A note on the boundedness of sublinear operators on grand variable Herz spaces

Abstract

In this paper, we introduce grand variable Herz type spaces using discrete grand spaces and prove the boundedness of sublinear operators on these spaces.

1 Introduction

The Herz spaces $\dot{K}^{\alpha,p}_{q}(\mathbb{R}^{n})$ and $K^{\alpha ,p}_{q}(\mathbb{R}^{n})$ were introduced in [10] being known under the names of homogeneous and non-homogeneous Herz spaces and they are defined by the norms

$$\begin{aligned}& \Vert f \Vert _{\dot{K}^{\alpha,p}_{q}}:= \biggl\{ \sum_{k\in\mathbb{Z}}2^{k\alpha p} \biggl( \int_{R_{2^{k-1},2^{k}}} \bigl\vert f(x) \bigr\vert ^{q}\,dx \biggr)^{p/q} \biggr\} ^{1/p}, \end{aligned}$$

(1)

$$\begin{aligned}& \Vert f \Vert _{{K}^{\alpha,p}_{q}}:= \Vert f \Vert _{L^{q}(B(0.1))}+ \biggl\{ \sum_{k\in \mathbb{N}}2^{k\alpha p} \biggl( \int_{R_{2^{k-1},2^{k}}} \bigl\vert f(x) \bigr\vert ^{q}\, dx \biggr)^{p/q} \biggr\} ^{1/p}, \end{aligned}$$

(2)

respectively, where $R_{t,\tau}$ stands for the annulus $R_{t,\tau}:= B(0,\tau)\setminus B(0,t)$. These spaces were studied in many papers; see for instance [5, 7, 9, 12,13,14,15, 22] and the references therein.

Last two decades, under the influence of some applications revealed in [32], there was a vast boom of research in the so called variable exponent spaces (see e.g. [30]). For the time being, the theory of such variable exponent Lebesgue, Orlicz, Lorentz, and Sobolev function spaces is widely developed, we refer to the books [2,3,4, 20, 21]. Herz spaces with variables exponent have been recently introduced in [1, 12, 13]. Samko in [33] used variable exponent Herz spaces (with variable parameters), known as continual Herz spaces. Another approach regarding variable smoothness and integrability to study Herz type Hardy spaces was used in [29].

Grand Lebesgue spaces on bounded sets have been widely studied. They were introduced in [8, 11], cf. [2]. Grand spaces proved to be useful in application to partial differential equations. Various operators of harmonic analysis were intensively studied in the last years, cf. [6, 16,17,18,19,20, 27, 28] and the references therein.

Grand Lebesgue sequence spaces were introduced in [31], where various operators of harmonic analysis were studied in these spaces, e.g. maximal, convolutions, Hardy, Hilbert, and fractional operators, among others.

The aim of this paper is to introduce grand variable Herz spaces $\dot {K}^{\alpha,p),\theta}_{q}(\mathbb{R}^{n})$ and obtain the boundedness of sublinear operators on $\dot{K}^{\alpha,p),\theta}_{q}(\mathbb{R}^{n})$. The present article has three sections apart from the Introduction. Section 2 deals with some basic notions regarding grand Lebesgue sequence spaces. In Sect. 3 we give the definition of grand variable Herz spaces and prove Hölder’s inequality. In Sect. 4 we discuss boundedness of sublinear operators on grand variable Herz spaces. Throughout the paper, constants (often different constant in the same series of inequalities) will mainly be denoted by c or C. $f\lesssim g$ means that $f\le Cg$ and $f\approx g$ means that $f\lesssim g\lesssim f$.

2 Preliminaries

2.1 Lebesgue space with variable exponent

For the current section we refer to [4, 23] unless and until stated otherwise. Let $X\subseteq\mathbb{R}^{n}$ be an open set and $p(\cdot)$ be a real-valued measurable function on X with values in $[1,\infty)$. We suppose that

$$\begin{aligned} 1\le p_{-}(X)\le p_{+}(X)< \infty, \end{aligned}$$

(3)

where $p_{-}(X):= \operatorname{ess} \inf_{x\in X}p(x)$ and $p_{+}(X):= \operatorname{ess} \sup_{x\in X}p(x)$. By $L^{p(\cdot)}(X)$ we denote the space of measurable function f on X such that

$$\begin{aligned} I_{p(\cdot)}(f)= \int_{X} \bigl\vert f(x) \bigr\vert ^{p(x)}\,dx< \infty. \end{aligned}$$

It is a Banach space equipped with the norm (see e.g. [4]):

$$\begin{aligned} \Vert f \Vert _{L^{p(\cdot)}(X) }=\inf \biggl\{ \eta>0 : I_{p(\cdot)} \biggl(\frac {f}{\eta} \biggr)\le1 \biggr\} . \end{aligned}$$

By $p'(x)=p(x)/(p(x)-1)$, we denote the conjugate exponent of $p(\cdot )$. For the following lemma we refer to e.g. [3].

