Parametrized Littlewood–Paley Operators on Herz-Type Hardy Spaces with Variable Exponent

Abstract

In this paper, the authors establish the boundedness of parametrized Littlewood–Paley area integral and $g^*_\lambda$-function, and their high-order commutators on Herz-type Hardy spaces with variable exponent. These results are also new even for classical Herz-type Hardy space.

1 Introduction

Suppose that $S^{n-1}$ is the unit sphere in the n-dimensional Euclidean space ${{{{\mathbb {R}}}}^n}\ (n\ge 2)$. Let $\Omega$ be a homogeneous function of degree zero on ${{{{\mathbb {R}}}}^n}$ which is locally integrable and satisfies the cancellation condition

$$\begin{aligned} \int _{S^{n-1}}\Omega (x')\,{\mathrm{d}}\sigma (x')=0, \end{aligned}$$

(1.1)

where ${\mathrm{d}}\sigma$ is the Lebesgue measure and $x'=x/{|x|}$ for any $x\ne {\mathbf {0}}$. For a function f on ${{{{\mathbb {R}}}}^n}$, the parametrized Littlewood–Paley area integral $\mu ^\rho _{\Omega ,\, S}$ and $g^*_\lambda$-function $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }$ are, respectively, defined by setting, for any $x\in {{{{\mathbb {R}}}}^n}$,

$$\begin{aligned} \mu ^\rho _{\Omega ,\, S}(f)(x):=\left( \iint _{\Gamma (x)}\left| \frac{1}{t^p}\int _{|y-z|< t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}f(z)\,{{\mathrm{d}}z}\right| ^2\, \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+1}}\right) ^{1/2} \end{aligned}$$

and

$$\begin{aligned}&\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(f)(x)\\&\quad :=\left[ \iint _{{\mathbb {R}}^{n+1}_+}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \frac{1}{t^p}\int _{|y-z|< t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}f(z)\,{{\mathrm{d}}z}\right| ^2 \,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+1}}\right] ^{1/2}, \end{aligned}$$

where $\rho \in (0,\,\infty )$, $\lambda \in (1,\,\infty )$, $\Gamma (x):=\{(y,\,t)\in {{\mathbb {R}}^{n+1}_+}:\,|x-y|<t\}$ and ${{\mathbb {R}}^{n+1}_+}:={{\mathbb {R}}^{n}}\times (0,\,\infty )$. For an integer $m\ge 1$, let b be a locally integrable function on ${\mathbb {R}}^n$, the commutators $[b^m,\,\mu ^\rho _{\Omega ,\, S}]$ and $[b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }]$ are, respectively, defined by setting, for any $x\in {{{{\mathbb {R}}}}^n}$,

$$\begin{aligned}&{[}b^m,\,\mu ^\rho _{\Omega ,\, S}](f)(x)\\&\quad :=\left( \iint _{\Gamma (x)}\left| \frac{1}{t^p}\int _{|y-z|< t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}[b(x)-b(z)]^m f(z)\,{{\mathrm{d}}z}\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+1}}\right) ^{1/2} \end{aligned}$$

and

$$\begin{aligned}&{[}b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)(x)\\&\quad :=\left[ \iint _{{\mathbb {R}}^{n+1}_+}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n} \left| \frac{1}{t^p}\int _{|y-z|< t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}[b(x)-b(z)]^m f(z)\,{{\mathrm{d}}z}\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+1}}\right] ^{1/2}, \end{aligned}$$

where $\rho \in (0,\,\infty )$ and $\lambda \in (1,\,\infty )$. It is well known that the Littlewood–Paley function is a very important tool in harmonic analysis and PDE (see [6, 12]). Therefore, as a more general variant, the parametrized Littlewood–Paley area integral $\mu ^\rho _{\Omega ,\, S}$ and $g^*_\lambda$-function $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }$ were first studied by Sakamoto and Yabuta [11] in 1999. They showed that if $\Omega \in \mathrm{Lip}_\alpha (S^{n-1})$ with $\alpha \in (0,\,1]$, then $\mu ^\rho _{\Omega ,\, S}$ and $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }$ are bounded on $L^p({{\mathbb {R}}^{n}})$ with $p\in (1,\,\infty )$. As for their boundedness on Herz spaces with variable exponent, in 2017, Wang et al. [17] further showed that, if $\Omega \in L^2(S^{n-1})$, then the boundedness of the parametrized Littlewood–Paley operators and their high-order commutators generated, respectively, by $\mathrm{BMO}$ functions and Lipschitz functions are established.

On the other hand, in 2012, Wang et al. [14] introduced the Herz-type Hardy spaces with variable exponent and gave their atomic decomposition characterizations. Meanwhile, Wang et al. also proved the boundedness of singular integrals and Marcinkiewicz integral on Herz-type Hardy spaces with variable exponent in [13, 18], respectively. More conclusions of the Herz-type Hardy spaces with variable exponent are referred to [15, 16, 19, 21].

In light of Wang [13, 14, 17, 18], it is a natural and interesting problem to ask whether the parametrized Littlewood–Paley operators and their high-order commutators are bounded on Herz-type Hardy spaces with variable exponent. In this paper, we shall answer this problem affirmatively.

Precisely, this paper is organized as follows.

In Sect. 2, we first recall the notion concerning the Herz-type Hardy space with variable exponent and then present the boundedness of parametrized Littlewood–Paley operators and their high-order commutators on Herz-type Hardy spaces with variable exponent (Theorems 2.7, 2.8, 2.9, 2.10, 2.11 and 2.12), the proofs of which are given in Sect. 4. These results are also new even for Herz-type Hardy space.

Section 3 is devoted to show the $L^{q(\cdot )}$ norm estimates of $\mu ^{\rho }_{\Omega ,\, S}(a)\chi _k$ and $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(a)\chi _k$, where a is a multiple of a $(\alpha ,\,q(\cdot ))$-atom, i.e. Lemmas 3.5 and 3.6, which play key roles in the proofs of main results of this paper. In the process of the proofs of Lemmas 3.5 and 3.6, it is worth pointing out that, in order to obtain the $L^{q(\cdot )}$ norm estimates of $\mu ^{\rho }_{\Omega ,\, S}(a)\chi _k$ and $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(a)\chi _k$, we cannot directly use the method of Tao et al. [20, Theorems 1 and 3] since their method is based on $(\alpha ,\,\infty )$-atom only. To overcome this obstacle, we employ a different method, namely we first establish the subtle pointwise estimates for $\mu ^{\rho }_{\Omega ,\, S}(a)$ and $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(a)$ with the help of some ideas from the proof of [2, Theorem 1], and then apply the generalized Hölder’s inequality for variable exponent (Lemma 3.3), and we immediately induce the desired the $L^{q(\cdot )}$ norm estimates of $\mu ^{\rho }_{\Omega ,\, S}(a)\chi _k$ and $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(a)\chi _k$.

Finally, we make some conventions on notation. Let ${{\mathbb {Z}}}_+:=\{1,\, 2,\,\ldots \}$ and ${{\mathbb {N}}}:=\{0\}\cup {{\mathbb {Z}}}_+$. For any $\beta :=(\beta _1,\ldots ,\beta _n)\in {{\mathbb {N}}}^n$, let $|\beta |:=\beta _1+\cdots +\beta _n$. Throughout this paper, the letter C will denote a positive constant that may vary from line to line but will remain independent of the main variables. For any sets $E \subset {{{{\mathbb {R}}}}^n}$, we use $E^\complement$ to denote the set ${{{{\mathbb {R}}}}^n}{\setminus } E$, |E| its n-dimensional Lebesgue measure, $\chi _E$ its characteristic function. For any $s\in {{\mathbb {R}}}$, $\lfloor s\rfloor$ denotes the unique integer such that $s-1<\lfloor s\rfloor \le s$. If there are no special instructions, any space ${\mathcal {X}}({{{{\mathbb {R}}}}^n})$ is denoted simply by ${\mathcal {X}}$. For instance, $L^2({{{{\mathbb {R}}}}^n})$ is simply denoted by $L^2$. For any index $q\in [1,\,\infty ]$, $q'$ denotes the conjugate index of q, namely $1/q+1/{q'}=1$.

2 Notions and Main Results

In this section, we first recall the notion concerning the Herz-type Hardy space with variable exponent and then present the boundedness of parametrized Littlewood–Paley operators and their high-order commutators on Herz-type Hardy spaces with variable exponent.

Given a measurable function $p(\cdot ):{\mathbb {R}}^n\rightarrow [1,\,\infty )$, $L^{p(\cdot )}$ denotes the set of measurable functions f on ${\mathbb {R}}^n$ such that for some $\lambda >0$,

$$\begin{aligned} \int _{{\mathbb {R}}^n}\left( \frac{|f(x)|}{\lambda }\right) ^{p(x)}{\mathrm{d}}x<\infty . \end{aligned}$$

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

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}}:=\inf \left\{ \lambda \in (0,\,\infty ):\ \int _{{\mathbb {R}}^n}\left( \frac{|f(x)|}{\lambda }\right) ^{p(x)}{\mathrm{d}}x\le 1\right\} . \end{aligned}$$

These spaces are referred to as variable $L^p$ spaces, since they generalized the standard $L^p$ spaces: if $p(x)=p$ is constant, then $L^{p(\cdot )}$ is isometrically isomorphic to $L^{p}$.

