Variable Herz–Morrey estimates for rough fractional Hausdorff operator

Abstract

As a first attempt, we obtain the boundedness of the rough fractional Hausdorff operator on variable exponent Herz-type spaces. The method used in this paper enables us to study the operator on some other function spaces with variable exponents.

1 Introduction

The Hausdorff operator has a fascinating history of its development from simple to present form. The laying stone of this development is the one-dimensional Hausdorff operator

$$ \mathcal{H}_{\Phi }g(y)= \int _{0}^{\infty }\frac{\Phi (t)}{t} g \bigl(yt^{-1}\bigr)\,dt,$$

(1.1)

where $\Phi \in L^{1}_{\mathrm{{loc}}}(\mathbb{R})$. A change of variables in (1.1) results in the following form:

$$ \mathsf {H}_{\Phi }g(y)= \int _{0}^{\infty }\frac{\Phi (yt^{-1})}{t} g(t) \,dt.$$

(1.2)

The boundedness of these operators, an essential part of the analysis, has been discussed in [10, 29, 30]. For a detailed history and the recent development of the Hausdorff operator, we refer the interested reader to the review papers [6, 28].

The Hausdorff operator finds its applications in the study of one-dimensional Fourier analysis. In particular, it plays a significant role in the summability of the classical Fourier series. Therefore extensions of the one-dimensional Hausdorff operators to multidimensional spaces become extremely important. Several authors have spared their studies for such an extension. In this regard, some contributions are [5, 26, 27, 31]. Here we are mainly interested in the rough Hausdorff operator [5]

$$ {H}_{\Phi ,\Omega} g(y)= \int _{\mathbb{R}^{n}} \frac{\Phi (y \vert t \vert ^{-1})}{ \vert t \vert ^{n}} \Omega \bigl(t^{\prime} \bigr) g(t)\,dt$$

(1.3)

and the fractional Hausdorff Hausdorff operator [31]

$$ {H}_{\Phi}^{\beta }g(y)= \int _{\mathbb{R}^{n}} \frac{\Phi (y \vert t \vert ^{-1})}{ \vert t \vert ^{n-\beta}} g(t)\,dt, \quad 0\le \beta < n.$$

(1.4)

The rough fractional Hausdorff operator, which is a combination of (1.3) and (1.4),

$$ {H}_{\Phi ,\Omega}^{\beta }g(y)= \int _{\mathbb{R}^{n}} \frac{\Phi (y \vert t \vert ^{-1})}{ \vert t \vert ^{n-\beta}} \Omega \bigl(t^{\prime} \bigr) g(t)\,dt,\quad 0\le \beta < n,$$

(1.5)

was first studied in [14]. It is obvious that by taking $\beta =0$ in (1.5) we obtain ${H}_{\Phi ,\Omega}$, and by taking $\Omega =1$ in the same identity, we get the fractional Hausdorff operator ${H}_{\Phi}^{\beta}$. Thus discussing the boundedness of ${H}_{\Phi ,\Omega}^{\beta}$ on variable exponent Herz-type spaces automatically includes the same results for ${H}_{\Phi ,\Omega}$ and ${H}_{\Phi}^{\beta}$ on these spaces. Some studies containing boundedness results regarding ${H}_{\Phi ,\Omega}$, ${H}_{\Phi ,1}$, and ${H}_{\Phi}^{\beta}$ along with their commutators on different function spaces include [2, 3, 7, 11–13, 17, 19, 20, 35].

Another important aspect of the Hausdorff operator is that it contains other classical operators as its particular cases. For example, in $H^{\beta}_{\Phi ,1}$, if we take $\Phi (z)=\Phi _{1}(z)=z^{-n+\beta}\chi _{(1,\infty )}(z)$ and $\Phi (z)=\Phi _{2}(z)=\chi _{(0,1]}(z)$, respectively, then we obtain the n-dimensional fractional Hardy operator

$$ P_{\beta }g( y)=\frac{1}{ \vert y \vert ^{n-\beta}} \int _{ \vert y \vert < \vert t \vert }g(t)\,dt $$

and its adjoint operator:

$$ Q_{\beta }g(y)= \int _{ \vert y \vert \geq \vert t \vert }\frac{g(t)}{ \vert t \vert ^{n-\beta}}\,dt. $$

During the recent past, the boundedness of Hardy-type operators on variable exponent function spaces also drew great attention of the research community [18, 39]. Besides the Hardy operators, the Hausdorff operator also contains the n-dimensional version of the Caldrón operator

$$ C g(y)=\frac{1}{ \vert y \vert ^{n}} \int _{ \vert y \vert < \vert t \vert }g(t)\,dt+ \int _{ \vert t \vert \geq \vert y \vert } \frac{g(t)}{ \vert t \vert ^{n}}\,dt $$

if we choose $\Phi (t)=\min \{1,\frac{1}{|t|^{n}} \}$ in the definition of $H_{\Phi ,1}$.

Variable exponent function spaces have an increasing impact on the recent advances in harmonic analysis. This is mainly because of their frequent appearance in analysis and applications of different functional analysis tools for partial differential equations. This results in an increase in research publications in this field. The theory of variable Lebesgue spaces first appeared in the pioneering work of Kováčik and Rákosník [25]. Later on, several monograms appeared in the literature to strengthen the theory. Operator theory on function spaces also finds new dimensions, and many researchers discuss the boundedness of different operators on variable-exponent function spaces [24, 37]. In this paper, we inquire about the continuity of the rough fractional Hausdorff operator from this perspective.

On function spaces with constant exponents, the boundedness of the Hausdorff operator is accomplished by a scaling argument followed by the polar decomposition of integral on $\mathbb{R}^{n}$. However, such a direct approach on function spaces with variable exponent does not work well. It needs some modification in the later case. In this paper, we aim to tackle this problem by establishing the boundedness of the rough fractional Hausdorff operator on variable-exponent Herz-type spaces. Our strategy enables us to study the operator on other function spaces with variable exponents. As corollaries of our main results, we prove the boundedness of the rough Hausdorff operator on variable-exponent Herz-type spaces, which are also new to the best of the author’s knowledge.