Lemma 2.1

(Generalized Hölder’s inequality)

LetXbe a measurable subset of $\mathbb{R}^{n}$. Suppose that $1\le p_{-}(X)\le p_{+}(X)<\infty$. Then

$$\begin{aligned} \Vert fg \Vert _{L^{r(\cdot)}(X)}\le c \Vert f \Vert _{L^{p(\cdot)}(X)} \Vert g \Vert _{L^{q(\cdot)}(X)} \end{aligned}$$

holds, where $f\in L^{p(\cdot)}(X)$, $g\in L^{q(\cdot)}(X)$and $\frac{1}{r(x)}=\frac{1}{p(x)}+\frac{1}{q(x)}$for every $x\in X$.

In the sequel we use the well known log-condition

$$\begin{aligned} \bigl\vert q(x)-q(y) \bigr\vert \le\frac{A}{-\ln \vert x-y \vert }, \quad \vert x-y \vert \le\frac{1}{2}, x,y \in X, \end{aligned}$$

(4)

where $A=A(q)>0$ does not depend on x, y, and the decay condition: there exists a number $q_{\infty}\in(1,\infty)$, such that

$$\begin{aligned} \bigl\vert q(x)-q_{\infty}\bigr\vert \le\frac{A}{\ln(e+ \vert x \vert )}, \end{aligned}$$

(5)

and also the decay condition

$$\begin{aligned} \bigl\vert q(x)-q_{0} \bigr\vert \le\frac{A}{\ln \vert x \vert }, \quad \vert x \vert \le\frac{1}{2}, \end{aligned}$$

(6)

holds for some $q_{0}\in(1,\infty)$ in the case of homogeneous Herz spaces.

With respect to classes of variable exponents used in this paper, we adopt the following notations:

• (i) Given a function $f\in L^{1}_{\text{loc}}(X)$, the Hardy–Littlewood maximal operator M is defined by

  $$\begin{aligned} Mf(x):=\sup_{r>0}r^{-n} \int_{B(x,r)} \bigl\vert f(y) \bigr\vert \,dy \quad(x\in X), \end{aligned}$$

  where

  $$\begin{aligned} B(x,r):= \bigl\{ y\in X : \vert x-y \vert < r \bigr\} . \end{aligned}$$

• (ii) $L_{\text{loc}}^{q(\cdot)}(X):= \{f: f\in L^{q(\cdot)}(K)\text{ for all compact subsets }K\subset X \}$.

• (iii) The set $\mathcal{P}(X)$ consists of all $q(\cdot)$ satisfying $q_{-}>1$ and $q_{+}<\infty$.

• (iv) $\mathcal{P}^{\log}=\mathcal{P}^{\log}(X)$ is the class of functions $q\in\mathcal{P}(X)$ satisfying the conditions (3) and (4).

• (v) In the case X is unbounded, $\mathcal{P}_{\infty}(X)$ and $\mathcal{P}_{0,\infty}(X)$ are subsets of exponents in $\mathcal {P}(X)$ with values in $[1,\infty)$ which satisfy condition (5) and the two conditions (5) and (6), respectively.

• (vi) $\mathcal{B}(X)$ is the set of $q(\cdot)\in\mathcal{P}(X)$ satisfying the condition that M is bounded on $L^{q(\cdot)}(X)$.

The following lemma appears in [33].

Lemma 2.2

Let $D>1$and $q\in\mathcal{P}_{0,\infty}(\mathbb{R}^{n})$. Then

$$\begin{aligned} \frac{1}{c_{0}}r^{\frac{n}{q(0)}}\le \Vert \chi_{R_{r,Dr}} \Vert _{q(\cdot)}\le c_{0}r^{\frac{n}{q(0)}}, \quad\textit{for }0< r\le1, \end{aligned}$$

(7)

and

$$\begin{aligned} \frac{1}{c_{\infty}}r^{\frac{n}{q_{\infty}}}\le \Vert \chi_{R_{r,Dr}} \Vert _{q(\cdot )}\le c_{\infty}r^{\frac{n}{q_{\infty}}}, \quad\textit{for }r\ge1, \end{aligned}$$

(8)

respectively, where $c_{0}\ge1$and $c_{\infty}\ge1$depend onD, but do not depend onr.