Let E be a subset of ${{{{\mathbb {R}}}}^n}$ and $L^{p(\cdot )}(E)$ is a Banach space with the norm defined by

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}(E)}:=\inf \left\{ \lambda \in (0,\,\infty ):\ \int _{E}\left( \frac{|f(x)|}{\lambda }\right) ^{p(x)}{\mathrm{d}}x\le 1\right\} . \end{aligned}$$

The space $L^{p(\cdot )}_{{\mathrm{loc}}}({{{{\mathbb {R}}}}^n})$ is defined by setting

$$\begin{aligned} L^{p(\cdot )}_{{\mathrm{loc}}}({{{{\mathbb {R}}}}^n}):=\{f: f\in L^{p(\cdot )}(E) \mathrm { \ for \ all \ compact\ subsets\ { E\subset {\mathbb {R}}^n}}\}. \end{aligned}$$

The space $L^{p(\cdot )}_{{\mathrm{loc}}}({{{{\mathbb {R}}}}^n}\backslash \{0\})$ is defined by setting

$$\begin{aligned} L^{p(\cdot )}_{{\mathrm{loc}}}({{{{\mathbb {R}}}}^n}\backslash \{0\}):=\{f: f\in L^{p(\cdot )}(E) \mathrm { \ for \ all \ compact\ subsets\ { E\subset ({\mathbb {R}}^n}}\backslash \{0\})\}. \end{aligned}$$

Define ${\mathcal {P}}$ to be set of $p(\cdot ):{\mathbb {R}}^n\rightarrow [1,\,\infty )$ such that

$$\begin{aligned} p^- :={\mathrm{ess}}\,{\mathrm{inf}} \{p(x):x\in {\mathbb {R}}^n\}>1,\,\,\,\,\,\, p^+ :={\mathrm{ess}}\,{\mathrm{sup}} \{p(x):x\in {\mathbb {R}}^n\}<\infty . \end{aligned}$$

Denote $p'(x):=p(x)/(p(x)-1)$ if $p(x)>1$ and $p'(x)=+\infty$ if $p(x)=1$.

For $f\in L^1_{{\mathrm{loc}}}$, the Hardy–Littlewood maximal operator Mf of f is defined by setting, for any $x\in {\mathbb {R}}^n$,

$$\begin{aligned} Mf(x):=\displaystyle \sup _{r>0}\frac{1}{|B_r (x)|}\int _{B_r (x)} |f(y)|\,{\mathrm{d}}y, \end{aligned}$$

where $B_r (x):=\{y\in {\mathbb {R}}^n:|x-y|<r\}$. Let ${\mathcal {B}}$ be the set of $p(\cdot )\in {\mathcal {P}}$ such that the Hardy–Littlewood maximal operator M is bounded on $L^{p(\cdot )}$.

Proposition 2.1

([4, Remark 2.8]) Let $q(\cdot )\in {\mathcal {B}}$. Then there exists a positive constant C such that for all balls B in ${\mathbb {R}}^n$ and all measurable subsets $S\subset B$,

$$\begin{aligned} \frac{\Vert \chi _B\Vert _{L^{q(\cdot )}}}{\Vert \chi _S\Vert _{L^{q(\cdot )}}}\le C\frac{|B|}{|S|},\,\,\,\, \frac{\Vert \chi _S\Vert _{L^{q(\cdot )}}}{\Vert \chi _B\Vert _{L^{q(\cdot )}}}\le C\left( \frac{|S|}{|B|}\right) ^{\delta _1},\,\,\,\, \frac{\Vert \chi _S\Vert _{L^{q'(\cdot )}}}{\Vert \chi _B\Vert _{L^{q'(\cdot )}}}\le C\left( \frac{|S|}{|B|}\right) ^{\delta _2}, \end{aligned}$$

where $\delta _1$, $\delta _2$ are constants with $0<\delta _1,\,\delta _2<1$.

Throughout this paper, $\delta _2$ is the same as in Proposition 2.1.

Next we recall the definitions of the Herz spaces with variable exponent. Let $B_k:=\{x\in {\mathbb {R}}^n:|x|\le 2^k\}$ and $A_k:=B_k {\setminus } {B_{k-1}}$ for $k\in {\mathbb {Z}}$. Denote $\chi _k:=\chi _{A_k}$ for $k\in {\mathbb {Z}}$, $\widetilde{\chi }_k:=\chi _{k}$ if $k\in {\mathbb {Z}}_+$ and $\widetilde{\chi }_0:=\chi _{B_0}$.

Definition 2.2

[4, Definition 3.2] Let $\alpha \in {\mathbb {R}}$, $0<p\le \infty$ and $q(\cdot )\in {\mathcal {P}}$. The homogeneous Herz space with variable exponent ${\dot{K}}^{\alpha ,\,p}_{q(\cdot )}$ is defined by setting

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

where

$$\begin{aligned} \Vert f\Vert _{{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}}:= \left( \sum _{k=-\infty }^{\infty }2^{k\alpha p}\Vert f\chi _k\Vert ^p_{L^{q(\cdot )}}\right) ^{1/p}. \end{aligned}$$

The non-homogeneous Herz space with variable exponent $K^{\alpha ,\,p}_{q(\cdot )}$ is defined by setting

$$\begin{aligned} K^{\alpha ,\,p}_{q(\cdot )}:=\left\{ f\in L^{q(\cdot )}_{{\mathrm{loc}}}({\mathbb {R}}^n ):\Vert f\Vert _{K^{\alpha ,\,p}_{q(\cdot )}}<\infty \right\} , \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{K^{\alpha ,\,p}_{q(\cdot )}}:=\left( \sum _{k=0}^{\infty }2^{k\alpha p}\Vert f\widetilde{\chi }_k\Vert ^p_{L^{q(\cdot )}}\right) ^{1/p}. \end{aligned}$$

Remark 2.3

• (i) When $\alpha =0$, $p=1$, for any $q(\cdot )\in {\mathcal {P}}$, we have ${{\dot{K}}^{0,\,1}_{q(\cdot )}}={K^{0,\,1}_{q(\cdot )}}={L^{q(\cdot )}}$.

• (ii) When $\alpha =0$ and $q(x)=p\in (0,\,\infty )$, we have ${\dot{K}}^{0,\,p}_{p}=K^{0,\,p}_{p}=L^{p}$ and ${\dot{K}}^{\alpha /p,\,p}_{p}=L^{p}_{|x|^\alpha }$, where $L^{p}_{|x|^\alpha }$ is a Banach space with the norm defined by

  $$\begin{aligned} \Vert f\Vert _{L^{p}_{|x|^\alpha }}:= \left( \int _{{\mathbb {R}}^n}|f(x)|^p|x|^\alpha \,{\mathrm{d}}x\right) ^{1/p}<\infty . \end{aligned}$$

  The latter is not true for the non-homogeneous Herz spaces.

• (iii) When $q(x) = q\in (0,\,\infty )$, we have ${\dot{K}}^{\alpha ,\,p}_{q(\cdot )}={\dot{K}}^{\alpha ,\,p}_{q}$ and $K^{\alpha ,\,p}_{q(\cdot )}=K^{\alpha ,\,p}_{q}$.

In what follows, we denote by ${{\mathcal {S}}}$ the set of all Schwartz functions and by ${{\mathcal {S}}}'$ its dual space (namely, the set of all tempered distributions). Let $G_N (f)(x)$ be the grand maximal function of $f\in {{\mathcal {S}}}'$ defined by setting, for all $x\in {\mathbb {R}}^n$,

$$\begin{aligned} G_N (f)(x):=\sup _{\phi \in {\mathcal {A}}_N}\left| \phi ^*_\bigtriangledown (f)(x)\right| , \end{aligned}$$

where ${\mathcal {A}}_N:=\{\phi \in {{\mathcal {S}}}:\sup _{|\alpha |,\,|\beta |\le N}|x^\alpha D^\beta \phi (x)|\le 1\}$ and $N>n+1$, $\phi ^*_\bigtriangledown$ is the non-tangential maximal operator of $f\in {{\mathcal {S}}}'$ defined by setting, for all $x\in {\mathbb {R}}^n$,

$$\begin{aligned} \phi ^*_\bigtriangledown (f)(x):=\sup _{|y-x|<t,\,t\in (0,\,\infty )} \left| \phi _t*f(y)\right| . \end{aligned}$$

Here and hereafter, $\phi _t(x):=t^{-n}\phi (x/t)$.

Definition 2.4

[14, Definition 2.1] Let $\alpha \in {\mathbb {R}}$, $0<p<\infty$, $q(\cdot )\in {\mathcal {P}}$ and $N>n+1$.

• (i) The homogeneous Herz-type Hardy space with variable exponent $H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}$ is defined by setting

  $$\begin{aligned} H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}:=\left\{ f\in {{\mathcal {S}}}': G_N(f)\in {\dot{K}}^{\alpha ,\,p}_{q(\cdot )}\right\} \end{aligned}$$

  and $\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}}:=\Vert G_N(f)\Vert _{{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}}$.

• (ii) The non-homogeneous Herz-type Hardy space with variable exponent $HK^{\alpha ,\,p}_{q(\cdot )}$ is defined by setting

  $$\begin{aligned} HK^{\alpha ,\,p}_{q(\cdot )}:=\left\{ f\in {{\mathcal {S}}}': G_N(f)\in K^{\alpha ,\,p}_{q(\cdot )}\right\} \end{aligned}$$

  and $\Vert f\Vert _{HK^{\alpha ,\,p}_{q(\cdot )}}:=\Vert G_N(f)\Vert _{K^{\alpha ,\,p}_{q(\cdot )}}$.

Remark 2.5

• (i) When $\alpha =0$, $p=1$, for any $q(\cdot )\in {\mathcal {P}}$, we have ${H{\dot{K}}^{0,\,1}_{q(\cdot )}}={HK^{0,\,1}_{q(\cdot )}}={H^{q(\cdot )}}$.

• (ii) When $\alpha =0$ and $q(x) = q\in (0,\,\infty )$, we have $H{\dot{K}}^{0,\,p}_{p}=HK^{0,\,p}_{p}=H^{p}$.