In the next section, we present some definitions and preliminary lemmas. Section 3 contains the theorems stating the boundedness criterion for rough fractional Hausdorff operators on variable-exponent Herz-type spaces and their detailed proofs.

2 Variable-exponent function spaces

Let $O\subseteq \mathbb{R}^{n}$ be an open set, and let $q(\cdot ): O\rightarrow [1,\infty )$ be a measurable function. We denote by $q^{\prime}(\cdot )$ the conjugate exponent of $q(\cdot )$ satisfying

$$ \frac{1}{q(\cdot )}+\frac{1}{q^{\prime}(\cdot )}=1. $$

The set $\mathcal {P}(O)$ consists of all $q(\cdot )$ such that

$$ 1< q^{-}=\operatorname{essinf}\bigl\{ q(y):y\in O\bigr\} \le q^{+}= \operatorname{esssup}\bigl\{ q(y):y\in O\bigr\} < \infty . $$

The Lebesgue space with variable exponent $L^{q(\cdot )}(O)$ is a set of all measurable functions g such that for a positive σ,

$$ \int _{O} \biggl(\frac{ \vert g(y) \vert }{\sigma} \biggr)^{q(y)} \,dy< \infty . $$

Equipped with the Luxemburg norm

$$ \Vert g \Vert _{L^{q(\cdot )}(O)}=\inf \biggl\{ \sigma >0: \int _{O} \biggl( \frac{ \vert g(y) \vert }{\sigma} \biggr)^{q(y)} \,dy\leq 1 \biggr\} , $$

the space $L^{q(\cdot )}(O)$ becomes a Banach function space. The set of functions

$$ L^{q(\cdot )}_{\mathrm{{loc}}}(O)= \bigl\{ g:g\in L^{q(\cdot )}(E)\ \forall \text{ compact subset } \mathrm{E}\subset \mathrm{O} \bigr\} $$

serves to define local version of the variable-exponent Lebesgue space $L^{q(\cdot )}_{\mathrm{{loc}}}(O)$.

Variable-exponent function spaces bear a deep connection with the boundedness of Hardy–Littlewood maximal operator $\mathcal {M}$ defined by

$$ \mathcal {M}g(x)=\sup_{r>0}\frac{1}{ \vert B(x,r) \vert } \int _{B(x,r)} \bigl\vert g(t) \bigr\vert \,dt $$

on $L^{q(\cdot )}(\mathbb{R}^{n})$. We denote by $\mathcal{B}(\mathbb{R}^{n})$ the set consisting of $q(\cdot )\in \mathcal {P}(\mathbb{R}^{n})$ such that $\mathcal {M}$ is bounded on $L^{q(\cdot )}(\mathbb{R}^{n})$.

Proposition 2.1

([8, 34])

Let $O\subset \mathbb{R}^{n}$ be an open set, and let $q(\cdot )\in \mathcal {P}(O)$ satisfy

$$\begin{aligned}& \bigl\vert q(\xi )-q(\eta ) \bigr\vert \leq \frac{-C}{\ln ( \vert \xi -\eta \vert )}, \quad \frac{1}{2} \geq \vert \xi -\eta \vert , \end{aligned}$$

(2.1)

$$\begin{aligned}& \bigl\vert q(\xi )-q(\eta ) \bigr\vert \leq \frac{C}{\ln ( \vert \xi \vert +e)},\quad \vert \xi \vert \leq \vert \eta \vert , \end{aligned}$$

(2.2)

then $q(\cdot )\in \mathcal{B}(O)$, where C is a positive constant independent of ξ and η.

Lemma 2.2

([25] Generalized Hölder inequality)

Let $q(\cdot ),q_{1}(\cdot ),q_{2}(\cdot )\in \mathcal {P}(O)$.

• (a) If $g\in L^{q(\cdot )}(O)$ and $h\in L^{q^{\prime}(\cdot )}(O)$, then

  $$ \int _{O} \bigl\vert g(y)h(y) \bigr\vert \,dy\leq r_{q} \Vert g \Vert _{L^{q(\cdot )}(O)} \Vert h \Vert _{L^{q^{ \prime}(\cdot )}(O)},$$

  where $r_{q}=1+\frac{1}{q_{-}}-\frac{1}{q_{+}}$.

• (b) If $g\in L^{q_{1}(\cdot )}(O)$, $h\in L^{q_{2}(\cdot )}(O)$, and $\frac{1}{q(\cdot )}=\frac{1}{q_{1}(\cdot )}+\frac{1}{q_{2}(\cdot )}$, then

  $$ \Vert gh \Vert _{L^{q(\cdot )}(O)}\leq r_{q,q_{1}} \Vert g \Vert _{L^{q_{1}(\cdot )}(O)} \Vert h \Vert _{L^{q_{2}(\cdot )}(O)},$$

  where $r_{q,q_{1}}= (1+\frac{1}{(q_{1})_{-}}-\frac{1}{(q_{1})_{+}} )^{1/q_{-}}$.