2.2 Herz spaces with variable exponent

In this subsection, we introduce variable exponent Herz spaces. In what follows, we denote $\chi_{k}=\chi_{{R}_{k}}$, $R_{k}=B_{k}\setminus B_{k-1}$ and $B_{k}=\{x\in\mathbb{R}^{n}: |x|\le 2^{k}\}$ for all $k\in\mathbb{Z}$.

Definition 2.1

Let $1< p<\infty$, $\alpha\in\mathbb{R}$ and $q(\cdot) \in\mathcal {P}(\mathbb{R}^{n})$. The homogeneous Herz space ${\dot{K}}_{q(\cdot )}^{\alpha,p} (\mathbb{R}^{n} )$ is defined by

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

where

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

Definition 2.2

Let $1< p<\infty$, $\alpha\in\mathbb{R}$ and $q(\cdot) \in\mathcal {P}(\mathbb{R}^{n})$. The non-homogeneous Herz space $K_{q(\cdot)}^{\alpha ,p} (\mathbb{R}^{n} )$ is defined by

$$K_{q(\cdot)}^{\alpha,p}\bigl(\mathbb{R}^{n}\bigr)= \bigl\{ f \in L_{\text{loc}}^{q(\cdot)}\bigl(\mathbb{R}^{n}\bigr): \Vert f \Vert _{{K}^{\alpha,p}_{q(\cdot)} (\mathbb{R}^{n} )}< \infty \bigr\} , $$

where

$$\Vert f \Vert _{{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}= \Biggl(\sum_{k=1}^{\infty } \bigl\Vert 2^{k\alpha}f\chi_{k} \bigr\Vert _{L^{q(\cdot)}}^{p} \Biggr)^{\frac{1}{p}}+ \Vert f \Vert _{L^{q(\cdot)}B(0,1)}. $$

2.3 Grand Lebesgue sequence space

In this subsection we introduce grand Lebesgue sequence space. For the following definitions and statements, see [31]. The letter $\mathbb {X}$ stands for one of the sets $\mathbb{Z}^{n}$, $\mathbb{Z}$, $\mathbb{N}$ and $\mathbb{N}_{0}$.

Definition 2.3

Let $1\le p<\infty$ and $\theta>0$. The grand Lebesgue sequence space $l^{p),\theta}$ is defined by the norm

$$ \begin{aligned} \bigl\Vert \{x_{k}\}_{k\in\mathbb {X}} \bigr\Vert _{l^{p),\theta}(\mathbb{X})} &= \Vert \mathbf {x} \Vert _{l^{p),\theta}(\mathbb{X})} \\ &:= \sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb{X}} \vert x_{k} \vert ^{p(1+\varepsilon )} \biggr)^{\frac{1}{p(1+\varepsilon)}} =\sup_{\varepsilon>0}\varepsilon ^{\frac{\theta}{p(1+\varepsilon)}} \Vert x \Vert _{{l}^{p(1+\varepsilon)}(\mathbb{X})}, \end{aligned} $$

where $\mathbf{x}=\{x_{k}\}_{k\in\mathbb {X}}$.

Note that the following nesting properties hold:

$$\begin{aligned} l^{p(1-\varepsilon)}\hookrightarrow l^{p}\hookrightarrow l^{p),\theta _{1}} \hookrightarrow l^{p),\theta_{2}}\hookrightarrow l^{p(1+\delta)} \end{aligned}$$

(9)

for $0<\varepsilon<\frac{1}{p}$, $\delta>0$ and $0<\theta_{1}\le\theta_{2}$.

3 Grand variable Herz space

In this section, we introduce grand variable Herz space in a natural way using the discrete space from Definition 2.3.

Definition 3.1

Let $\alpha\in\mathbb{R}$, $1\le p<\infty$, $q:\mathbb{R}^{n}\rightarrow [1,\infty)$, $\theta>0$. We define the homogeneous grand variable Herz space by

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

where

$$ \begin{aligned} \Vert f \Vert _{\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})} & =\sup _{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Vert f\chi_{k} \Vert ^{p(1+\varepsilon )}_{L^{q(\cdot)}} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=\sup_{\varepsilon>0}\varepsilon^{\frac{1}{p(1+\varepsilon)}} \Vert f \Vert _{\dot{K}^{\alpha,p(1+\varepsilon)}_{q(\cdot)}(\mathbb{R}^{n})}. \end{aligned} $$

In a similar way, non-homogeneous grand variable Herz spaces can be introduced.

In the following theorem, we prove that Herz space is contained in grand variable Herz space.

Theorem 3.1

For $p>1$, we have $\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})\subset \dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})$, $\theta>0$.