• (iii) When $q(x) = q\in (0,\,\infty )$, we have $H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}=H{\dot{K}}^{\alpha ,\,p}_{q}$ and $HK^{\alpha ,\,p}_{q(\cdot )}=HK^{\alpha ,\,p}_{q}$. Particularly, when $q\in (1,\,\infty )$, $\alpha \in (-n/q,\,n(1-1/q))$ and $p\in (0,\,\infty )$, we have $H{\dot{K}}^{\alpha ,\,p}_{q}={\dot{K}}^{\alpha ,\,p}_{q}$ and $HK^{\alpha ,\,p}_{q}=K^{\alpha ,\,p}_{q}$; while when $q\in (1,\,\infty )$, $\alpha \in [n(1-1/q),\,\infty )$ and $p\in (0,\,\infty )$, these are not true anymore [9, 10]. Thus, on the Herz-type Hardy spaces, the interesting cases are the cases $\alpha \in [n(1-1/q),\,\infty )$.

Definition 2.6

[14, Definition 2.2] Let $n\delta _2\le \alpha <\infty$, $q(\cdot )\in {\mathcal {B}}$, $b\in L^1_{{\mathrm{loc}}}$ and nonnegative integer $s\ge \lfloor {\alpha -n\delta _2}\rfloor$.

• (i) A function a(x) on ${\mathbb {R}}^n$ is said to be a central $(\alpha ,\,q(\cdot ))$-atom, if it satisfies

  • (1) supp $a\subset B(0,\,r)=\left\{ x\in {\mathbb {R}}^n:\,|x|<r\right\}$.

  • (2) $\Vert a\Vert _{L^{q(\cdot )}}\le |B(0,\,r)|^{-\alpha /n}$.

  • (3) $\int _{{{{\mathbb {R}}}}^n}a(x)x^\beta {\mathrm{d}}x=\int _{{{{\mathbb {R}}}}^n}a(x)b(x)x^\beta {\mathrm{d}}x=0$ for any $\beta \in {{\mathbb {N}}}^n$ with $|\beta |\le s$.

• (ii) A function a(x) on ${\mathbb {R}}^n$ is said to be a central $(\alpha ,\,q(\cdot ))$-atom of restricted type, if it satisfies the conditions $(2),\,(3)$ above and $\quad (1)'$ supp $a\subset B(0,\,r)$, $r\ge 1$.

If $r=2^k$ for some $k\in {\mathbb {Z}}$ in Definition 2.6, then the corresponding central $(\alpha ,\,q(\cdot ))$-atom is called a dyadic central $(\alpha ,\,q(\cdot ))$-atom.

Throughout the paper, we always assume that $\Omega$ is homogeneous of degree zero and satisfies (1.1).

Recall that, for any $\beta \in (0,\,1]$, a function $\Omega \in L^2(S^{n-1})$ is said to satisfy the $L^{2,\,\beta }$-Dini condition if

$$\begin{aligned} \int _0^1\frac{\omega _2(\delta )}{\delta ^{1+\beta }}\,{\mathrm{d}}\delta <\infty , \end{aligned}$$

where ${\omega _2(\delta )}$ is the integral modulus of continuity of order 2 of $\Omega$ defined by setting, for any $\delta \in (0,\,1]$,

$$\begin{aligned} {\omega _2(\delta )}:=\sup _{\Vert {\gamma }\Vert <\delta } \left( \int _{S^{n-1}}|\Omega ({\gamma }x')-\Omega (x')|^2\,{\mathrm{d}}\sigma (x')\right) ^{1/2} \end{aligned}$$

and ${\gamma }$ denotes a rotation on $S^{n-1}$ with $\Vert {\gamma }\Vert :=\sup _{y'\in S^{n-1}}|{\gamma }y'-y'|$.

The space $\mathrm{BMO}$ consists of all locally integrable functions f such that

$$\begin{aligned} \Vert f\Vert _*:= \sup _{Q}\frac{1}{|Q|}\int _{Q}|f(x)-f_Q|\,{\mathrm{d}}x<\infty , \end{aligned}$$

where $f_Q=|Q|^{-1}\int _{Q}f(y)\,{\mathrm{d}}y$, the supremum is taken over all cubes $Q\subset {\mathbb {R}}^n$ with sides parallel to the coordinate axes and |Q| denotes the Lebesgue measure of Q.

For $0<\varepsilon \le 1$, the Lipschitz space ${\mathrm{Lip}}_{\varepsilon }$ is defined as

$$\begin{aligned} {{\mathrm{Lip}}_{\varepsilon }} :=\left\{ f:\, \Vert f\Vert _{{\mathrm{Lip}}_{\varepsilon }}=\sup _{x,\,y\in {\mathbb {R}}^n;\,x\ne y}\frac{|f(x)-f(y)|}{|x-y|^\varepsilon }<\infty \right\} . \end{aligned}$$

The main results of this paper are as follows, the proofs of which are given in Sect. 4.

Theorem 2.7

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\delta _2\in (0,\,1)$, $q(\cdot )\in {\mathcal {B}}$ and $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2\})$. Suppose $0<p_1\le p_2<\infty$ and $n\delta _2\le \alpha <n\delta _2+\xi$ (resp. $0<\max \{n\delta _2,\,\alpha _2\}\le \alpha _1<n\delta _2+\xi$ ). If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition, then there exists a positive constant C independent of f such that

$$\begin{aligned} \left\| \mu ^\rho _{\Omega ,\,S}(f)\right\| _{{\dot{K}}^{\alpha ,\,p_2}_{q(\cdot )}} \le C\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}}\,\,\,\,\, \left( {\mathrm{resp.}} \left\| \mu ^\rho _{\Omega ,\,S}(f)\right\| _{K^{\alpha _2,\,p_2}_{q(\cdot )}} \le C\Vert f\Vert _{HK^{\alpha _1,\,p_1}_{q(\cdot )}}\right) . \end{aligned}$$

Theorem 2.8

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\lambda \in (2,\,\infty )$, $\delta _2\in (0,\,1)$, $q(\cdot )\in {\mathcal {B}}$ and $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2,\,(\lambda -2)n/3\})$. Suppose $0<p_1\le p_2<\infty$ and $n\delta _2\le \alpha <n\delta _2+\xi$ (resp. $0<\max \{n\delta _2,\,\alpha _2\}\le \alpha _1<n\delta _2+\xi$). If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition, then there exists a positive constant C independent of f such that

$$\begin{aligned} \left\| \mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(f)\right\| _{{\dot{K}}^{\alpha ,\,p_2}_{q(\cdot )}} \le C\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}}\,\,\,\,\, \left( {\mathrm{resp.}} \left\| \mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(f)\right\| _{K^{\alpha _2,\,p_2}_{q(\cdot )}} \le C\Vert f\Vert _{HK^{\alpha _1,\,p_1}_{q(\cdot )}}\right) . \end{aligned}$$

Theorem 2.9

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\delta _2\in (0,\,1)$, $q(\cdot )\in {\mathcal {B}}$ and $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2\})$. Suppose $0<p_1\le p_2<\infty$ and $n\delta _2\le \alpha <n\delta _2+\xi$ (resp. $0<\max \{n\delta _2,\,\alpha _2\}\le \alpha _1<n\delta _2+\xi$). If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition, $b\in \mathrm{BMO}$ and $m\in {{\mathbb {Z}}}_+$, then there exists a positive constant C independent of f such that

$$\begin{aligned}&\left\| [b^m,\,\mu ^\rho _{\Omega ,\, S}](f)\right\| _{{\dot{K}}^{\alpha ,\,p_2}_{q(\cdot )}} \le C\Vert b\Vert ^{mp_1}_*\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}} \\&\left( {\mathrm{resp.}} \left\| [b^m,\,\mu ^\rho _{\Omega ,\, S}](f)\right\| _{K^{\alpha _2,\,p_2}_{q(\cdot )}} \le C\Vert b\Vert ^{mp_1}_*\Vert f\Vert _{HK^{\alpha _1,\,p_1}_{q(\cdot )}}\right) . \end{aligned}$$

Theorem 2.10

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\lambda \in (2,\,\infty )$, $\delta _2\in (0,\,1)$, $q(\cdot )\in {\mathcal {B}}$ and $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2,\,(\lambda -2)n/3\})$. Suppose $0<p_1\le p_2<\infty$ and $n\delta _2\le \alpha <n\delta _2+\xi$ (resp. $0<\max \{n\delta _2,\,\alpha _2\}\le \alpha _1<n\delta _2+\xi$). If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition, $b\in \mathrm{BMO}$ and $m\in {{\mathbb {Z}}}_+$, then there exists a positive constant C independent of f such that

$$\begin{aligned}&\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\right\| _{{\dot{K}}^{\alpha ,\,p_2}_{q(\cdot )}} \le C\Vert b\Vert ^{mp_1}_*\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}} \\&\left( {\mathrm{resp.}} \left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\right\| _{K^{\alpha _2,\,p_2}_{q(\cdot )}} \le C\Vert b\Vert ^{mp_1}_*\Vert f\Vert _{HK^{\alpha _1,\,p_1}_{q(\cdot )}}\right) . \end{aligned}$$

Theorem 2.11

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $m\in {{\mathbb {Z}}}_+$, $0<\varepsilon < \min \{1,\,n/m\}$, $\delta _2\in (0,\,1)$, $q_1(\cdot ),\,q_2(\cdot )\in {\mathcal {B}}$ be such that $q^+_1<\frac{n}{m\varepsilon }$, and $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2\})$. Suppose $0<p_1\le p_2<\infty$, $q_2(\cdot )(n-m\varepsilon )/n\in {\mathcal {B}}$, $1/q_1(x)-1/q_2(x)=m\varepsilon /n$ and $n\delta _2\le \alpha <n\delta _2+\xi$ (resp. $0<\max \{n\delta _2,\,\alpha _2\}\le \alpha _1<n\delta _2+\xi$). If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition and $b\in {\mathrm{Lip}}_{\varepsilon }$, then there exists a positive constant C independent of f such that

$$\begin{aligned}&\left\| [b^m,\,\mu ^\rho _{\Omega ,\, S}](f)\right\| _{{\dot{K}}^{\alpha ,\,p_2}_{q_2(\cdot )}} \le C\Vert b\Vert ^{mp_1}_{\mathrm{Lip_\varepsilon }}\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q_1(\cdot )}} \\&\left( {\mathrm{resp.}} \left\| [b^m,\,\mu ^\rho _{\Omega ,\, S}](f)\right\| _{K^{\alpha _2,\,p_2}_{q_2(\cdot )}} \le C\Vert b\Vert ^{mp_1}_{\mathrm{Lip_\varepsilon }}\Vert f\Vert _{HK^{\alpha _1,\,p_1}_{q_1(\cdot )}}\right) . \end{aligned}$$

