Vert \chi_{\{Mf>\sigma\}} \Vert _{X} \lesssim \sigma^{-1} \Vert f \Vert _{X} $$

is true for \(\sigma>0\) and for all \(f\in X\). Then we have

$$\sup_{B:\mathrm{ball}}\frac{1}{ \vert B \vert } \Vert \chi_{B} \Vert _{X} \Vert \chi_{B} \Vert _{X^{\prime}}< \infty. $$

Lemma 3.4

([37])

  1. (1)

    \(X(\mathbb{R}^{n},W)\) is a Banach function space equipped with the norm

    $$\Vert f \Vert _{X(\mathbb{R}^{n},W)}= \Vert fW \Vert _{X}; $$
  2. (2)

    The associate space \(X^{\prime}(\mathbb{R}^{n},W^{-1})\) of \(X (\mathbb{R}^{n},W)\) is also a Banach function space.

Lemma 3.5

([34])

Let X be a Banach function space and M be bounded on \(X^{\prime}\), then there exists a constant \(\delta\in(0,1)\) for all \(B\subset\mathbb{R}^{n}\) and \(E\subset B\),

$$\frac{ \Vert \chi_{E} \Vert _{X}}{ \Vert \chi_{B} \Vert _{X}}\le \biggl(\frac{ \vert E \vert }{ \vert B \vert } \biggr)^{\delta}. $$

The paper [12] shows that \(L^{p(\cdot)}(\mathbb{R}^{n})\) is a Banach function space and the associated space \(L^{p^{\prime}(\cdot) }(R^{n})\) with equivalent norm.

Remark 3.6

Let \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and by comparing the Lebesgue space \(L^{p(\cdot)}(w^{p(\cdot)})\) and \(L^{p^{\prime}(\cdot)}(w^{-p^{\prime}(\cdot)})\) with the definition of \(X(\mathbb{R}^{n},W)\), we have:

  1. 1.

    If we take \(W=w\) and \(X=L^{p(\cdot)}(\mathbb{R}^{n})\), then we get \(L^{p(\cdot)}(\mathbb{R}^{n},w)=L^{p(\cdot)}(w^{p(\cdot)})\).

  2. 2.

    If we consider \(W=w^{-1}\) and \(X=L^{p'(\cdot)}(\mathbb{R}^{n})\), then we have \(L^{p'(\cdot)}(w^{-p^{\prime}(\cdot)})=L^{p'(\cdot)}(\mathbb {R}^{n},w^{-1})\).

By virtue of Lemma 3.4, we get

$$\bigl(L^{p(\cdot)}\bigl(\mathbb{R}^{n},w\bigr) \bigr)^{\prime}=\bigl(L^{p(\cdot)}\bigl(w^{p(\cdot)}\bigr) \bigr)^{\prime}=L^{p'(\cdot)}\bigl(w^{-p^{\prime}(\cdot)} \bigr)=L^{p'(\cdot)}\bigl( \mathbb{R}^{n},w^{-1} \bigr). $$

Lemma 3.7

([48])

Let \(p(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\cap\mathcal {C}^{\log}( \mathbb{R}^{n})\) be a Log Hölder continuous function both at infinity and at origin, if \(w^{p_{2}(\cdot)}\in A_{p_{2}(\cdot)}\) implies \(w^{-p^{\prime}_{2}(\cdot)}\in A_{p^{\prime}_{2}(\cdot)}\). Thus the Hardy–Littlewood operator is bounded on \(L^{p^{\prime}_{2}(\cdot)}(w^{-p^{\prime}_{2}(\cdot)})\) and there exist constants \(\delta_{1},\delta_{2}\in(0,1)\) such that

$$ \frac{ \Vert \chi_{E} \Vert _{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}}{ \Vert \chi _{B} \Vert _{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}}= \frac{ \Vert \chi_{E} \Vert _{(L^{p^{\prime}_{2}(\cdot)}(w^{-p^{\prime}_{2}(\cdot )}))^{\prime}}}{ \Vert \chi_{B} \Vert _{(L^{p^{\prime}_{2}(\cdot)}(w^{-p^{\prime}_{2}(\cdot)}))^{\prime}}} \lesssim \biggl( \frac{ \vert E \vert }{ \vert B \vert } \biggr)^{\delta_{1}}$$
(3)

and

$$ \frac{ \Vert \chi_{E} \Vert _{(L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)}))^{\prime}}}{ \Vert \chi_{B} \Vert _{(L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)}))^{\prime}}} \lesssim \biggl(\frac{ \vert E \vert }{ \vert B \vert } \biggr)^{\delta_{2}},$$
(4)

for all balls B and all measurable sets \(E\subset B\).