Proof

Let $f\in\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$. Then

$$\begin{aligned} \Vert f \Vert _{\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})} &=\sup_{\varepsilon>0} \biggl({ \varepsilon}^{\theta}\sum_{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Vert f\chi_{k} \Vert ^{p(1+\varepsilon )}_{L^{q(\cdot)}} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &= \bigl\Vert 2^{k\alpha} \Vert f\chi_{k} \Vert _{L^{q(\cdot)}} \bigr\Vert _{l^{p),\theta }} \\ &\le C \bigl\Vert 2^{k\alpha} \Vert f\chi_{k} \Vert _{L^{q(\cdot)}} \bigr\Vert _{l^{p}}= C \biggl(\sum _{k\in\mathbb{Z}}2^{k\alpha p} \Vert f\chi_{k} \Vert ^{p}_{L^{q(\cdot )}} \biggr)^{\frac{1}{p}} \\ &=C \Vert f \Vert _{\dot{K}^{\alpha,p}_{q(\cdot )}(\mathbb{R}^{n})}, \end{aligned}$$

where we used (9). □

Now we prove the Hölder inequality for a grand variable Herz space.

Theorem 3.2

If $0< p_{i}\le\infty$, $1\le q_{-}\le q_{+}<\infty$, $-\infty<\alpha _{i}<\infty$, $i=1,2$, $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$, $1=\frac {1}{q(\cdot)}+\frac{1}{q'(\cdot)}$and $\alpha=\alpha_{1}+\alpha_{2}$. Then

$$\Vert fg \Vert _{\dot{K}^{\alpha,p),\theta}_{1}(\mathbb{R}^{n})}\le \Vert f \Vert _{\dot {K}^{\alpha_{1},p_{1}),\theta}_{q(\cdot)}(\mathbb{R}^{n})} \Vert g \Vert _{\dot {K}^{\alpha_{2},p_{2}),\theta}_{q'(\cdot)}(\mathbb{R}^{n})}. $$

Proof

We have

$$\begin{aligned} \Vert fg \Vert _{\dot{K}^{\alpha,p),\theta}_{1}(\mathbb{R}^{n})}&=\sup_{\varepsilon >0} \biggl({ \varepsilon}^{\theta}\sum_{k\in\mathbb{Z}}2^{k\alpha p(1+\varepsilon)} \Vert fg\chi_{k} \Vert ^{p(1+\varepsilon)}_{L^{1}(\mathbb {R}^{n})} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \biggl( \int_{2^{k}}^{2^{k+1}} \vert fg \vert \biggr)^{p(1+\varepsilon)} \biggr)^{\frac{1}{p(1+\varepsilon)}}. \end{aligned}$$

By using Hölder’s inequality

$$\begin{aligned} &\le C\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k(\alpha_{1}+\alpha_{2}) p(1+\varepsilon)} \bigl( \Vert f \chi_{k} \Vert _{L^{q(\cdot)}}^{p(1+\varepsilon)} \bigr) \bigl( \Vert g\chi_{k} \Vert _{L^{q'(\cdot)}}^{p(1+\varepsilon)} \bigr) \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=C\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}} \bigl(2^{k\alpha_{1}} \Vert f \chi_{k} \Vert _{L^{q(\cdot)}} \bigr)^{p(1+\varepsilon)} \bigl(2^{k\alpha_{2}} \Vert g\chi_{k} \Vert _{L^{q'(\cdot)}} \bigr)^{p(1+\varepsilon)} \biggr)^{\frac{1}{p(1+\varepsilon)}}, \end{aligned}$$

by generalized Hölder’s inequality

$$\begin{aligned} &\le C\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta} \biggl( \sum_{k\in \mathbb{Z}} \bigl(2^{k\alpha_{1}} \Vert f \chi_{k} \Vert _{L^{q(\cdot)}} \bigr)^{p_{1}(1+\varepsilon)} \biggr)^{1/p_{1}(1+\varepsilon)} \\ &\quad{} \times\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta} \sum_{k\in\mathbb{Z}} \bigl(2^{k\alpha_{2}} \bigl( \Vert g \chi_{k} \Vert _{L^{q'(\cdot )}} \bigr) \bigr)^{p_{2}(1+\varepsilon)} \biggr)^{1/p_{2}(1+\varepsilon )} \biggr) \\ &= C\Vert f \Vert _{\dot{K}^{\alpha_{1},p_{1}),\theta}_{q(\cdot)}(\mathbb{R}^{n})} \Vert g \Vert _{\dot{K}^{\alpha_{2},p_{2}),\theta}_{q'(\cdot)}(\mathbb{R}^{n})}. \end{aligned}$$