Theorem 2.12

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\lambda \in (2,\,\infty )$, $m\in {{\mathbb {Z}}}_+$, $0<\varepsilon < \min \{1,\,n/m\}$, $\delta _2\in (0,\,1)$, $q_1(\cdot ),\,q_2(\cdot )\in {\mathcal {B}}$ be such that $q^+_1<\frac{n}{m\varepsilon }$, and $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2,\,(\lambda -2)n/3\})$. Suppose $0<p_1\le p_2<\infty$, $q_2(\cdot )(n-m\varepsilon )/n\in {\mathcal {B}}$, $1/q_1(x)-1/q_2(x)=m\varepsilon /n$ and $n\delta _2\le \alpha <n\delta _2+\xi$ (resp. $0<\max \{n\delta _2,\,\alpha _2\}\le \alpha _1<n\delta _2+\xi$). If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition and $b\in {\mathrm{Lip}}_{\varepsilon }$, then there exists a positive constant C independent of f such that

$$\begin{aligned}&\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\right\| _{{\dot{K}}^{\alpha ,\,p_2}_{q_2(\cdot )}} \le C\Vert b\Vert ^{mp_1}_{\mathrm{Lip_\varepsilon }}\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q_1(\cdot )}} \\&\left( {\mathrm{resp.}} \left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\right\| _{K^{\alpha _2,\,p_2}_{q_2(\cdot )}} \le C\Vert b\Vert ^{mp_1}_{\mathrm{Lip_\varepsilon }}\Vert f\Vert _{HK^{\alpha _1,\,p_1}_{q_1(\cdot )}}\right) . \end{aligned}$$

Remark 2.13

When $q(x) = q\in (0,\,\infty )$, we have $H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}=H{\dot{K}}^{\alpha ,\,p}_{q}$ $({\mathrm{resp.}} \,HK^{\alpha ,\,p}_{q(\cdot )}=HK^{\alpha ,\,p}_{q})$. In this case, Theorems 2.7–2.12 are also new. On the other hand, when $q(\cdot )\in {\mathcal {P}}$, $\alpha \in (-n\delta _1,\,n\delta _2)$ and $p\in (0,\,\infty )$, we have $H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}={\dot{K}}^{\alpha ,\,p}_{q(\cdot )}$ $({\mathrm{resp.}}\,HK^{\alpha ,\,p}_{q(\cdot )}= K^{\alpha ,\,p}_{q(\cdot )})$, Theorems 2.7, 2.8, 2.9, 2.10, 2.11 and 2.12 are reduced to [17, Theorem 2.2, Theorem 2.1, Theorem 3.2, Theorem 3.1 Theorem 4.2 and Theorem 4.1], respectively.

3 Proofs of Main Lemmas

In this section, we first recall the well-known lemma and then present the $L^{q(\cdot )}$ norm estimates of $\mu ^{\rho }_{\Omega ,\, S}(a_j)\chi _k$ and $\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }(a_j)\chi _k$.

Lemma 3.1

([14, Theorem 2.1] Let $n\delta _2\le \alpha <\infty$, $0<p<\infty$ and $q(\cdot )\in {\mathcal {B}}$. Then $f\in {H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}}$ (or ${H{K}^{\alpha ,\,p}_{q(\cdot )}}$ ) if and only if

$$\begin{aligned} f=\sum ^\infty _{k=-\infty } \lambda _ka_k \ \left( \mathrm {or}\sum ^\infty _{k=0} \lambda _ka_k\right) \ \ {\mathrm{in~the~sense~of~}} \ {{\mathcal {S}}}', \end{aligned}$$

where each $a_k$ is a central $(\alpha ,\,q(\cdot ))$-atom (or central $(\alpha ,\,q(\cdot ))$-atom of restricted type) with support contained in $B_k$ and $\sum ^\infty _{k=-\infty } |\lambda _k|^p<\infty$ ($\mathrm {or} \sum ^\infty _{k=0} |\lambda _k|^p<\infty$). Moreover,

$$\begin{aligned} \Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p}_{q(\cdot )}}\approx \inf \left( \sum ^\infty _{k=-\infty } |\lambda _k|^p\right) ^{1/p}\,\,\, \left( \mathrm {or}\,\,\,\Vert f\Vert _{H{K}^{\alpha ,\,p}_{q(\cdot )}}\approx \inf \left( \sum ^\infty _{k=0} |\lambda _k|^p\right) ^{1/p}\right) , \end{aligned}$$

where the infimum is taken over all above decompositions of f.

Lemma 3.2

([2, Lemma 2.1] or [8, Lemma 4.11]) Let $\rho \in (0,\,\infty )$. Suppose that $\Omega \in L^2(S^{n-1})$ satisfies the $L^2$-Dini condition. Then, there exists a positive constant C such that, for any $y\in B(\mathbf {0},\,{R/2})$ with $R\in (0,\,\infty )$,

$$\begin{aligned} \left( \int _{R\le |x|<2R}\left| \frac{\Omega (x-y)}{|x-y|^{n-\rho }} -\frac{\Omega (x)}{|x|^{n-\rho }}\right| ^2 {\mathrm{d}}x\right) ^{1/2} \le CR^{\rho -n/2}\left( {\frac{|y|}{R}}+ \int _{|y|/2R}^{|y|/R}\frac{\omega _2(\delta )}{\delta }{\mathrm{d}}\delta \right) . \end{aligned}$$

Lemma 3.3

([7, Theorem 2.1]) (Generalized Hölder’s inequality) Let $p(\cdot ),\, p_1(\cdot ), p_2(\cdot )\in {\mathcal {P}}$. For any $f\in L^{p(\cdot )}$, $g\in L^{p'(\cdot )}$, then fg is integrable on ${\mathbb {R}}^n$ and

$$\begin{aligned} \int _{{\mathbb {R}}^n}|f(x)g(x)|\,{\mathrm{d}}x\le C_p\Vert f\Vert _{L^{p(\cdot )}}\Vert g\Vert _{L^{p'(\cdot )}}, \end{aligned}$$

where $C_p=1+1/p^{-}-1/p^{+}$.

Lemma 3.4

([4, Lemma 2.9]) Suppose $q(\cdot )\in {\mathcal {B}}$. Then, there exists a positive constant C such that for all balls B in ${\mathbb {R}}^n$,

$$\begin{aligned} \frac{1}{|B|}{\Vert \chi _B\Vert _{L^{q(\cdot )}}}{\Vert \chi _B\Vert _{L^{q'(\cdot )}}}\le C. \end{aligned}$$

Lemma 3.5 shows that $\mu ^{\rho }_{\Omega ,\, S}$ maps all multiple of an atoms into uniformly bounded elements of $L^{q(\cdot )}$.

Lemma 3.5

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\delta _2\in (0,\,1)$, $n\delta _2\le \alpha <\infty$, $q(\cdot )\in {\mathcal {B}}$, $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2\})$, $j,\,k\in {{\mathbb {Z}}}_+$ and $j\le k-3$. Suppose $a_j$ is a dyadic $(\alpha ,\,q(\cdot ))$-atom with the support $B_j:=B(0,\,2^j)$. If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition, then there exists a positive constant C independent of $a_j$ such that for any $x\in A_k$,

$$\begin{aligned} \left\| \mu ^{\rho }_{\Omega ,\, S}(a_j)\chi _k\right\| _{L^{q(\cdot )}}\le C \, 2^{(j-k)(n\delta _2+\xi )} \left\| a_j\right\| _{L^{q(\cdot )}} \end{aligned}$$

Proof

We show this lemma by borrowing some ideas from the proof of [2, Theorem 1]. $2B_j:=2B_{j+1}$. The trick of the proof is to find a subtle segmentation. For any $x\in A_k$, write

$$\begin{aligned} \mu ^\rho _{\Omega ,\,S}(a_j)(x)&=\left( \int ^{\infty }_{0}\int _{|y-x|<t}\left| \int _{|y-z|<t} \frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z) \,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{1/2}\\&\le \left( \int ^{\infty }_{0}\int _{\begin{array}{c} |y-x|<t \\ y\in 2B_j \end{array}}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }} a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{1/2}\\&\quad +\,\left( \int ^{\infty }_{0}\int _{\begin{array}{c} |y-x|<t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \end{array}}\cdot \cdot \cdot \right) ^{1/2} +\left( \int ^{\infty }_{0}\int _{\begin{array}{c} |y-x|<t \\ y\in (2B_j)^\complement \\ t>|y|+2^{j} \end{array}}\cdot \cdot \cdot \right) ^{1/2}\\&=:{\mathrm{I_1+I_2+I_3}}. \end{aligned}$$

For ${\mathrm{I_1}}$, by $x\in A_k$, $y\in 2B_j$, $z\in B_j$ and $j\le k-3$, it is easy to see that

$$\begin{aligned} t>|y-x|\ge |x|-|y|>|x|-|x|/2=|x|/2 \ {\mathrm{and}} \ |y-z|<2^{j+2}. \end{aligned}$$