Lemma 2.3

([22])

If $q(\cdot )\in \mathcal{B}(\mathbb{R}^{n})$, then there exist constants $0<\delta <1$ and $C>0$ such that for all balls B in $\mathbb{R}^{n}$ and all measurable subsets $S\subset B$,

$$ \frac{ \Vert \chi _{B} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}}{ \Vert \chi _{S} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}} \leq C \biggl(\frac{ \vert B \vert }{ \vert S \vert } \biggr)^{\delta}. $$

Lemma 2.4

([22])

Assuming that $q(\cdot )\in \mathcal{B}(\mathbb{R}^{n})$, for all balls $B\subset \mathbb{R}^{n}$ and for a positive constant C, we have

$$ C^{-1}\leq \frac{1}{ \vert B \vert } \Vert \chi _{B} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \Vert \chi _{B} \Vert _{L^{q^{\prime}(\cdot )}(\mathbb{R}^{n})}\leq C.$$

Lemma 2.5

([9])

Define a variable exponent $\tilde{p}(\cdot )$ such that $\frac{1}{q(t)}=\frac{1}{\tilde{p}(t)}+\frac{1}{p}$ ($t\in \mathbb{R}^{n}$). Then we have

$$ \Vert gh \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}\leq C \Vert g \Vert _{L^{\tilde{p}(\cdot )}( \mathbb{R}^{n})} \Vert h \Vert _{L^{p}(\mathbb{R}^{n})}.$$

Lemma 2.6

([33])

Let $p(\cdot )\in \mathcal {P}(\mathbb{R}^{n})$ satisfy conditions (2.1) and (2.2) in Proposition 2.1. Then

$$ \Vert \chi _{Q} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}\approx \textstyle\begin{cases} \vert Q \vert ^{\frac{1}{q(x)}} & \textit{if } \vert Q \vert < 2^{n} \textit{ and } x\in Q, \\ \vert Q \vert ^{\frac{1}{q(\infty )}} & \textit{if } \vert Q \vert \ge 1,\end{cases} $$

for all cubes (or balls) $Q\subset \mathbb{R}^{n}$, where $q(\infty )=\lim_{x\rightarrow \infty}q(x)$.

The boundedness of the fractional integral operator $I_{\beta}$ defined by

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

(2.3)

on variable Lebesgue space (see [4]) takes a crucial part in proving our main result.

Proposition 2.7

Let $q_{1}(\cdot )\in \mathcal{P}(\mathbb{R}^{n})$ and $0<\beta <\frac{n}{(q_{1})_{+}}$, and define $q_{2}(\cdot )$ by

$$ \frac{1}{q_{2}(\cdot )}=\frac{1}{q_{1}(\cdot )}-\frac{\beta}{n}. $$

(2.4)

Then

$$ \Vert I_{\beta}f \Vert _{L^{q_{2}(\cdot )}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{L^{q_{1}( \cdot )}(\mathbb{R}^{n})}. $$

(2.5)

Using Proposition (2.7), Wu [38] established the following result.

Lemma 2.8

Let β, $q_{1}(\cdot )$, and $q_{2}(\cdot )$ be as defined in Proposition 2.7. Then

$$ \Vert \chi _{B_{l}} \Vert _{L^{q_{2}(\cdot )}(\mathbb{R}^{n})}\leq C 2^{-l \beta} \Vert \chi _{B_{l}} \Vert _{L^{q_{1}(\cdot )}(\mathbb{R}^{n})} $$

(2.6)

for all balls $B_{l}=\{x\in \mathbb{R}^{n}:|x|\le 2^{l}\}$ with $l\in \mathbb{Z}$.

Let $C_{l}=\{x\in \mathbb{R}^{n}:2^{l-1}<|x|\le 2^{l}\}$ and $\chi _{l}=\chi _{C_{l}}$ for $l\in \mathbb{Z}$. Then the homogenous Herz space with variable exponent is defined as follows.

Definition 2.9

([23])

Let $\alpha \in \mathbb{R}$, $0< q<\infty$, and $p(\cdot )\in \mathcal {P}(\mathbb{R}^{n})$. The homogenous Herz space with variable exponent $\dot{K}^{\alpha ,q}_{p(\cdot )}(\mathbb{R}^{n})$ is the set of all measurable functions f such that

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

where

$$ \Vert f \Vert _{ \dot {K}^{\alpha ,q}_{p(\cdot )}(\mathbb{R}^{n})}= \Biggl(\sum_{k=- \infty}^{\infty}2^{k\alpha q} \Vert f\chi _{l} \Vert ^{q}_{L^{p (\cdot )}( \mathbb{R}^{n})} \Biggr)^{\frac{1}{q}}. $$

If $p(\cdot )=p$, then we have the classical Herz space $\dot{K}_{q}^{\alpha ,p}$ defined in [32]. Some generalizations of Herz spaces were made in [1, 15, 16, 36] shortly after their first appearance.

Definition 2.10

Let $\alpha \in \mathbb{R}$, $0< q<\infty$, $\lambda \in [0,\infty )$, and $p(\cdot )\in \mathcal{P}(\mathbb{R}^{n})$. The space $M\dot{K}^{\alpha ,\lambda}_{q,p(\cdot )}(\mathbb{R}^{n})$ is the set of all measurable functions f given by

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

where

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

Obviously, $M\dot{K}^{\alpha ,0}_{q,p(\cdot )}(\mathbb{R}^{n})= \dot{K}^{\alpha ,q}_{p(\cdot )}(\mathbb{R}^{n})$ is the Herz space with variable exponent. The Herz–Morrey spaces with variable exponent $M\dot{K}^{\alpha ,\lambda}_{q,p(\cdot )}$ are first defined in [21, 22].