 □

4 Boundedness of sublinear operators

In this section, we show that sublinear operators are bounded on $\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})$. Hernandez, Li, Lu and Yang [9, 24, 26] have proved that if a sublinear operator T is bounded on $L^{p}(\mathbb{R}^{n})$ and satisfies the size condition

$$ \bigl\vert Tf(x) \bigr\vert \le C \int_{\mathbb{R}^{n}} \vert x-y \vert ^{-n} \bigl\vert f(y) \bigr\vert \,dy,\quad x\notin\operatorname{spt }f $$

(10)

for all $f\in L^{1}(\mathbb{R}^{n})$ with compact support then T is bounded on the homogeneous Herz space ${\dot{K}}_{q}^{\alpha,p}$ and on the non-homogeneous Herz space ${K}_{q}^{\alpha,p}$. The condition (11) is satisfied by many interesting operators in harmonic analysis, such as Calderón–Zygmund operators, Carleson’s maximal operator, the Hardy–Littlewood maximal operator, Fefferman’s singular multipliers, Fefferman’s singular integrals, Ricci–Stein’s oscillatory singular integrals, and the Bochner–Riesz means (for details see [25, 34]).

Theorem 4.1

Let $1< p<\infty$, $q(\cdot)\in\mathcal{P}_{0,\infty}(\mathbb{R}^{n})$such that ${-n}/{q(0)}<\alpha<{n}/{q'(0)}$and ${-n}/{q_{\infty}}<\alpha <{n}/{q'_{\infty}}$. Suppose thatTis a sublinear operator and bounded on $L^{q(\cdot)}(\mathbb{R}^{n})$satisfying the size condition (10). ThenTis bounded on $\dot{K}^{\alpha ,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})$.

Proof

Since T is sublinear, we have for every $f\in\dot{K}^{\alpha ,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})$

$$\begin{aligned}& \Vert Tf \Vert _{\dot{K}^{\alpha, p),\theta}_{q(\cdot)}(\mathbb{R}^{n})} \\& \quad=\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Vert \chi_{k}Tf \Vert ^{p(1+\varepsilon )}_{L^{q(\cdot)}(\mathbb{R}^{n})} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\& \quad\le\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=-\infty}^{\infty} \bigl\Vert \chi _{k}T(f \chi_{l}) \bigr\Vert ^{p(1+\varepsilon)}_{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr) \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\& \quad\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=-\infty}^{k-2} \bigl\Vert \chi _{k}T(f \chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\& \qquad{}+ c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=k-1}^{k+1} \bigl\Vert \chi_{k}T(f\chi _{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\& \qquad{}+ c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=k+2}^{\infty} \bigl\Vert \chi _{k}T(f \chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\& \quad =: E_{1}+E_{2}+E_{3}. \end{aligned}$$

For $E_{2}$, using the $L^{q(\cdot)}(\mathbb{R}^{n})$ boundedness of T we obtain

$$\begin{aligned} E_{2}&\le c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\sum_{k\in \mathbb{Z}}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum_{l=k-1}^{k+1} \bigl\Vert T(f \chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=k-1}^{k+1} \Vert f\chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac {1}{p(1+\varepsilon)}} \\ &\le c\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{k\in\mathbb {Z}}2^{k\alpha p(1+\varepsilon)} \Vert f\chi_{k} \Vert ^{p(1+\varepsilon )}_{L^{q}(\cdot)(\mathbb{R}^{n})} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=c \Vert f \Vert _{\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}. \end{aligned}$$

For $E_{1}$, we use the facts that, for each $k\in\mathbb{Z}$ and $l\le k-2$ and a.e. $x\in R_{k}$, size condition (10) and Hölder’s inequality imply

$$ \begin{aligned} [b]\bigl\vert T(f\chi_{l}) (x) \bigr\vert &\le C \int_{R_{l}} \vert x-y \vert ^{-n} \bigl\vert f(y) \bigr\vert \,dy \\ &\le c2^{-kn} \int_{R_{l}} \bigl\vert f(y) \bigr\vert \,dy\le c2^{-kn} \Vert f\chi_{l} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})}. \end{aligned} $$

(11)

Moreover, splitting $E_{1}$ by means of Minkowski’s inequality we have

$$\begin{aligned} E_{1}&\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum_{k=-\infty}^{-1}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=-\infty }^{k-2} \bigl\Vert \chi_{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\quad{}+c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty }2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=-\infty}^{k-2} \bigl\Vert \chi _{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &:= E_{11}+E_{12}. \end{aligned}$$