From this, Minkowski’s inequality for integrals, $\Omega \in L^2(S^{n-1})$, generalized Hölder’s inequality (Lemma 3.3) and $0<\xi <\rho -n/2$, we deduce that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I_1}}&=\left( \int ^{\infty }_{0}\int _{\begin{array}{c} |y-x|<t \\ y\in 2B_j \end{array}}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z) \,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{{1/2}}\\&\le \int _{B_j}\left| a_j(z)\right| \left( \int ^{\infty }_{0}\int _{\begin{array}{c} |y-x|<t \\ y\in 2B_j \\ |y-z|<t \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\, \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{{1/2}}\,{\mathrm{d}}z \\&\le \int _{B_j}\left| a_j(z)\right| \left[ \left( \int ^\infty _{\frac{|x|}{2}} \frac{{\mathrm{d}}t}{t^{n+2\rho +1}}\right) \left( \int _{\begin{array}{c} y\in 2B_j \\ |y-z|<2^{j+2} \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\, {\mathrm{d}}y\right) \right] ^{{1/2}}\,{\mathrm{d}}z \\&\le C \int _{B_j}\left| a_j(z)\right| \left[ \frac{1}{|x|^{n+2\rho }}\left( \int _{S^{n-1}} \int ^{2^{j+2}}_0\frac{|\Omega (y')|^2}{u^{2n-2\rho }}u^{n-1} \,{\mathrm{d}}u{\mathrm{d}}\sigma (y')\right) \right] ^{{1/2}}\,{\mathrm{d}}z \\&\le C \int _{B_j}\left| a_j(z)\right| \left( \frac{1}{|x|^{n+2\rho }}2^{(j+2)(-n+2\rho )}\right) ^{{1/2}}\,{\mathrm{d}}z \\&\le C \frac{2^{j(\rho -n/2)}}{|x|^{\rho +n/2}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \le C \frac{2^{j(\rho -n/2)}}{2^{k(\rho +n/2)}} \int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)(\rho -n/2)}\left\| a_j \right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}} \left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}, \end{aligned}$$

which is wished.

For ${\mathrm{I_2}}$, by $x\in A_k$, $y\in (2B_j)^\complement$, $z\in B_j$, $j\le k-3$ and the mean value theorem, we have

$$\begin{aligned}&|y-z|\sim |y|; \end{aligned}$$

(3.1)

$$\begin{aligned}&|y|-2^{j}\le |y|-|z|\le |y-z|<t\le |y|+2^{j}; \end{aligned}$$

(3.2)

$$\begin{aligned}&|x|\le |x-y|+|y|\le t+|y|\le 2|y|+2^{j}\le 3|y|; \end{aligned}$$

(3.3)

$$\begin{aligned}&\left| \frac{1}{\left( |y|-2^{j}\right) ^{n+2\rho }}-\frac{1}{\left( |y|+2^{j}\right) ^{n+2\rho }}\right| \le C\frac{2^j}{|y|^{n+2\rho +1}}. \end{aligned}$$

(3.4)

By Minkowski’s inequality for integrals, (3.1)–(3.4), $0<\xi <\min \{1/2,\,\rho -n/2\}$, $\Omega \in L^2(S^{n-1})$ and generalized Hölder’s inequality (Lemma 3.3), we know that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I_2}}&=\left( \iint _{\begin{array}{c} |y-x|<t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \end{array}}\left| \int _{|y-z|<t} \frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\, \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{{1/2}}\\&\le \int _{B_j}|a_j(z)| \left( \iint _{\begin{array}{c} |y-x|<t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \\ |y-z|<t \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\, \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{{1/2}}\,{\mathrm{d}}z \\&\le C \int _{B_j}|a_j(z)| \left[ \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|\le 3|y| \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }} \left( \int ^{|y|+2^{j}}_{|y|-2^{j}}\frac{{\mathrm{d}}t}{t^{n+2\rho +1}} \right) \,{\mathrm{d}}y\right] ^{{1/2}}\,{\mathrm{d}}z\\&\le C \int _{B_j}|a_j(z)| \left( \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|\le 3|y| \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }} \frac{2^j}{|y|^{n+2\rho +1}}\,{\mathrm{d}}y\right) ^{{1/2}}\,{\mathrm{d}}z \\&\le C \int _{B_j}|a_j(z)| \left( \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|\le 3|y| \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{n-2\xi +1}} \frac{2^j}{|x|^{2n+2\xi }}\,{\mathrm{d}}y\right) ^{{1/2}}\,{\mathrm{d}}z \\&\le C \frac{(2^j)^{1/2}}{|x|^{n+\xi }}\int _{B_j}|a_j(z)| \left( \int _{y\in (2B_j)^\complement }\frac{|\Omega (y-z)|^2}{|y-z|^{n-2\xi +1}} \,{\mathrm{d}}y\right) ^{{1/2}}\,{\mathrm{d}}z \\&\le C \frac{(2^j)^{1/2}}{|x|^{n+\xi }}\int _{B_j}|a_j(z)| \left( \int _{S^{n-1}}\int ^\infty _{2^{j+1}} \frac{|\Omega (y')|^2}{u^{n-2\xi +1}}u^{n-1} \,{\mathrm{d}}u{\mathrm{d}}\sigma (y')\right) ^{1/2}\,{\mathrm{d}}z\\&\le C \frac{2^{j\xi }}{|x|^{n+\xi }}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \le C \frac{2^{j\xi }}{2^{k(n+\xi )}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}} \left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}, \end{aligned}$$

which is also wished.

Now let us consider ${\mathrm{I_3}}$. It is easy to check that $B_j\subset \{z\in {{{{\mathbb {R}}}}^n}: \ |y-z|<t\}$ because of $y\in (2B_j)^\complement$ and $t>|y|+2^{j}$. From this, the vanishing moments of $a_j$ and Minkowski’s inequality for integrals, it follows that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I_3}}&=\left[ \iint _{\begin{array}{c} |y-x|<t \\ y\in (2B_j)^\complement \\ t>|y|+2^{j} \end{array}}\left| \int _{|y-z|<t} \left( \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }} \right) a_j(z)\,{\mathrm{d}}z\right| ^2\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{1/2}\\&\le \int _{B_j} |a_j(z)| \left( \iint _{\begin{array}{c} y\in (2B_j)^\complement \\ t>\max \{|y-x|,\,\\ |y|+2^{j},\,|y-z|\} \end{array}} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }} -\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{1/2}{\mathrm{d}}z\\&\le \int _{B_j} |a_j(z)| \left( \iint _{\begin{array}{c} y\in (2B_j)^\complement \\ t>\max \{|y-x|,\,\\ |y|+2^{j},\,|y-z|\} \\ |x|\le 2|y| \end{array}} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }} -\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right) ^{1/2}{\mathrm{d}}z\\&\quad +\int _{B_j} |a_j(z)| \left( \iint _{\begin{array}{c} y\in (2B_j)^\complement \\ t>\max \{|y-x|,\,|y|+2^{j},\,|y-z|\} \\ |x|>2|y| \end{array}} \cdot \cdot \cdot \right) ^{1/2}{\mathrm{d}}z=:{\mathrm{I_{31}}+\mathrm{I_{32}}}. \end{aligned}$$

Below, we will give the estimates of ${\mathrm{I_{31}}}$ and ${\mathrm{I_{32}}}$, respectively.

For ${\mathrm{I_{31}}}$, from $0<\xi <\min \{1/2,\,\rho -n/2\}$ and Lemma 3.2, we deduce that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I_{31}}}&\le \int _{B_j} |a_j(z)| \left[ \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|\le 2|y| \end{array}} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \left( \int ^{\infty }_{|y|+2^{j}}\frac{{\mathrm{d}}t}{t^{n+2\rho +1}}\right) \,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le \int _{B_j} |a_j(z)| \left[ \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|\le 2|y| \end{array}} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{1}{(|y|+2^{j})^{n+2\rho }}\,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }} \left[ \int _{y\in (2B_j)^\complement } \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{1}{(|y|+2^{j})^{2\rho -n-2\xi }}\,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty \left[ \int _{2^i2^j\le |y|<2^{i+1}2^j} \right. \\&\quad \left. \times \, \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{1}{(|y|+2^{j})^{2\rho -n-2\xi }}\,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty \frac{1}{(2^i2^j+2^{j})^{\rho -n/2-\xi }}\\&\quad \times \, \left( \int _{2^i2^j\le |y|<2^{i+1}2^j} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \,{\mathrm{d}}y\right) ^{1/2}{\mathrm{d}}z \\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty \frac{(2^i2^j)^{n/2-(n-\rho )}}{(2^i2^j+2^{j})^{\rho -n/2-\xi }} \left( \frac{|z|}{2^i2^j}+\int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j}\frac{\omega _2(\delta )}{\delta }\,{\mathrm{d}}\delta \right) {\mathrm{d}}z\\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty (2^i2^j)^{\xi } \left( \frac{1}{2^i}+\int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j} \frac{\omega _2(\delta )}{\delta }\,{\mathrm{d}}\delta \right) {\mathrm{d}}z\\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty \frac{(2^i2^j)^{\xi }}{2^i} \,{\mathrm{d}}z\\&\quad +\, C \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty (2^i2^j)^{\xi } \int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j}\frac{\omega _2(\delta )}{\delta }\,{\mathrm{d}}\delta \, {\mathrm{d}}z =:C({\mathrm{I'_{31}}+\mathrm{I''_{31}}}). \end{aligned}$$

For ${\mathrm{I'_{31}}}$, by $0<\xi <\min \{1/2,\,\rho -n/2\}$ and generalized Hölder’s inequality (Lemma 3.3), we know that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I'_{31}}}&= \int _{Bj} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty \frac{(2^i2^j)^{\xi }}{2^i} \,{\mathrm{d}}z\\&\le C \frac{2^{j\xi }}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty 2^{i\,(\xi -1)} \int _{B_j}|a_j(z)| \,{\mathrm{d}}z\\&\le C \frac{2^{j\xi }}{2^{k(n+\xi )}} \int _{B_j}|a_j(z)| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