Remark 2.11

Let $p(\cdot ), p_{1}(\cdot ), p_{2}(\cdot )\in \mathcal {P}(\mathbb{R}^{n})$ and meet conditions (2.1) and (2.2) in Proposition 2.1, then so does $p^{\prime}(\cdot )$ and $p_{1}^{\prime}(\cdot )$. This implies that $p_{1}(\cdot ), p_{1}^{\prime}(\cdot ), p_{2}(\cdot )\in \mathcal{B}( \mathbb{R}^{n})$. Therefore, using Lemma 2.2, we have the constants $\delta _{2}\in (0,\frac{1}{(p_{2})_{+}})$, $\delta _{1}\in (0, \frac{1}{(p^{\prime}_{1})_{+}})$, $\delta ^{*}_{2}\in (0,\frac{1}{(p)_{+}})$, and $\delta ^{*}_{1}\in (0,\frac{1}{(p^{\prime})_{+}})$ such that the inequalities

$$\begin{aligned}& \frac{ \Vert \chi _{S} \Vert _{L^{p_{2}(\cdot )}(\mathbb{R}^{n})}}{ \Vert \chi _{B} \Vert _{L^{p_{2}(\cdot )}(\mathbb{R}^{n})}} \leq C \biggl(\frac{ \vert S \vert }{ \vert B \vert } \biggr)^{\delta _{2}}, \qquad \frac{ \Vert \chi _{S} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})}}{ \Vert \chi _{B} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})}} \leq C \biggl(\frac{ \vert S \vert }{ \vert B \vert } \biggr)^{\delta _{1}},\\& \frac{ \Vert \chi _{S} \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})}}{ \Vert \chi _{B} \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})}} \leq C \biggl(\frac{ \vert S \vert }{ \vert B \vert } \biggr)^{\delta _{2}^{*}}, \qquad \frac{ \Vert \chi _{S} \Vert _{L^{p^{\prime}(\cdot )}(\mathbb{R}^{n})}}{ \Vert \chi _{B} \Vert _{L^{p^{\prime}(\cdot )}(\mathbb{R}^{n})}} \leq C \biggl(\frac{ \vert S \vert }{ \vert B \vert } \biggr)^{\delta _{1}^{*}}, \end{aligned}$$

hold for all balls $B\subset \mathbb{R}^{n}$ and $S\subset B$.

3 Main results and proofs

In this section, we present theorems on the boundedness of the rough fractional Hausdorff operator on the Herz-type spaces. We denote

$$ A_{\Phi ,s}= \biggl( \int _{0}^{\infty} \bigl\vert \Phi (t) \bigr\vert ^{s}t^{(n-\beta )s-n} \frac{dt}{t} \biggr)^{\frac{1}{s}}. $$

Theorem 3.1

Let $0<\beta <\frac{n}{(p_{1})_{+}}$, $0< q_{1}\le q_{2}<\infty$, and $\Omega \in L^{s}(S^{n-1})$, and let $p_{1}(\cdot ), p_{2}(\cdot )\in \textit{P}(\mathbb{R}^{n})$ satisfy conditions (2.1) and (2.2) in Proposition 2.1with

$$ \frac{1}{p_{1}(\cdot )}=\frac{1}{p_{2}(\cdot )}+\frac{\beta}{n}, $$

where $p_{1}(\cdot )<\frac{n}{\beta}$. Suppose that for $0<\delta _{1}$, $\delta _{2}<1$, α satisfies $\frac{n}{p_{1}^{-}}-n\delta _{2}-\beta <\alpha <n\delta _{1}- \frac{n}{(p_{1}^{\prime})^{+}}$ and $0\le{\lambda}<\alpha +\beta +n\delta _{2}-\frac{n}{p_{1}^{-}}$. If Φ is a radial function and $A_{\Phi ,s}<\infty$, then for $f\in M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}(\cdot )}(\mathbb{R}^{n})$, we have

$$ \begin{aligned}[b] \bigl\Vert H_{\Phi ,\Omega}^{\beta}f \bigr\Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}( \cdot )}(\mathbb{R}^{n})} \leq CA_{\Phi ,s} \Vert f \Vert _{M\dot{K}^{\alpha , \lambda}_{q_{1},p_{1}(\cdot )}(\mathbb{R}^{n})} .\end{aligned} $$

(3.1)

Proof

Since $q_{1}\le q_{2}$, by the definition of the Morrey–Herz space we have

$$\begin{aligned} \bigl\Vert H_{\Phi ,\Omega}^{\beta}f \bigr\Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}( \cdot )}(\mathbb{R}^{n})} \leq& \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{k=-\infty}^{k_{0}}2^{k\alpha q_{1}} \bigl\Vert \bigl(H_{\Phi , \Omega}^{\beta}f\bigr)\chi _{k} \bigr\Vert ^{q_{1}}_{L^{p_{2}(\cdot )}(\mathbb{R}^{n})} \\ \leq& \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}2^{k\alpha q_{1}} \Biggl(\sum _{j=-\infty}^{\infty} \bigl\Vert \bigl(H_{ \Phi ,\Omega}^{\beta}(f \chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \Biggr)^{q_{1}} \\ \leq& C\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}2^{k\alpha q_{1}} \Biggl(\sum _{j=-\infty}^{k-1} \bigl\Vert \bigl(H_{ \Phi ,\Omega}^{\beta}(f \chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \Biggr)^{q_{1}} \\ &{}+ C\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}}2^{k \alpha q_{1}} \Biggl(\sum _{j=k}^{\infty} \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f \chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}(\mathbb{R}^{n})} \Biggr)^{q_{1}} \\ =:& I_{1}+I_{2}. \end{aligned}$$

For both $I_{1}$ and $I_{2}$, we have to approximate the inner norm $\|(H_{\Phi ,\Omega}^{\beta}(f\chi _{j}))\chi _{k}\|_{L^{p_{2}(\cdot )}( \mathbb{R}^{n})}$. For this, we proceed as follows:

$$ \begin{aligned}[b] \bigl\vert H_{\Phi ,\Omega}^{\beta}(f\chi _{j}) (x) \bigr\vert &\le \int _{C_{j}} \biggl\vert \frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{\prime}\bigr) f(y) \biggr\vert \,dy \\ &\le \biggl\Vert \biggl(\frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{ \prime}\bigr) \biggr)\chi _{j} \biggr\Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