For $E_{11}$ by Lemma 2.2 we have

$$\begin{aligned} 2^{-kn} \Vert \chi_{k} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})}&\le c2^{-kn}2^{ (\frac {kn}{q(0)} )}2^{ (\frac{ln}{q'(0)} )} \le c2^{\frac {(l-k)n}{q'(0)}}. \end{aligned}$$

(12)

Applying (11) and (12) to $E_{11}$, we get

$$\begin{aligned}[b] E_{11} &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty}^{-1}2^{k\alpha p(1+\varepsilon)} \\ & \quad{}\times \Biggl(\sum_{l=-\infty}^{k-2} \Vert \chi_{k} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}2^{-kn} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac {1}{p(1+\varepsilon)}} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1} \Biggl(\sum _{l=-\infty}^{k-2}2^{\alpha l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}2^{b(l-k)} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}},\end{aligned} $$

(13)

where $b:=\frac{n}{q'(0)}-\alpha>0$. Then we use Hölder’s inequality, Fubini’s theorem for series and $2^{-p(1+\varepsilon)}< 2^{-p}$ to obtain

$$\begin{aligned} &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1} \Biggl(\sum _{l=-\infty}^{k-2}2^{\alpha p(1+\varepsilon) l} \Vert f\chi _{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}^{p(1+\varepsilon )}2^{bp(1+\varepsilon)(l-k)/2} \Biggr) \\ &\quad\times \Biggl(\sum_{l=-\infty}^{k-2}2^{b(p(1+\varepsilon ))'(l-k)/2} \Biggr)^{p(1+\varepsilon)/(p(1+\varepsilon))'} \Biggr)^{\frac {1}{p(1+\varepsilon)}} \\ &=c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1}\sum _{l=-\infty}^{k-2}2^{\alpha p(1+\varepsilon) l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}^{p(1+\varepsilon)}2^{bp(1+\varepsilon )(l-k)/2} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{l=-\infty }^{-1}2^{\alpha p(1+\varepsilon) l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}^{p(1+\varepsilon)}\sum _{k=l+2}^{-1} 2^{bp(1+\varepsilon )(l-k)/2} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &< c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{l=-\infty }^{-1}2^{\alpha p(1+\varepsilon) l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}^{p(1+\varepsilon)}\sum _{k=l+2}^{-1} 2^{bp(l-k)/2} \Biggr)^{\frac {1}{p(1+\varepsilon)}} \\ &\leq c\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{l\in\mathbb {Z}}2^{\alpha p(1+\varepsilon) l} \Vert f\chi_{l} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}^{p(1+\varepsilon)} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\le c \Vert f \Vert _{\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}. \end{aligned}$$

Now for $E_{12}$ using Minkowski’s inequality we have

$$\begin{aligned} E_{12}&\le c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\sum_{k=0}^{\infty}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum_{l=-\infty }^{-1} \bigl\Vert \chi_{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\quad+c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty }2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=0}^{k-2} \bigl\Vert \chi_{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac {1}{p(1+\varepsilon)}} \\ &:=A_{1}+A_{2}. \end{aligned}$$

The estimate for $A_{2}$ follows in a similar manner to $E_{11}$ with $q'(0)$ replaced by $q'_{\infty}$ and using the fact that $\frac {n}{q'_{\infty}}-\alpha>0$. For $A_{1}$ using Lemma 2.2 we have

$$\begin{aligned} 2^{-kn} \Vert \chi_{k} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})}&\le c2^{-kn}2^{ (\frac{kn}{q_{\infty}} )}2^{ (\frac{ln}{q' (0)} )} \le c2^{ (\frac {-kn}{q'_{\infty}} )}2^{ (\frac{ln}{q^{\prime}(0)} )}. \end{aligned}$$

(14)

Now using (11) and (14) and the fact that $\alpha-\frac{n}{q'_{\infty }}<0$ we have

$$\begin{aligned} A_{1}&\le\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum_{k=0}^{\infty}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=-\infty }^{-1} \bigl\Vert \chi_{k}Tf(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty}2^{k\alpha p(1+\varepsilon)} \\ & \quad \times \Biggl(\sum_{l=-\infty}^{-1}2^{-kn}2^{(kn/q_{\infty})}2^{(ln/q'(0))} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty}2^{k\alpha p(1+\varepsilon)} \\ & \quad\times \Biggl(\sum_{l=-\infty}^{-1}2^{(-kn/q'_{\infty})}2^{(ln/q'(0))} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty}2^{(k\alpha-kn/q'_{\infty}) p(1+\varepsilon)} \\ & \quad\times \Biggl(\sum_{l=-\infty}^{-1}2^{(ln/q'(0))} \bigl\Vert f(\chi _{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\Biggl( \sum_{l=-\infty}^{-1}2^{(ln/q'(0))} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\Biggl( \sum_{l=-\infty}^{-1}2^{l\alpha} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}2^{(ln/q'(0)-l\alpha)} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)}. \end{aligned}$$