For ${\mathrm{I''_{31}}}$, by $0<\xi <\min \{1/2,\,\beta \}$, $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition and generalized Hölder’s inequality (Lemma 3.3), we know that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I''_{31}}}&= \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty (2^i2^j)^{\xi } \int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j}\frac{\omega _2(\delta )}{\delta }\,{\mathrm{d}}\delta \, {\mathrm{d}}z \\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty \frac{(2^i2^j)^{\xi }}{2^{i\beta }} \int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j}\frac{\omega _2(\delta )}{\delta ^{1+\beta }}\,{\mathrm{d}}\delta \, {\mathrm{d}}z \\&\le C \frac{2^{j\xi }}{(|x|+2^{j})^{n+\xi }}\sum _{i=1}^\infty 2^{i(\xi -\beta )}\int _0^1\frac{\omega _2(\delta )}{\delta ^{1+\beta }}\,{\mathrm{d}}\delta \int _{B_j}|a_j(z)|\, {\mathrm{d}}z \\&\le C \frac{2^{j\xi }}{2^{k(n+\xi )})}\int _{B_j}|a_j(z)|\, {\mathrm{d}}z \\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

Combining the estimates of ${\mathrm{I_{31}}}$, ${\mathrm{I'_{31}}}$ and ${\mathrm{I''_{31}}}$, we obtain that

$$\begin{aligned} {\mathrm{I_{31}}} \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _j\right\| _{L^{q'(\cdot )}}, \end{aligned}$$

which is wished.

Now we give the estimate for ${\mathrm{I_{32}}}$. Noticing that $t>\max \{|y-x|,\,|y|+2^{j},\,|y-z|\}$ and $|x|>2|y|$, it follows that

$$\begin{aligned} t>|y-x|\ge |x|-|y|>|x|/2. \end{aligned}$$

From this and $\xi <\rho -n/2$, it follows that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{I_{32}}}&\le \int _{B_j} |a_j(z)| \left[ \iint _{\begin{array}{c} y\in (2B_j)^\complement \\ t>\max \{|y-x|,\, |y|+2^{j},\,|y-z|\} \\ |x|>2|y| \end{array}} \frac{1}{(|y|+2^{j})^{2\rho -n-2\xi }} \right. \\&\quad \left. \times \, \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{2n+2\xi +1}}\right] ^{1/2}{\mathrm{d}}z \\&\le \int _{B_j} |a_j(z)| \left[ \int _{y\in (2B_j)^\complement } \frac{1}{(|y|+2^{j})^{2\rho -n-2\xi }} \right. \\&\quad \left. \times \,\left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \left( \int ^{\infty }_{\frac{|x|}{2}}\frac{{\mathrm{d}}t}{t^{2n+2\xi +1}}\right) \,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{|x|^{n+\xi }} \left[ \int _{y\in (2B_j)^\complement } \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{1}{(|y|+2^{j})^{2\rho -n-2\xi }}\,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z . \end{aligned}$$

By this and repeating this process which is similar to the one of estimating ${\mathrm{I_{31}}}$ of Lemma 3.5, we have

$$\begin{aligned} {\mathrm{I_{32}}}&\le C \frac{2^{j\xi }}{|x|^{n+\xi }}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \le C \frac{2^{j\xi }}{2^{k(n+\xi )}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

Combining the estimates of ${\mathrm{I_1}}$, ${\mathrm{I_2}}$, ${\mathrm{I_{31}}}$ and ${\mathrm{I_{32}}}$, we obtain that, for any $x\in A_k$,

$$\begin{aligned} \mu ^\rho _{\Omega ,\,S}(a_j)(x) \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

From this, Lemma 3.4 and Proposition 2.1, it follows that, for any $x\in A_k$,

$$\begin{aligned} \left\| \mu ^{\rho }_{\Omega ,\, S}(a_j)\chi _k\right\| _{L^{q(\cdot )}}&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _k\right\| _{L^{q(\cdot )}}\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _{B_k}\right\| _{L^{q(\cdot )}}\\&\le C 2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\frac{\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}}{\left\| \chi _{B_k}\right\| _{L^{q'(\cdot )}}}\\&\le C 2^{(j-k)(n\delta _2+\xi )}\left\| a_j\right\| _{L^{q(\cdot )}}. \end{aligned}$$

This finishes the proof of Lemma 3.5. $\square$

Lemma 3.6 shows that $\mu ^{\rho ,\,*}_{\Omega ,\,\lambda }$ maps all multiple of an atoms into uniformly bounded elements of $L^{q(\cdot )}$.

Lemma 3.6

Let $\beta \in (0,\,1]$, $\rho \in (n/2,\,\infty )$, $\lambda \in (2,\,\infty )$, $\delta _2\in (0,\,1)$, $n\delta _2\le \alpha <\infty$, $q(\cdot )\in {\mathcal {B}}$, $\xi \in (0,\,\min \{1/2,\,\beta ,\,\rho -n/2,\,(\lambda -2)n/3\})$, $j,\,k\in {{\mathbb {Z}}}_+$ and $j\le k-3$. Suppose $a_j$ is a dyadic $(\alpha ,\,q(\cdot ))$-atom with the support $B_j:=B(0,\,2^j)$. If $\Omega$ satisfies the $L^{2,\,\beta }$-Dini condition, then there exists a positive constant C independent of $a_j$ such that for any $x\in A_k$,

$$\begin{aligned} \left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)\chi _k\right\| _{L^{q(\cdot )}}\le C \, 2^{(j-k)(n\delta _2+\xi )} \left\| a_j\right\| _{L^{q(\cdot )}} \end{aligned}$$

Proof

We show this lemma by borrowing some ideas from the proof of [3, Theorem 1.1]. By Lemma 3.5, we know that, for any $x\in A_k$,

$$\begin{aligned}&\mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)(x)\\&\quad =\left[ \iint _{{{\mathbb {R}}}^{n+1}_+}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}(a_j)(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\quad \le \left[ \iint _{|y-x|<t}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}(a_j)(z)\,{\mathrm{d}}z \right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\qquad +\,\left[ \iint _{|y-x|\ge t}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}(a_j)(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\quad \le \mu ^\rho _{\Omega ,\,S}(a_j)(x) \\&\qquad +\,\left[ \iint _{|y-x|\ge t}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}(a_j)(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\quad \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}} \\&\qquad +\,\left[ \iint _{|y-x|\ge t}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}(a_j)(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\quad =:C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}+{\mathrm{J}}. \end{aligned}$$

Thus, in order to get what we want, it suffices to prove that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{J}} \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

For any $x\in A_k$, write

$$\begin{aligned} {\mathrm{J}}&\le \left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in 2B_j \end{array}} \left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\quad +\,\left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \end{array}}\cdot \cdot \cdot \right] ^{{1/2}} +\left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t>|y|+2^{j} \end{array}}\cdot \cdot \cdot \right] ^{{1/2}}=:{\mathrm{J_1+J_2+J_3}}. \end{aligned}$$

For ${\mathrm{J_1}}$, notice that $|y-x|\ge t$, $x\in A_k$, $y\in 2B_j$, $z\in B_j$ and $j\le k-3$, it is easy to see that

$$\begin{aligned} |y-x|\ge |x|-|y|>|x|-|x|/2=|x|/2 \ {\mathrm{and}} \ |y-z|<2^{j+2}. \end{aligned}$$

By this, Minkowski’s inequality for integrals, $\xi <\min \{(\lambda -2)n/3,\,\rho -n/2\}$, $\Omega \in L^2(S^{n-1})$ and generalized Hölder’s inequality (Lemma 3.3), we obtain that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{J_1}}&\le \left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in 2B_j \\ |y-x|>|x|/2 \end{array}}\left( \frac{t}{t+|x-y|}\right) ^{\lambda n} \left| \int _{\begin{array}{c} |y-z|<t \\ |y-z|<2^{j+2} \end{array}}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\le \int _{B_j}|a_j(z)|\left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in 2B_j \\ |y-x|>|x|/2 \\ |y-z|<2^{j+2},\,|y-z|<t \end{array}} \left( \frac{t}{t+\frac{|x|}{2}}\right) ^{2n+3\xi }\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}}{\mathrm{d}}z \\&\le C \left( \frac{1}{|x|}\right) ^{n+\xi } \int _{B_j}|a_j(z)|\left( \int _{\begin{array}{c} y\in 2B_j \\ |y-x|>|x|/2 \\ |y-z|<2^{j+2} \end{array}}\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\right. \\&\quad \left. \times \, \int _0^{|y-x|}\frac{t^{2n+3\xi }t^{2\rho -n-2\xi }}{|x|^\xi |y-z|^{2\rho -n-2\xi }t^{n+2\rho +1}} {\mathrm{d}}y{\mathrm{d}}t\right) ^{{1/2}}{\mathrm{d}}z \\&\le C \left( \frac{1}{|x|}\right) ^{n+\xi } \int _{B_j}|a_j(z)|\left( \int _{\begin{array}{c} y\in 2B_j \\ |y-x|>|x|/2 \\ |y-z|<2^{j+2} \end{array}}\frac{|\Omega (y-z)|^2|y-x|^\xi }{|y-z|^{n-2\xi }|x|^\xi }{\mathrm{d}}y\right) ^{{1/2}}{\mathrm{d}}z \\&\le C \frac{1}{|x|^{n+\xi }} \int _{B_j}|a_j(z)|\left( \int _{|y-z|<2^{j+2}}\frac{|\Omega (y-z)|^2}{|y-z|^{n-2\xi }}{\mathrm{d}}y\right) ^{{1/2}}{\mathrm{d}}z \\&\le C \frac{1}{|x|^{n+\xi }}\int _{B_j}|a_j(z)| \left( \int _{S^{n-1}}\int ^{2^{j+2}}_0\frac{|\Omega (y')|^2}{u^{n-2\xi }}u^{n-1}\,\mathrm{d}u\mathrm{d}\sigma (y')\right) ^{1/2}\,{\mathrm{d}}z \\&\le C \frac{2^{j\xi }}{|x|^{n+\xi }}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \le C \frac{2^{j\xi }}{2^{k(n+\xi )}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}, \end{aligned}$$

which is wished.