Lemma 3.8

([34])

Let \(p_{1}(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\cap\mathcal {C}^{\log}( \mathbb{R}^{n})\) and \(0<\beta<\frac{n}{p_{1+}}\), and \(\frac{1}{p_{2}(\cdot)}=\frac{1}{p_{1}(\cdot)}-\frac{\beta}{n}\). If \(w\in A(p_{1}(\cdot),p_{2}(\cdot))\), then \(I^{\beta}\) is bounded from \(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) to \(L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})\).

4 Main results and proofs

The following proposition was proved in [36].

Proposition 4.1

Let \(q(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\), \(0< p<\infty\) and \(0\leq\lambda<\infty\). If \(\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap\mathcal {C}^{\log}( \mathbb{R}^{n})\), then

$$ \begin{aligned} \Vert f \Vert ^{p}_{M\dot{K}^{\alpha(\cdot),\lambda}_{p,q(\cdot)}(w^{q( \cdot)})}={}&{ \sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda p}\sum _{j=-\infty}^{k_{0}} 2^{j\alpha(\cdot)p} \Vert f \chi_{j} \Vert ^{p}_{L^{q(\cdot)}(w^{q(\cdot)})} \\ \leq{}&\max \Biggl\{ \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0} \lambda p} \Biggl(\sum_{j=-\infty}^{k_{0}} 2^{j\alpha(0)p} \Vert f\chi_{j} \Vert ^{p}_{L^{q(\cdot)}(w^{q(\cdot)})} \Biggr), \\ &{}\underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}} \Biggl(2^{-k_{0}\lambda p} \Biggl(\sum_{j=-\infty}^{-1} 2^{j\alpha(0)p} \Vert f\chi_{j} \Vert ^{p}_{L^{q( \cdot)}(w^{q(\cdot)})} \Biggr) \\ &{}+2^{-k_{0}\lambda p} \Biggl(\sum_{j=0}^{k_{0}} 2^{j\alpha(\infty)p} \Vert f\chi_{j} \Vert ^{p}_{L^{q(\cdot)}(w^{q(\cdot)})} \Biggr) \Biggr) \Biggr\} . \end{aligned} $$

One of the main results of this study is as follows.

Theorem 4.2

Let \(0< q_{1}\leq q_{2}<\infty\), \(p_{2}(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\cap\mathcal {C}^{\log}( \mathbb{R}^{n})\), and \(p_{1}(\cdot)\) be such that \(\frac{1}{p_{2}(\cdot)}=\frac{1}{p_{1}(\cdot)}-\frac{\beta}{n}\). Also, let \(w^{p_{2}(\cdot)}\in A_{1}\), \(\lambda>0\) and \(\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap\mathcal {C}^{\log}( \mathbb{R}^{n})\) be log Hölder continuous at the origin, with \(\alpha(0)\leq\alpha(\infty)<\lambda+n\delta_{2}-\beta\), where \(\delta_{2}\in(0,1)\) is the constant appearing in (4), then

$$\Vert H_{\beta}f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{2},p_{2}(\cdot )}(w^{p_{2}( \cdot)})}\leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}( \cdot)}(w^{p_{1}(\cdot)})}. $$

Proof

For any \(f\in{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot)}(w^{p_{1}( \cdot)})}\), if we represent \(f_{j}=f\cdot\chi_{j}=f\cdot\chi_{A_{j}}\) for each \(j\in\mathbb{Z}\), then we write

$$f(x)=\sum_{j=-\infty}^{\infty}f(x)\cdot \chi_{j}(x)=\sum_{j=- \infty}^{\infty}f_{j}(x). $$

The generalized Hölder inequality yields

$$ \begin{aligned}[b] \bigl\vert H_{\beta}f(x)\cdot \chi_{k}(x) \bigr\vert &\leq\frac{1}{ \vert x \vert ^{n-\beta}} \int _{B_{k}} \bigl\vert f(t) \bigr\vert \,dt \cdot \chi_{k}(x) \\ &\leq C2^{-kn}\sum_{j=-\infty}^{k} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}}2^{k\beta} \chi_{k}(x).\end{aligned} $$
(5)