Now applying Hölder’s inequality and using the fact that $\frac {n}{q'(0)}-\alpha>0$ we have

$$\begin{aligned} A_{1}&\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\Biggl(\sum_{l=-\infty}^{-1}2^{l\alpha p(1+\varepsilon)} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} ^{p(1+ \varepsilon)} \Biggr) \\ & \quad \times \Biggl(\sum_{l=-\infty}^{-1}2^{l(n/q'(0)-\alpha) (p(1+\varepsilon))' } \Biggr)^{p(1+\varepsilon)/(p(1+\varepsilon ))'} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\biggl( \sum_{l\in \mathbb{Z}}2^{l\alpha p(1+\varepsilon)} \bigl\Vert f( \chi_{l}) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}^{p(1+ \varepsilon)} \biggr) \biggr)^{1/p(1+\varepsilon )} \\ &\le c \Vert f \Vert _{\dot{K}^{\alpha, p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}. \end{aligned}$$

Next, we estimate $E_{3}$. For each $k\in\mathbb{Z}$ and $l\ge k+2$ and a.e. $x\in R_{k}$; the size condition (11) and Hölder’s inequality imply

$$ \begin{aligned} [b]\bigl\vert T(f\chi_{l}) (x) \bigr\vert &\le C \int_{R_{l}} \vert x-y \vert ^{-n} \bigl\vert f(y) \bigr\vert \,dy \\ &\le c2^{-ln} \int_{R_{l}} \bigl\vert f(y) \bigr\vert \,dy\le c2^{-ln} \Vert f\chi_{l} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})}. \end{aligned} $$

(15)

Similar to $E_{1}$, splitting $E_{3}$ by means of Minkowski’s inequality we have

$$\begin{aligned} E_{3}&\le c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\sum_{k=-\infty}^{-1}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum_{l=k+2}^{\infty } \bigl\Vert \chi_{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\quad+c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty }2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=k+2}^{\infty} \bigl\Vert \chi_{k}T(f\chi _{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &:= E_{31}+E_{32}. \end{aligned}$$

For $E_{32}$ Lemma 2.2 yields

$$\begin{aligned} 2^{-ln} \Vert \chi_{k} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})}&\le c2^{-ln}2^{ (\frac{kn}{q_{\infty}} )}2^{ (\frac{ln}{q'_{\infty}} )} \le c2^{\frac {(k-l)n}{q_{\infty}}}. \end{aligned}$$

(16)

Using (15) and (16) for $E_{32}$, we get

$$\begin{aligned}[b] E_{32}&\le c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\sum_{k=0}^{\infty}2^{k\alpha p(1+\varepsilon)} \\ &\quad \times \Biggl(\sum_{l=k+2}^{\infty} \Vert \chi_{k} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}2^{-ln} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty} \Biggl(\sum _{l=k+2}^{\infty}2^{\alpha l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}2^{d(k-l)} \Biggr)^{p(1+\varepsilon)} \Biggr)^{\frac{1}{p(1+\varepsilon)}},\end{aligned}$$

where $d:=\frac{n}{q_{\infty}}+\alpha>0$. Then we use Hölder’s inequality, Fubini’s theorem for series and $2^{-p(1+\varepsilon)}< 2^{-p}$ to obtain

$$\begin{aligned} &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty} \Biggl(\sum _{l=k+2}^{\infty}2^{\alpha p(1+\varepsilon) l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}^{p(1+\varepsilon )}2^{dp(1+\varepsilon)(k-l)/2} \Biggr) \\ &\quad\times \Biggl(\sum_{l=k+2}^{\infty}2^{d(p(1+\varepsilon ))'(k-l)/2} \Biggr)^{p(1+\varepsilon)/(p(1+\varepsilon))'} \Biggr)^{\frac {1}{p(1+\varepsilon)}} \\ &=c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=0}^{\infty }\sum _{l=k+2}^{\infty}2^{\alpha p(1+\varepsilon) l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}^{p(1+\varepsilon)}2^{dp(1+\varepsilon )(k-l)/2} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{l=0}^{\infty }2^{\alpha p(1+\varepsilon) l} \Vert f \chi_{l} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}^{p(1+\varepsilon)}\sum _{k=0}^{l-2} 2^{dp(1+\varepsilon )(k-l)/2} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &< c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\sum_{l\in\mathbb {Z}}2^{\alpha p(1+\varepsilon) l} \Vert f\chi_{l} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}^{p(1+\varepsilon)}\sum _{k=-\infty}^{l-2} 2^{dp(k-l)/2} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &=c\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{l\in\mathbb {Z}}2^{\alpha p(1+\varepsilon) l} \Vert f\chi_{l} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}^{p(1+\varepsilon)} \biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\le c \Vert f \Vert _{\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}. \end{aligned}$$