For ${\mathrm{J_2}}$, write

$$\begin{aligned} {\mathrm{J_2}}&\le \left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \\ |x|>2|y| \end{array}} \left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\quad +\,\left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \\ |x|\le 2|y| \end{array}} \left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&=:{\mathrm{J_{21}+J_{22}}}. \end{aligned}$$

The estimate of ${\mathrm{J_{22}}}$ is quite similar to that given earlier for the estimate of ${\mathrm{I_{2}}}$ in Lemma 3.5 and so is omitted. We are now turning to the estimate of ${\mathrm{J_{21}}}$.

For ${\mathrm{J_{21}}}$, by $x\in A_k$, $y\in (2B_j)^\complement$, $|x|>2|y|$, $z\in B_j$, $j\le k-3$ and the mean value theorem, we know that

$$\begin{aligned}&|y-z|\sim |y|; \end{aligned}$$

(3.5)

$$\begin{aligned}&|y|-2^{j}\le |y|-|z|\le |y-z|<t\le |y|+2^{j}; \end{aligned}$$

(3.6)

$$\begin{aligned}&|x-y|\ge |x|-|y|>|x|/2. \end{aligned}$$

(3.7)

By Minkowski’s inequality for integrals, $\xi<\min \{(\lambda -2)n/3,\,1/2\}<(\lambda -2)n/2$, (3.5)–(3.7), Lemma 3.2, $\Omega \in L^2(S^{n-1})$ and generalized Hölder’s inequality (Lemma 3.3), we know that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{J_{21}}}&=\left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \\ |x|>2|y| \end{array}} \left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\le \int _{B_j}|a_j(z)|\left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t\le |y|+2^{j} \\ |x|>2|y| \end{array}} \left( \frac{t}{t+|x-y|}\right) ^{2n+2\xi }\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{1/2}{\mathrm{d}}z \\&\le C \int _{B_j}|a_j(z)|\left[ \iint _{\begin{array}{c} y\in (2B_j)^\complement \\ |x-y|>|x|/2 \\ |y|-2^{j}\le t \\ t\le |y|+2^{j} \end{array}} \left( \frac{t}{|x-y|}\right) ^{2n+2\xi }\frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{1/2}{\mathrm{d}}z \\&\le C \int _{B_j}|a_j(z)|\left[ \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|>2|y| \end{array}} \frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }}\left( \int ^{|y|+2^{j}}_{|y|-2^{j}}\frac{t^{n+2\xi -2\rho -1}}{|x-y|^{2n+2\xi }}{\mathrm{d}}t\right) {\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le C \int _{B_j}|a_j(z)|\left( \int _{\begin{array}{c} y\in (2B_j)^\complement \\ |x|>2|y| \end{array}} \frac{|\Omega (y-z)|^2}{|y-z|^{2n-2\rho }} \frac{2^j}{|x-y|^{2n+2\xi }|y|^{2\rho -n-2\xi +1}}\,{\mathrm{d}}y\right) ^{1/2}{\mathrm{d}}z \\&\le C\frac{(2^j)^{1/2}}{|x|^{n+\xi }}\int _{B_j}|a_j(z)|\left( \int _{y\in (2B_j)^\complement } \frac{|\Omega (y-z)|^2}{|y-z|^{n-2\xi +1}}\,{\mathrm{d}}y\right) ^{1/2}{\mathrm{d}}z \\&\le C\frac{(2^j)^{1/2}}{|x|^{n+\xi }}\int _{B_j}|a_j(z)|\left( \int _{S^{n-1}} \int ^\infty _{2^{j+1}}\frac{|\Omega (y')|^2}{u^{n-2\xi +1}}u^{n-1}\,\mathrm{d}u\mathrm{d}\sigma (y')\right) ^{1/2}{\mathrm{d}}z \\&\le C \frac{2^{j\xi }}{|x|^{n+\xi }}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \le C \frac{2^{j\xi }}{2^{k(n+\xi )}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}, \end{aligned}$$

which is also wished.

For ${\mathrm{J_3}}$, noticing that $t>|y|+2^{j}$, we know that, for any $y\in (2B_j)^\complement$,

$$\begin{aligned}&B\subset \{z\in {{{{\mathbb {R}}}}^n}: \ |z-y|<t\}; \end{aligned}$$

(3.8)

$$\begin{aligned}&t+|x-y|\ge t+|x|-|y|\ge |x|+2^{j}>|x|. \end{aligned}$$

(3.9)

From (3.8), the vanishing moments of $a_j$, Minkowski’s inequality for integrals, (3.9), $\xi<\min \{\beta ,\,\rho -n/2,\,(\lambda -2)n/3\}<(\lambda -2)n/2$ and Lemma 3.2, we deduce that, for any $x\in A_k$,

$$\begin{aligned} {\mathrm{J_{3}}}&= \left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t>|y|+2^{j} \end{array}} \left( \frac{t}{t+|x-y|}\right) ^{\lambda n}\left| \int _{|y-z|<t}\frac{\Omega (y-z)}{|y-z|^{n-\rho }}a_j(z)\,{\mathrm{d}}z\right| ^2\,\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{{1/2}} \\&\le \int _{B_j} |a_j(z)| \left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t>|y|+2^{j} \\ |y-z|<t \end{array}} \frac{t^{\lambda n}}{(t+|x-y|)^{2n+2\xi }} \frac{1}{(t+|x-y|)^{\lambda n-2n-2\xi }}\right. \\&\quad \left. \times \, \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{|x|^{n+\xi }} \left[ \iint _{\begin{array}{c} |y-x|\ge t \\ y\in (2B_j)^\complement \\ t>|y|+2^{j} \\ |y-z|<t \end{array}} \frac{t^{\lambda n}}{(t+|x-y|)^{\lambda n-2n-2\xi }} \right. \\&\quad \left. \times \, \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2\frac{{\mathrm{d}}y{\mathrm{d}}t}{t^{n+2\rho +1}}\right] ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{|x|^{n+\xi }} \left[ \iint _{\begin{array}{c} y\in (2B_j)^\complement \\ |y-x|>|y|+2^{j} \end{array}} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2\right. \\&\quad \left. \times \, \left( \int ^{|y-x|}_{|y|+2^{j}}\frac{t^{\lambda n-2n-2\xi }}{(t+|x-y|)^{\lambda n-2n-2\xi }t^{2\rho -n+1-2\xi }}\,{\mathrm{d}}t\right) \,{\mathrm{d}}y\right] ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{|x|^{n+\xi }} \left( \int _{y\in (2B_j)^\complement } \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2\frac{1}{|y|^{2\rho -n-2\xi }}\,{\mathrm{d}}y\right) ^{1/2}{\mathrm{d}}z \\&\le C\int _{B_j} \frac{|a_j(z)|}{|x|^{n+\xi }} \left( \sum _{i=1}^\infty \int _{2^i2^{j}\le |y|<2^{i+1}2^{j}} \left| \frac{\Omega (y-z)}{|y-z|^{n-\rho }}-\frac{\Omega (y)}{|y|^{n-\rho }}\right| ^2 \frac{1}{|y|^{2\rho -n-2\xi }}\,{\mathrm{d}}y\right) ^{1/2}{\mathrm{d}}z \\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|)^{n+\xi }}\sum _{i=1}^\infty \frac{(2^i2^j)^{n/2-(n-\rho )}}{(2^i2^j)^{\rho -n/2-\xi }} \left( \frac{|z|}{2^i2^j}+\int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j}\frac{\omega _2(\delta )}{\delta }\,{\mathrm{d}}\delta \right) {\mathrm{d}}z \\&\le C \int _{B_j} \frac{|a_j(z)|}{(|x|)^{n+\xi }}\sum _{i=1}^\infty (2^i2^j)^{\xi } \left( \frac{1}{2^i}+\int _{\frac{|z|}{2^{i+1}2^j}}^\frac{|z|}{2^i2^j}\frac{\omega _2(\delta )}{\delta }\,{\mathrm{d}}\delta \right) {\mathrm{d}}z. \end{aligned}$$

By this and repeating this process which is similar to the one of estimating ${\mathrm{I'_{31}}}$ and ${\mathrm{I''_{31}}}$ of Lemma 3.5, we have

$$\begin{aligned} {\mathrm{J_{3}}}&\le C \frac{2^{j\xi }}{|x|^{n+\xi }}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \le C \frac{2^{j\xi }}{2^{k(n+\xi )}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

Combining the estimates of ${\mathrm{J_1}}$, ${\mathrm{J_{2}}}$ and ${\mathrm{J_3}}$, we know that, for any $x\in A_k$,

$$\begin{aligned} \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)(x) \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}. \end{aligned}$$

From this, Lemma 3.4 and Proposition 2.1, it follows that, for any $x\in A_k$,

$$\begin{aligned} \left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)\chi _k\right\| _{L^{q(\cdot )}}&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _k\right\| _{L^{q(\cdot )}}\\&\le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _{B_k}\right\| _{L^{q(\cdot )}}\\&\le C 2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\frac{\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}}{\left\| \chi _{B_k}\right\| _{L^{q'(\cdot )}}}\\&\le C 2^{(j-k)(n\delta _2+\xi )}\left\| a_j\right\| _{L^{q(\cdot )}}. \end{aligned}$$

This finishes the proof of Lemma 3.6. $\square$

4 Proofs of Main Results

In this section, we will prove the main results of this paper. Noticing that

$$\begin{aligned} \mu ^\rho _{\Omega ,\,S}(f)(x)\le & {} 2^{n\lambda }\mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(f)(x),\,\,\,\, [b^m,\,\mu ^\rho _{\Omega ,\, S}](f)(x)\\\le & {} 2^{n\lambda }[b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)(x),\quad \,\, m\in {{\mathbb {Z}}}_+. \end{aligned}$$

Therefore, it is enough to consider the operators $\mu ^\rho _{\Omega ,\,S}$ and $[b^m,\,\mu ^\rho _{\Omega ,\, S}]$ in the proofs of our results. That is to say, we only prove Theorems 2.8, 2.10 and 2.12.