Since $p^{\prime}_{1}(\cdot )\in \textit{P}(\mathbb{R}^{n})$, we can fix s such that $s>p_{1}^{\prime}(\cdot )$. We define a new variable $p(\cdot )$ such that $\frac{1}{p_{1}^{\prime}(\cdot )}=\frac{1}{s}+\frac{1}{p(\cdot )}$. So by Lemma 2.5 we obtain

$$ \begin{aligned}[b] \bigl\vert H_{\Phi ,\Omega}^{\beta}(f \chi _{j}) (x) \bigr\vert &\le \biggl\Vert \biggl( \frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{\prime}\bigr) \biggr) \chi _{j} \biggr\Vert _{L^{s}(\mathbb{R}^{n})} \Vert \chi _{B_{j}} \Vert _{L^{p(\cdot )}( \mathbb{R}^{n})} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

(3.2)

By polar decomposition it is easy to see that

$$ \begin{aligned}[b] \biggl\Vert \biggl(\frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{ \prime}\bigr) \biggr)\chi _{j} \biggr\Vert ^{s}_{L^{s}(\mathbb{R}^{n})}&= \int _{C_{j}} \biggl\vert \frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{\prime}\bigr) \biggr\vert ^{s}\,dy \\ &= \int _{2^{j-1}}^{2^{j}} \int _{S^{n-1}} \biggl\vert \frac{\Phi ( \vert x \vert r^{-1})}{r^{n-\beta}} \biggr\vert ^{s} \bigl\vert \Omega \bigl(y^{\prime}\bigr) \bigr\vert ^{s}\,d\sigma \bigl(y^{\prime}\bigr)r^{n} \frac{dr}{r}, \end{aligned} $$

where $d\sigma (y^{\prime})$ denotes the normalized Lebesgue measure on the unit sphere $S^{n-1}$. A change of variables results in the following inequality:

$$ \begin{aligned}[b] \biggl\Vert \biggl(\frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{ \prime}\bigr) \biggr)\chi _{j} \biggr\Vert ^{s}_{L^{s}(\mathbb{R}^{n})} &= \int _{S^{n-1}} \bigl\vert \Omega \bigl(y^{\prime}\bigr) \bigr\vert ^{s}\,d\sigma \bigl(y^{\prime}\bigr) \int _{\frac{ \vert x \vert }{2^{j}}}^{ \frac{ \vert x \vert }{2^{j-1}}} \bigl\vert \Phi (t) \bigr\vert ^{s}\bigl( \vert x \vert t^{-1}\bigr)^{n-(n-\beta )s} \frac{dt}{t} \\ &\le \Vert \Omega \Vert ^{s}_{L^{s}(S^{n-1})} \vert x \vert ^{n-(n-\beta )s} \int _{0}^{ \infty} \bigl\vert \Phi (t) \bigr\vert ^{s}t^{(n-\beta )s-n}\frac{dt}{t} \\ &= A^{s}_{\Phi ,s} \Vert \Omega \Vert ^{s}_{L^{s}(S^{n-1})} \vert x \vert ^{s\beta +n-ns} .\end{aligned} $$

Therefore

$$ \begin{aligned}[b] \biggl\Vert \biggl( \frac{\Phi (x \vert y \vert ^{-1})}{ \vert y \vert ^{n-\beta}} \Omega \bigl(y^{ \prime}\bigr) \biggr)\chi _{j} \biggr\Vert _{L^{s}(\mathbb{R}^{n})} &\le CA_{\Phi ,s} \vert x \vert ^{ \beta -\frac{n}{s^{\prime}}} .\end{aligned} $$

(3.3)

Also, when $|B_{j}|\le 2^{n}$ and $y\in B_{j}$, by $\frac{1}{p_{1}^{\prime}(y)}=\frac{1}{s}+\frac{1}{p(y)}$ and Lemma 2.6 we have

$$ \begin{aligned}[b] \Vert \chi _{B_{j}} \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})}\thickapprox \vert B_{j} \vert ^{ \frac{1}{p(\cdot )}} \thickapprox \vert B_{j} \vert ^{-\frac{1}{s}} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})} .\end{aligned} $$

When $|B_{j}|\ge 1$, we have

$$ \begin{aligned}[b] \Vert \chi _{B_{j}} \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})}\thickapprox \vert B_{j} \vert ^{ \frac{1}{p(\infty )}} \thickapprox \vert B_{j} \vert ^{-\frac{1}{s}} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

Hence we obtain

$$ \begin{aligned}[b] \Vert \chi _{B_{j}} \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})}\thickapprox \vert B_{j} \vert ^{- \frac{1}{s}} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

(3.4)

Substituting (3.3) and (3.4) into (3.2), we get

$$ \begin{aligned}[b] \bigl\vert H_{\Phi ,\Omega}^{\beta}(f\chi _{j}) (x) \bigr\vert &\le C A_{\Phi ,s} \vert x \vert ^{ \beta -\frac{n}{s^{\prime}}} \vert B_{j} \vert ^{-\frac{1}{s}} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{ \prime}(\cdot )}(\mathbb{R}^{n})} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}( \mathbb{R}^{n})}. \end{aligned} $$

Multiplying both sides with $\chi _{k}(x)$ and taking the $L^{p_{2}(\cdot )}$ norm, the last inequality becomes

$$ \begin{aligned}[b] & \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f\chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \\ &\quad \le C A_{\Phi ,s} \vert B_{k} \vert ^{\frac{\beta}{n}-\frac{1}{s^{\prime}}} \vert B_{j} \vert ^{- \frac{1}{s}} \Vert \chi _{k} \Vert _{L^{p_{2}(\cdot )}(\mathbb{R}^{n})} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})} \Vert f\chi _{j} \Vert _{L^{p_{1}( \cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

(3.5)

Having estimating the inner norm, we are now in position to approximate $I_{1}$ and $I_{2}$. Let us first approximate $I_{1}$. For $j< k$, in view of Remark 2.11, from (3.5) we obtain the following inequality:

$$ \begin{aligned}[b] & \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f\chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \\ &\quad \le C A_{\Phi ,s} \vert B_{k} \vert ^{\frac{\beta}{n}-1} 2^{n(j-k)(\delta _{1}- \frac{1}{s})} \Vert \chi _{B_{k}} \Vert _{L^{p_{2}(\cdot )}(\mathbb{R}^{n})} \Vert \chi _{B_{k}} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