Now for $E_{31}$ using Minkowski’s inequality we have

$$\begin{aligned} E_{31}&\le c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\sum_{k=-\infty}^{-1}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum_{l=k+2}^{-1} \bigl\Vert \chi_{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &\quad+c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=0}^{\infty} \bigl\Vert \chi _{k}T(f\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{\frac{1}{p(1+\varepsilon)}} \\ &:=B_{1}+B_{2}. \end{aligned}$$

The estimate for $B_{1}$ follows in a similar manner to $E_{32}$ with $q_{\infty}$ replaced by $q(0)$ and using the fact that $\frac {n}{q(0)}+\alpha>0$. For $B_{2}$ using Lemma 2.2 we have

$$\begin{aligned} 2^{-ln} \Vert \chi_{k} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{l} \Vert _{L^{q'(\cdot)}(\mathbb{R}^{n})}&\le c2^{-ln}2^{ (\frac {kn}{q(0)} )}2^{ (\frac{ln}{q'_{\infty}} )} \le c2^{ (\frac{kn}{q(0)} )}2^{ (\frac{-ln}{q_{\infty}} )}. \end{aligned}$$

(17)

Now using (15) and (17) and the fact that $\alpha+\frac{n}{q(0)}>0$ we have

$$\begin{aligned} B_{2}&\le\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum_{k=-\infty }^{-1}2^{k\alpha p(1+\varepsilon)} \Biggl(\sum _{l=0}^{\infty} \bigl\Vert \chi _{k}Tf(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon )} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1}2^{k\alpha p(1+\varepsilon)} \\ & \quad \times \Biggl(\sum_{l=0}^{\infty }2^{-ln}2^{(kn/q(0))}2^{(ln/q'_{\infty}))} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon )} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1}2^{k\alpha p(1+\varepsilon)} \\ & \quad\times \Biggl(\sum_{l=0}^{\infty }2^{(kn/q(0))}2^{-(ln/q_{\infty})} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\sum _{k=-\infty }^{-1}2^{k(\alpha+n/q(0)) p(1+\varepsilon)} \\ & \quad \times \Biggl(\sum_{l=0}^{\infty}2^{-(ln/q_{\infty})} \bigl\Vert f(\chi _{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\Biggl( \sum_{l=0}^{\infty}2^{-(ln/q_{\infty})} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)} \\ &\le c\sup_{\varepsilon>0} \Biggl({\varepsilon}^{\theta}\Biggl( \sum_{l=0}^{\infty}2^{l\alpha} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}2^{-l(n/q_{\infty}+\alpha)} \Biggr)^{p(1+\varepsilon)} \Biggr)^{1/p(1+\varepsilon)}. \end{aligned}$$

Now applying Hölder’s inequality and using the fact that $\frac {n}{q_{\infty}}+\alpha>0$ we have

$$\begin{aligned} B_{2}&\le c\sup_{\varepsilon>0} \Biggl({ \varepsilon}^{\theta}\Biggl(\sum_{l=0}^{\infty}2^{l\alpha p(1+\varepsilon)} \bigl\Vert f(\chi_{l}) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}^{p(1+\varepsilon)} \Biggr) \\ & \quad\times \Biggl(\sum_{l=0}^{\infty}2^{-l(n/q_{\infty}+\alpha )(p(1+\varepsilon))'} \Biggr)^{p(1+\varepsilon)/(p(1+\varepsilon))'} \Biggr)^{1/p(1+\varepsilon )} \\ &\le c\sup_{\varepsilon>0} \biggl({\varepsilon}^{\theta}\sum _{l\in \mathbb{Z}}2^{l\alpha p(1+\varepsilon)} \bigl\Vert f( \chi_{l}) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})}^{p(1+\varepsilon)} \biggr)^{1/p(1+\varepsilon)} \\ &\le c \Vert f \Vert _{\dot{K}^{\alpha, p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}. \end{aligned}$$

Combining the estimates for $E_{1}$, $E_{2}$ and $E_{3}$ yields

$$\begin{aligned} \Vert Tf \Vert _{\dot{K}^{\alpha, p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}\le c \Vert f \Vert _{\dot{K}^{\alpha,p),\theta}_{q(\cdot)}(\mathbb{R}^{n})}, \end{aligned}$$

which ends the proof. □