Making use of Lemmas 3.3 and 3.7, respectively, we obtain

$$ \begin{aligned}[b] & \Vert H_{\beta}f\cdot\chi_{k} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}}2^{-kn} \Vert \chi_{k} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}} \Vert \chi_{k} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}}^{-1} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}} \Vert \chi_{j} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}}^{-1} \frac{ \Vert \chi_{j} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}}}{ \Vert \chi_{k} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}}} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k}2^{n\delta_{2}(j-k)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}} \Vert \chi_{j} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}}^{-1}.\end{aligned} $$
(6)

To proceed further, we take \(f=\chi_{B_{j}}\) in the definition of \(I_{\beta}\) to get

$$I_{\beta}(\chi_{B_{j}}) (x) \geq C 2^{j\beta} \chi_{B_{j}}(x), $$

which implies that

$$\chi_{B_{j}}(x)\leq C 2^{-j\beta}I_{\beta}( \chi_{B_{j}}) (x). $$

Taking the norm on both sides and using Lemmas 3.8 and 3.3, respectively, we get

$$ \begin{aligned}[b] \Vert \chi_{B_{j}} \Vert _{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}&\leq C 2^{-j \beta} \bigl\Vert I_{\beta}(\chi_{B_{j}}) \bigr\Vert _{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})} \\ & \leq C 2^{-j\beta} \bigl\Vert (\chi_{B_{j}}) \bigr\Vert _{L^{p_{1}(\cdot)}(w^{p_{1}( \cdot)})} \\ & \leq2^{j(n-\beta)} \bigl\Vert (\chi_{B_{j}}) \bigr\Vert _{(L^{p_{1}(\cdot)}(w^{p_{1}( \cdot)}))^{\prime}}^{-1}.\end{aligned} $$
(7)

Inserting (7) into (6), we are down to

$$ \begin{aligned}[b] & \Vert H_{\beta}f\cdot \chi_{k} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k}2^{n\delta_{2}(j-k)}2^{j(n- \beta)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}}^{-1} \Vert \chi_{j} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}}^{-1} \\ &\quad =C\sum_{j=-\infty}^{k}2^{(\beta-n\delta_{2})(k-j)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \bigl(2^{-jn} \Vert \chi_{j} \Vert _{{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}} \Vert \chi_{j} \Vert _{(L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})})^{\prime}} \bigr)^{-1} \\ &\quad \leq C\sum_{j=-\infty}^{k}2^{(\beta-n\delta_{2})(k-j)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}.\end{aligned} $$
(8)

In the rest of the proof, in order to estimate \(\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\), we consider two cases as below.

Case 1: We take \(j<0\) and start estimating as follows:

$$ \begin{aligned}[b] \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}&=2^{-j\alpha(0)} \bigl(2^{j \alpha(0)q_{1}} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}^{q_{1}} \bigr)^{\frac{1}{q_{1}}} \\ & \leq2^{-j\alpha(0)} \Biggl(\sum_{i=-\infty}^{j}2^{i\alpha(0)q_{1}} \Vert f_{i} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}^{q_{1}} \Biggr)^{ \frac{1}{q_{1}}} \\ & \leq2^{j(\lambda-\alpha(0))}2^{-j\lambda} \Biggl(\sum _{i=-\infty}^{j}2^{i \alpha(\cdot)q_{1}} \Vert f_{i} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}^{q_{1}} \Biggr)^{\frac{1}{q_{1}}} \\ & \leq C2^{j(\lambda-\alpha(0))} \Vert f \Vert _{M\dot{K}^{\alpha(\cdot), \lambda}_{q_{1},p_{1}(\cdot)}(w^{p_{1}(\cdot)})}.\end{aligned} $$
(9)

Case 2: For \(j\geq0\), we get

$$ \begin{aligned}[b] \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}&=2^{-j\alpha(\infty)} \bigl(2^{j\alpha(\infty)q_{1}} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} \bigr)^{\frac{1}{q_{1}}} \\ & \leq2^{-j\alpha(\infty)} \Biggl(\sum_{i=0}^{j}2^{i\alpha(\infty)q_{1}} \Vert f_{i} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}^{q_{1}} \Biggr)^{ \frac{1}{q_{1}}} \\ & \leq2^{j(\lambda-\alpha(\infty))}2^{-j\lambda} \Biggl(\sum _{i=- \infty}^{j}2^{i\alpha(\cdot)q_{1}} \Vert f_{i} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} \Biggr)^{\frac{1}{q_{1}}} \\ & \leq C2^{j(\lambda-\alpha(\infty))} \Vert f \Vert _{M\dot{K}^{\alpha(\cdot), \lambda}_{q_{1},p_{1}(\cdot)}(w^{p_{1}(\cdot)})}.\end{aligned} $$
(10)