The conditions $0<\beta <\frac{n}{(p_{1})_{+}}$ and $\frac{1}{p_{2}(\cdot )}+\frac{\beta}{n}=\frac{1}{p_{1}(\cdot )}$ ensure the applicability of Lemma 2.8 to obtain

$$ \begin{aligned}[b] & \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f\chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \\ &\quad \le C A_{\Phi ,s} 2^{n(j-k)(\delta _{1}-\frac{1}{s})} \vert B_{k} \vert ^{-1} \Vert \chi _{B_{k}} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \Vert \chi _{B_{k}} \Vert _{L^{p_{1}^{ \prime}(\cdot )}(\mathbb{R}^{n})} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}( \mathbb{R}^{n})}. \end{aligned} $$

Finally, Lemma 2.4 helps us in establishing the following inequality:

$$ \begin{aligned}[b] \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f \chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})}&\le C A_{\Phi ,s} 2^{n(j-k)(\delta _{1}-\frac{1}{s})} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} .\end{aligned} $$

(3.6)

Hence, using (3.6), we get

$$ \begin{aligned}[b] I_{1} &\leq C A^{q_{1}}_{\Phi ,s} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{k=-\infty}^{k_{0}}2^{k\alpha q_{1}} \Biggl(\sum _{j=- \infty}^{k-1} 2^{n(j-k)(\delta _{1}-\frac{1}{s})} \Vert f\chi _{j} \Vert _{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \Biggr)^{q_{1}} \\ &\leq C A^{q_{1}}_{\Phi ,s} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \Biggl( \sum _{j=-\infty}^{k-1} 2^{(k-j)( \alpha -n\delta _{1}+\frac{n}{s})}2^{j\alpha } \Vert f\chi _{j} \Vert _{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \Biggr)^{q_{1}}.\end{aligned} $$

(3.7)

The condition $\alpha < n\delta _{1}-\frac{n}{(p_{1}^{\prime})^{+}}$ implies that $\alpha < n(\delta _{1}-\frac{1}{s})$. So for $1< q_{1}<\infty$, we use the Hölder inequality to obtain

$$\begin{aligned} I_{1} \leq& C A^{q_{1}}_{\Phi ,s}\sup _{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \\ &{}\times \sum_{k=-\infty}^{k_{0}} \sum _{j=-\infty}^{k-1} 2^{ \frac{{q_{1}}}{2}(k-j)(\alpha -n\delta _{1}+\frac{n}{s})} 2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \Biggl(\sum _{j=- \infty}^{k-1} 2^{\frac{{q_{1}^{\prime}}}{2}(k-j)(\alpha -n\delta _{1}+ \frac{n}{s})} \Biggr)^{\frac{q_{1}}{q_{1}^{\prime}}} \\ \le& C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=-\infty}^{k-1} 2^{ \frac{{q_{1}}}{2}(k-j)(\alpha -n\delta _{1}+\frac{n}{s})}2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ =&C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})}\sum _{k=j+1}^{k_{0}} 2^{\frac{nq_{1}}{2}(k-j)( \alpha -n\delta _{1}+\frac{n}{s})} \\ \leq& C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ \le& C A^{q_{1}}_{\Phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})}. \end{aligned}$$

Similarly, for $0< q_{1}\le 1$, from (3.7) we have

$$ \begin{aligned}[b] I_{1} &\leq C A^{q_{1}}_{\phi ,s} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{j=-\infty}^{k-1} 2^{{q_{1}}(k-j)( \alpha -n\delta _{1}+\frac{n}{s})}2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \\ &\quad =C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})}\sum _{k=j+1}^{k_{0}} 2^{{q_{1}}(k-j)(\alpha -n \delta _{1}+\frac{n}{s})} \\ &\leq C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &\le C A^{q_{1}}_{\phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})}.\end{aligned} $$

Next, we turn to estimate $I_{2}$. For $j\ge k$, we again use inequality (3.5) and Remark 2.11 to obtain

$$ \begin{aligned}[b] & \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f\chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \\ &\le C A_{\Phi ,s} \vert B_{k} \vert ^{\frac{\beta}{n}-\frac{1}{s^{\prime}}} \vert B_{j} \vert ^{- \frac{1}{s}} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{\prime}(\cdot )}(\mathbb{R}^{n})} \Vert \chi _{B_{j}} \Vert _{L^{p_{2}(\cdot )}(\mathbb{R}^{n})}2^{n(k-j)\delta _{2}} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})}. \end{aligned} $$

Making use of the Lemma 2.8 once again, we get

$$ \begin{aligned}[b] & \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f\chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})} \\ &\quad \le C A_{\Phi ,s} \vert B_{k} \vert ^{\frac{\beta}{n}-\frac{1}{s^{\prime}}} \vert B_{j} \vert ^{- \frac{1}{s}-\frac{\beta}{n}} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}^{\prime}( \cdot )}(\mathbb{R}^{n})} \Vert \chi _{B_{j}} \Vert _{L^{p_{1}(\cdot )}( \mathbb{R}^{n})}2^{n(k-j)\delta _{2}} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}( \mathbb{R}^{n})}, \end{aligned} $$

which in view of Lemma 2.4 results in

$$ \begin{aligned}[b] \bigl\Vert \bigl(H_{\Phi ,\Omega}^{\beta}(f \chi _{j})\bigr)\chi _{k} \bigr\Vert _{L^{p_{2}(\cdot )}( \mathbb{R}^{n})}&\le C A_{\Phi ,s} \vert B_{k} \vert ^{\frac{\beta}{n}- \frac{1}{s^{\prime}}} \vert B_{j} \vert ^{\frac{1}{s^{\prime}}-\frac{\beta}{n}}2^{n(k-j) \delta _{2}} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &=C A_{\Phi ,s}2^{(k-j)(\beta +n\delta _{2}-\frac{n}{s^{\prime}})} \Vert f \chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})}.\end{aligned} $$