By the definition of variable exponent Morrey–Herz space along with the use of Proposition 4.1, we arrive at the following inequality:

$$\begin{aligned} \Vert H_{\beta}f \Vert ^{q_{1}}_{M\dot{K}^{\alpha(\cdot),\lambda }_{q_{2},p_{2}(\cdot)}(w^{p_{2}( \cdot)})}={}&\sup _{k_{0}\in{Z}}2^{-k_{0}\lambda q_{1}}\sum_{k=- \infty}^{k_{0}}2^{k\alpha(\cdot)q_{1}} \Vert H_{\beta}f\cdot\chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})} \\ \leq{}& \max \Biggl\{ \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0} \lambda q_{1}} \sum_{k=-\infty}^{k_{0}}2^{k\alpha(0)q_{1}} \Vert H_{\beta}f \cdot\chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}, \\ &{}\underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \Biggl(\sum_{k=-\infty}^{-1}2^{k\alpha(0)q_{1}} \Vert H_{\beta}f\cdot \chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})} \\ & {}+\sum_{k=0}^{k_{0}}2^{k\alpha(\infty)q_{1}} \Vert H_{\beta}f\cdot \chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})} \Biggr) \Biggr\} \\ ={}&\max\{Y_{1},Y_{2}+Y_{3}\}, \end{aligned}$$
(11)

where

$$\begin{aligned}& Y_{1}=\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}}2^{k\alpha(0)q_{1}} \Vert H_{\beta}f\cdot\chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}, \\& Y_{2}=\underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{-1}2^{k\alpha(0)q_{1}} \Vert H_{\beta}f\cdot\chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}, \\& Y_{3}=\underset{k_{0}\geq0}{\sup _{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=0}^{k_{0}}2^{k\alpha(\infty)q_{1}} \Vert H_{\beta}f\cdot\chi_{k} \Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}. \end{aligned}$$

First, we approximate \(Y_{1}\). Since \(\alpha(0)\leq\alpha({\infty})< n\delta_{2}+\lambda-\beta\),

$$ \begin{aligned}[b] Y_{1}&\leq C\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum _{k=-\infty}^{k_{0}}2^{k\alpha(0)q_{1}} \Biggl(\sum _{j=-\infty}^{k} 2^{(\beta-n\delta_{2})(k-j)} \Vert f \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \Biggr)^{q_{1}} \\ & \leq C\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}}2^{k\alpha(0)q_{1}} \Biggl(\sum_{j=-\infty}^{k} 2^{(\beta-n\delta_{2})(k-j)}2^{j(\lambda-\alpha(0))} \Vert f \Vert _{M \dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \Biggr)^{q_{1}} \\ & \leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=- \infty}^{k_{0}}2^{k\alpha(0)q_{1}} \Biggl(\sum_{j=-\infty}^{k} 2^{( \beta-n\delta_{2})(k-j)}2^{j(\lambda-\alpha(0))} \Biggr)^{q_{1}} \\ & \leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=- \infty}^{k_{0}}2^{k\lambda q_{1}} \Biggl(\sum_{j=-\infty}^{k} 2^{(j-k)(- \beta+n\delta_{2}-\alpha(0)+\lambda)} \Biggr)^{q_{1}} \\ &\leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} .\end{aligned} $$

The estimate of \(Y_{2}\) is similar to that of \(Y_{1}\). Lastly, we estimate \(Y_{3}\)

$$\begin{aligned} Y_{3}&\leq C\underset{K_{0} \geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0} \lambda q_{1}}\sum _{k=0}^{k_{0}}2^{k\alpha(\infty)q_{1}} \Biggl( \sum _{j=-\infty}^{k} 2^{(\beta-n\delta_{2})(k-j)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \Biggr)^{q_{1}} \\ & \leq C\underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=0}^{k_{0}}2^{k\alpha(\infty)q_{1}} \Biggl(\sum_{j=-\infty}^{k} 2^{(\beta-n\delta_{2})(k-j)}2^{j(\lambda-\alpha(\infty))} \Vert f \Vert _{M \dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \Biggr)^{q_{1}} \\ & \leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} \underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=0}^{k_{0}}2^{k\alpha(\infty)q_{1}} \Biggl(\sum_{j=-\infty}^{k} 2^{(\beta-n\delta_{2})(k-j)}2^{j(\lambda-\alpha(\infty))} \Biggr)^{q_{1}} \\ & \leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}} \underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=0}^{k_{0}}2^{k\lambda q_{1}} \Biggl(\sum_{j=-\infty}^{k} 2^{(j-k)(- \beta+n\delta_{2}-\alpha(\infty)+\lambda)} \Biggr)^{q_{1}} \\ &\leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot)})}^{q_{1}}. \end{aligned}$$