Hence $I_{2}$ can be approximated as

$$ \begin{aligned}[b] I_{2} &\leq C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum _{k=-\infty}^{k_{0}} \Biggl(\sum _{j=k}^{\infty} 2^{(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})}2^{j\alpha} \Vert f\chi _{j} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \Biggr)^{q_{1}}.\end{aligned} $$

(3.8)

Similarly, the condition $0\le{\lambda}<\alpha +\beta +n\delta _{2}-\frac{n}{p_{1}^{-}}$ implies that $\lambda <\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}}$ and $\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}}>0$. So we can choose a constant $\epsilon > 1$ such that $\lambda -\frac{1}{\epsilon}(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})<0$. For $1 < q_{1} < \infty$, by the Hölder inequality we get

$$\begin{aligned} I_{2} \leq& C A^{q_{1}}_{\Phi ,s}\sup _{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \\ &{}\times \sum_{k=-\infty}^{k_{0}} \sum _{j=k}^{\infty} 2^{ \frac{{q_{1}}}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \Biggl(\sum _{j=k}^{\infty} 2^{{q_{1}^{\prime}} \frac{(\epsilon -1)}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})} \Biggr)^{\frac{q_{1}}{q_{1}^{\prime}}} \\ \le& C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=k}^{\infty} 2^{ \frac{{q_{1}}}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \\ \leq& C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=k}^{k_{0}-1} 2^{ \frac{{q_{1}}}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \\ &{}+ C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=k_{0}}^{\infty}2^{ \frac{{q_{1}}}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \\ =&M_{1}+M_{2}. \end{aligned}$$

In view of the same condition $\frac{n}{s^{\prime}}-n\delta _{2}-\beta <\alpha$, $M_{1}$ is approximated as

$$ \begin{aligned}[b] M_{1} &\leq C A^{q_{1}}_{\Phi ,s} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f\chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \sum_{k=-\infty}^{j}2^{ \frac{{q_{1}}}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})} \\ &\leq C A^{q_{1}}_{\Phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &\le C A^{q_{1}}_{\Phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})}.\end{aligned} $$

Since $\lambda <\frac{1}{\epsilon}(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})$ and $\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}}>0$, we get

$$\begin{aligned} M_{2} \leq& C A^{q_{1}}_{\Phi ,s}\sup _{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{j=k_{0}}^{\infty} 2^{ \frac{{q_{1}}}{\epsilon}(k-j)(\alpha +\beta +n\delta _{2}- \frac{n}{s^{\prime}})}2^{j\lambda q_{1}}2^{-j\lambda q_{1}} \\ &{}\times \Biggl( \sum_{l=-\infty}^{j}2^{l\alpha q_{1}} \Vert f\chi _{l} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})} \Biggr) \\ \le& C A^{q_{1}}_{\Phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}}2^{\frac{{q_{1}}}{\epsilon}k( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})} \sum_{j=k_{0}}^{ \infty}2^{{q_{1}}j (\lambda -\frac{1}{\epsilon}(\alpha +\beta +n \delta _{2}-\frac{n}{s^{\prime}}) )} \\ \le& C A^{q_{1}}_{\Phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}2^{\frac{{q_{1}}}{\epsilon}k_{0}(\alpha +\beta +n \delta _{2}-\frac{n}{s^{\prime}})} 2^{{q_{1}}k_{0} (\lambda - \frac{1}{\epsilon}(\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}}) )} \\ \le& C A^{q_{1}}_{\Phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})} . \end{aligned}$$

This completes the approximation of $I_{2}$ in the case $1< q_{1}<\infty$.

Finally, it remains to estimate $I_{2}$ for the case $0< q_{1}\le 1$. For this, from (3.8) we get

$$\begin{aligned} I_{2} \leq& C A^{q_{1}}_{\phi ,s}\sup _{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{j=k}^{\infty} 2^{{q_{1}}(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ \leq& C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=k}^{k_{0}-1} 2^{{q_{1}}(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &{}+ C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=k_{0}}^{\infty} 2^{{q_{1}}(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ =&L_{1}+L_{2}. \end{aligned}$$

Since $\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}}>0$, for $L_{1}$, we get

$$ \begin{aligned}[b] L_{1} &\leq C A^{q_{1}}_{\phi ,s} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{j=k}^{k_{0}-1} 2^{{q_{1}}(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &=C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}( \cdot )}(\mathbb{R}^{n})}\sum _{k=-\infty}^{j} 2^{{q_{1}}(k-j)(\alpha + \beta +n\delta _{2}-\frac{n}{s^{\prime}})} \\ &\leq C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}} \sum_{j=-\infty}^{k_{0}-1} 2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &\le C A^{q_{1}}_{\phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})}.\end{aligned} $$

Lastly, for the estimation of $L_{2}$, we proceed as follows:

$$ \begin{aligned}[b] L_{2} &\leq C A^{q_{1}}_{\phi ,s} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{j=k_{0}}^{\infty} 2^{{q_{1}}(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})}2^{j\alpha q_{1}} \Vert f \chi _{j} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \\ &\le C A^{q_{1}}_{\phi ,s}\sup_{k_{0}\in \mathbb{Z}}2^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{j=k_{0}}^{\infty} 2^{{q_{1}}(k-j)( \alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})} \Biggl(\sum _{l=- \infty}^{j}2^{l\alpha q_{1}} \Vert f\chi _{l} \Vert ^{q_{1}}_{L^{p_{1}(\cdot )}( \mathbb{R}^{n})} \Biggr) \\ &\le C A^{q_{1}}_{\phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}(\cdot )}^{ \alpha ,\lambda}(\mathbb{R}^{n})} \sup_{k_{0}\in \mathbb{Z}}2^{-k_{0} \lambda q_{1}}\sum_{k=-\infty}^{k_{0}}2^{{q_{1}}k(\alpha +\beta +n \delta _{2}-\frac{n}{s^{\prime}})} \sum_{j=k_{0}}^{\infty} 2^{{q_{1}}j( \lambda -\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}})} . \end{aligned} $$

Since $0\le \lambda <\alpha +\beta +n\delta _{2}-\frac{n}{s^{\prime}}$, we have

$$ \begin{aligned}[b] L_{2} &\le C A^{q_{1}}_{\phi ,s} \Vert f \Vert ^{q_{1}}_{M\dot{K}_{q_{1},p_{1}( \cdot )}^{\alpha ,\lambda}(\mathbb{R}^{n})} .\end{aligned} $$

Combining all the estimates we arrive at (3.1). Thus we have completed the proof of the theorem. □

Taking $\lambda =0$ in Theorem 3.1, we obtain the boundedness of rough fractional Hausdorff operator on variable Herz space.

Corollary 3.2

Let $0<\beta <\frac{n}{(p_{1})_{+}}$, $0< q_{1}\le q_{2}<\infty$, and $\Omega \in L^{s}(S^{n-1})$, and let $p_{1}(\cdot ), p_{2}(\cdot )\in \mathcal{P}(\mathbb{R}^{n})$ satisfy conditions (2.1) and (2.2) in Proposition 2.1with

$$ \frac{1}{p_{1}(\cdot )}=\frac{1}{p_{2}(\cdot )}+\frac{\beta}{n}, $$

where $p_{1}(\cdot )<\frac{n}{\beta}$. Suppose that for $0<\delta _{1}$, $\delta _{2}<1$, α satisfies $\frac{n}{p_{1}^{-}}-n\delta _{2}-\beta <\alpha <n\delta _{1}- \frac{n}{(p_{1}^{\prime})^{+}}$ If Φ is a radial function and $A_{\Phi ,s}<\infty$, then for $f\in \dot{K}^{\alpha ,q_{1}}_{p_{1}(\cdot )}(\mathbb{R}^{n})$, we have

$$ \begin{aligned}[b] & \bigl\Vert H_{\Phi ,\Omega}^{\beta}f \bigr\Vert _{\dot{K}^{\alpha ,q_{2}}_{p_{2}(\cdot )}( \mathbb{R}^{n})}\leq CA_{\Phi ,s} \Vert f \Vert _{\dot{K}^{\alpha ,q_{1}}_{p_{1}( \cdot )}(\mathbb{R}^{n})} .\end{aligned} $$

The following theorem establishes the boundedness of the rough Hausdorff operator on the Morrey–Herz space with variable exponent.

Theorem 3.3

Let $0< q_{1}\le q_{2}<\infty$ and $\Omega \in L^{s}(S^{n-1})$, and let $p(\cdot )\in \mathcal{P}(\mathbb{R}^{n})$ satisfy conditions (2.1) and (2.2) in Proposition 2.1. Suppose that for $0<\delta ^{*}_{1}$, $\delta ^{*}_{2}<1$, α satisfies $\frac{n}{p_{1}^{-}}-n\delta ^{*}_{2}<\alpha <n\delta ^{*}_{1}- \frac{n}{(p_{1}^{\prime})^{+}}$ and $0\le{\lambda}<\alpha +n\delta _{2}-\frac{n}{p_{1}^{-}}$. If Φ is a radial function and $A_{\Phi ,s}<\infty$, then for $f\in M\dot{K}^{\alpha ,\lambda}_{q_{1},p(\cdot )}(\mathbb{R}^{n})$, we have

$$ \begin{aligned}[b] & \Vert H_{\Phi ,\Omega}f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p(\cdot )}( \mathbb{R}^{n})}\leq CA_{\Phi ,s} \Vert f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{1},p( \cdot )}(\mathbb{R}^{n})} .\end{aligned} $$

Proof

The proof is similar to that of Theorem 3.1, so we omit the details. □

As a corollary of Theorem 3.3, we obtain the boundedness of the rough Hausdorff operator on the variable Herz space.

Corollary 3.4

Let $0< q_{1}\le q_{2}<\infty$ and $\Omega \in L^{s}(S^{n-1})$, and let $p(\cdot )\in \mathcal{P}(\mathbb{R}^{n})$ satisfy conditions (2.1) and (2.2) in Proposition 2.1. Suppose that for $0<\delta ^{*}_{1}$, $\delta ^{*}_{2}<1$, α satisfies $\frac{n}{p_{1}^{-}}-n\delta ^{*}_{2}<\alpha <n\delta ^{*}_{1}- \frac{n}{(p_{1}^{\prime})^{+}}$. If Φ is a radial function and $A_{\Phi ,s}<\infty$, then for $f\in \dot{K}^{\alpha ,q_{1}}_{p(\cdot )}(\mathbb{R}^{n})$, we have

$$ \begin{aligned}[b] & \Vert H_{\Phi ,\Omega}f \Vert _{\dot{K}^{\alpha ,q_{2}}_{p(\cdot )}(\mathbb{R}^{n})} \leq CA_{\Phi ,s} \Vert f \Vert _{\dot{K}^{\alpha ,q_{1}}_{p(\cdot )}(\mathbb{R}^{n})} .\end{aligned} $$

4 Conclusions

In this paper, we have shown that the rough fractional Hausdorff operator is bounded on Herz-type spaces. The scaling argument commonly employed to establish the boundedness of the Hausdorff operator on function spaces with constant exponents makes the study of Hausdorff operators on function spaces with variable exponents unsuitable. We overcome this problem by employing a new strategy. Furthermore, this strategy will be helpful in studying Hausdorf-type operators on other function spaces with variable exponents.