The desired result is obtained by inserting the approximations of \(Y_{1},Y_{2}\), and \(Y_{3}\) into (11). □

Theorem 4.3

Let \(q_{1},q_{2},p_{1}(\cdot),p_{2}(\cdot),\beta, \alpha(\cdot)\) and w be as in Theorem 4.2. In addition, if \(-n\delta_{1}+\lambda<\alpha(0)\leq\alpha(\infty)\), where \(\delta_{1}\in(0,1)\) is the constant appearing in (3), then

$$\bigl\Vert H^{*}_{\beta}f \bigr\Vert _{M\dot{K}^{\alpha(\cdot),\lambda}_{q_{2},p_{2}( \cdot)}(w^{p_{2}(\cdot)})} \leq C \Vert f \Vert _{M\dot{K}^{\alpha(\cdot), \lambda}_{q_{1},p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. $$

Proof

An application of the Hölder inequality gives

$$ \begin{aligned}[b] \bigl\vert H^{*}_{\beta}f(x) \cdot\chi_{k}(x) \bigr\vert &\leq \int_{R^{n}\setminus B_{k}} \bigl\vert f(t) \|x \bigr\vert ^{\beta-n}\,dt\cdot\chi_{k}(x) \\ &\leq C\sum_{j=k+1}^{\infty}2^{j(\beta-n)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}}\chi _{k}(x).\end{aligned} $$

Now, using Lemma 3.3, we have

$$ \begin{aligned}[b] \bigl\Vert H^{*}_{\beta}f \cdot\chi_{k} \bigr\Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}}& \leq C\sum _{j=k+1}^{\infty}2^{j(\beta-n)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{({L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))^{\prime}}} \Vert \chi_{k} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}} \\ &\leq C\sum_{j=k+1}^{\infty}2^{j\beta} \Vert f_{j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot)})} \Vert \chi_{j} \Vert _{{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}}^{-1} \Vert \chi_{k} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}}.\end{aligned} $$
(12)

In view of inequality (7), we obtain

$$ \begin{aligned}[b] \bigl\Vert H^{*}_{\beta}f \cdot\chi_{k} \bigr\Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}}& \leq C\sum _{j=k+1}^{\infty} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \frac{ \Vert \chi_{k} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}}}{ \Vert \chi _{j} \Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}}} \\ &\leq C\sum_{j=k+1}^{\infty}2^{n\delta_{1}(k-j)} \Vert f_{j} \Vert _{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} ,\end{aligned} $$
(13)

where we used Lemma 3.7 in the last step.

In the remaining proof of this theorem, we follow the procedure as in Theorem 4.2 to have

$$ \bigl\Vert H^{*}_{\beta}f\cdot\chi_{k} \bigr\Vert _{L^{p_{2}(\cdot)}{(w^{p_{2}(\cdot)})}}= \max\{Z_{1},{Z_{2}+Z_{3}} \},$$
(14)

where

$$\begin{aligned}& Z_{1}=\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}}2^{k\alpha(0)q_{1}} \bigl\Vert H^{*}_{\beta}f\cdot \chi_{k} \bigr\Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}. \\& Z_{2}=\underset{k_{0}\geq0}{\sup_{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{-1}2^{k\alpha(0)q_{1}} \bigl\Vert H^{*}_{\beta}f\cdot\chi_{k} \bigr\Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}. \\& Z_{3}=\underset{k_{0}\geq0}{\sup _{k_{0}\in{Z}}}2^{-k_{0}\lambda q_{1}} \sum_{k=0}^{k_{0}}2^{k\alpha(\infty)q_{1}} \bigl\Vert H^{*}_{\beta}f\cdot \chi_{k} \bigr\Vert ^{q_{1}}_{L^{p_{2}(\cdot)}(w^{p_{2}(\cdot)})}. \end{aligned}$$

The estimates of \(Z_{i}\) (\(i=1,2,3\)) are similar to those of \(Y_{i}\) (\(i=1,2,3\)) of Theorem 4.2. Here we conclude our result. □