Proof of Theorem 2.8

We only prove homogeneous case. In [16], the authors proved ${K^{\alpha _1,\,p_2}_{q(\cdot )}} \subset {HK^{\alpha _2,\,p_2}_{q(\cdot )}}$ for $0<\alpha _2\le \alpha _1$. So the non-homogeneous case can be proved in the same way. For any $f\in H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}$, by Lemma 3.1, we know that $f=\sum _{j=-\infty }^{\infty }\lambda _ja_j$ in ${{\mathcal {S}}}'$, where $\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}}\approx \inf \left( \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1}\right) ^{1/p_1}$ (the infimum is taken over above decompositions of f), and $a_j$ is a dyadic central $(\alpha ,\,q(\cdot ))$-atom with the support $B_j$. Noticing that $p_1\le p_2$, we deduce that

$$\begin{aligned} \left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(f)\right\| ^{p_1}_{{\dot{K}}^{\alpha ,\,p_2}_{q(\cdot )}}&=\left\{ \sum ^\infty _{k=-\infty } 2^{k\alpha p_2}\left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(f)\chi _k\right\| ^{p_2}_{L^{q(\cdot )}}\right\} ^{p_1/p_2} \nonumber \\&\le \sum ^\infty _{k=-\infty } 2^{k\alpha p_1}\left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(f)\chi _k\right\| ^{p_1}_{L^{q(\cdot )}} \nonumber \\&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|\left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)\chi _k\right\| _{L^{q(\cdot )}}\right) ^{p_1}\nonumber \\&\quad +\, C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^\infty _{j=k-2} |\lambda _j|\left\| \mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)\chi _k\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&=:{\mathrm{I_{1}+I_{2}}}. \end{aligned}$$

(4.1)

First we estimate ${\mathrm{I_{1}}}$ by considering two cases: $1<p_1<\infty$ and $0<p_1\le 1$.

Case 1 $1<p_1<\infty$. For this case, take $1/p_1+1/p'_1=1$. Then, by Lemma 3.6, $n\delta _2+\xi -\alpha >0$ and Hölder’s inequality, we find that

$$\begin{aligned} {\mathrm{I_{1}}}&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty } 2^{(j-k)(n\delta _2+\xi )}|\lambda _j|\left\| a_j\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\le C \sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|2^{(j-k)(n\delta _2+\xi -\alpha )}\right) ^{p_1} \nonumber \\&\le C \sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|^{p_1}2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}/2}\right) \left( \sum ^{k-3}_{j=-\infty } 2^{(j-k)(n\delta _2+\xi -\alpha ){p'_1}/2}\right) ^{p_1/p'_1}\nonumber \\&\le C \sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|^{p_1}2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}/2}\right) \nonumber \\&\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} \left( \sum ^\infty _{k=j+3} 2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}/2}\right) \nonumber \\&\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} . \end{aligned}$$

(4.2)

Case 2 $0<p_1\le 1$. For this case, by Lemma 3.6 and $n\delta _2+\xi -\alpha >0$, we know that

$$\begin{aligned} {\mathrm{I_{1}}}&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty } 2^{(j-k)(n\delta _2+\xi )}|\lambda _j|\left\| a_j\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\le C \sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|2^{(j-k)(n\delta _2+\xi -\alpha )}\right) ^{p_1} \nonumber \\&\le C \sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|^{p_1}2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}}\right) \nonumber \\&\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} \left( \sum ^\infty _{k=j+3} 2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}}\right) \nonumber \\&\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} . \end{aligned}$$

(4.3)

Next we estimate ${\mathrm{I_{2}}}$. From the fact that $\mu ^{\rho ,\,*}_{\Omega ,\,\lambda }$ is bounded on $L^{q(\cdot )}$, it follows that

$$\begin{aligned} {\mathrm{I_{2}}}&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^\infty _{j=k-2} |\lambda _j|\left\| a_j\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\le C \sum ^\infty _{k=-\infty }\left( \sum ^\infty _{j=k-2} |\lambda _j|2^{(k-j)\alpha }\right) ^{p_1}\nonumber \\&\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} . \end{aligned}$$

(4.4)

Combining (4.1), (4.2), (4.3) and (4.4), we finish the proof of Theorem 2.8. $\square$

In order to Theorem 2.10, we need the following lemma.

Lemma 4.1

([5, Lemma 3]) Let $p(\cdot )\in {\mathcal {B}}$, m be a positive integer and B be a ball in ${\mathbb {R}}^n$. Then, for all $b\in \mathrm{BMO}$ and all $j,\,i\in {\mathbb {Z}}$ with $j>i$,

$$\begin{aligned}&\frac{1}{C}\Vert b\Vert ^m_*\le \sup _B\frac{1}{\left\| \chi _B\right\| _{L^{p(\cdot )}}}\left\| \left( b-b_B\right) ^m\chi _B\right\| _{L^{p(\cdot )}}\le C\Vert b\Vert ^m_*, \\&\left\| \left( b-b_{B_i}\right) ^m\chi _{B_j}\right\| _{L^{p(\cdot )}}\le C (j-i)^m\Vert b\Vert ^m_*\left\| \chi _{B_j}\right\| _{L^{p(\cdot )}}, \end{aligned}$$

where $B_i:=\{x\in {\mathbb {R}}^n:|x|\le 2^i\}$ and $B_j:=\{x\in {\mathbb {R}}^n:|x|\le 2^j\}$ .

Proof of Theorem 2.10

Similar to Theorem 2.8, we only prove the homogeneous case. For any $f\in H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}$, by Lemma 3.1, we know that $f=\sum _{j=-\infty }^{\infty }\lambda _ja_j$ in ${{\mathcal {S}}}'$, where $\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q(\cdot )}}\approx \inf \left( \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1}\right) ^{1/p_1}$ (the infimum is taken over above decompositions of f), and $a_j$ is a dyadic central $(\alpha ,\,q(\cdot ))$-atom with the support $B_j$. Noticing that $p_1\le p_2$, we deduce that

$$\begin{aligned} \left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\right\| ^{p_1}_{{\dot{K}}^{\alpha ,\,p_2}_{q(\cdot )}}&=\left\{ \sum ^\infty _{k=-\infty } 2^{k\alpha p_2}\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\chi _k\right\| ^{p_2}_{L^{q(\cdot )}}\right\} ^{p_1/p_2} \nonumber \\&\le \sum ^\infty _{k=-\infty } 2^{k\alpha p_1}\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\chi _k\right\| ^{p_1}_{L^{q(\cdot )}} \nonumber \\&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)\chi _k\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\quad + \,C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^\infty _{j=k-2} |\lambda _j|\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)\chi _k\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&=:{\mathrm{J_{1}+J_{2}}}. \end{aligned}$$

(4.5)

For ${\mathrm{J_{1}}}$, repeating this process which is similar to the one of estimating $\mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)$ of Lemma 3.6 and generalized Hölder’s inequality (Lemma 3.3), we know that, for any $x\in A_k$, $b\in \mathrm{BMO}$ and $m\in {{\mathbb {Z}}}_+$,

$$\begin{aligned}&{[}b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)(x) \\&\quad \le C \frac{2^{j\xi }}{2^{k(n+\xi )}}\int _{B_j}\left| a_j(z)\right| |b(x)-b(z)|^m\,{\mathrm{d}}z\\&\quad \le C 2^{-kn}2^{(j-k)\xi }\left( \left| b(x)-b_{B_j}\right| ^m\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z+\int _{B_j}\left| a_j(z)\right| \left| b_{B_j}-b(z)\right| ^m\,{\mathrm{d}}z\right) \\&\quad \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}}\left( \left| b(x)-b_{B_j}\right| ^m\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}+ \left\| (b_{B_j}-b(\cdot ))^m\chi _{B_j}(\cdot )\right\| _{L^{q'(\cdot )}}\right) . \end{aligned}$$

From this, Lemmas 4.1, 3.4 and Proposition 2.1, it follows that, for any $x\in A_k$, $b\in \mathrm{BMO}$ and $m\in {{\mathbb {Z}}}_+$,

$$\begin{aligned}&\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)\chi _k\right\| _{L^{q(\cdot )}} \nonumber \\&\quad \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}} \nonumber \\&\qquad \times \, \left( \left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \left( b(\cdot )-b_{B_j}\right) ^m\chi _k(\cdot )\right\| _{L^{q(\cdot )}}+ \Vert b\Vert ^m_*\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _{k}\right\| _{L^{q(\cdot )}}\right) \nonumber \\&\quad \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}} \nonumber \\&\qquad \times \, \left( (k-j)^m\Vert b\Vert ^m_*\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _{{B_k}}(\cdot )\right\| _{L^{q(\cdot )}}+ \Vert b\Vert ^m_*\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _{{B_k}}\right\| _{L^{q(\cdot )}}\right) \nonumber \\&\quad \le C 2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q(\cdot )}} (k-j)^m\Vert b\Vert ^m_*\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}\left\| \chi _{{B_k}}\right\| _{L^{q(\cdot )}} \nonumber \\&\quad \le C (k-j)^m2^{(j-k)\xi }\Vert b\Vert ^m_*\left\| a_j\right\| _{L^{q(\cdot )}}\frac{\left\| \chi _{B_j}\right\| _{L^{q'(\cdot )}}}{\left\| \chi _{B_k}\right\| _{L^{q'(\cdot )}}} \nonumber \\&\quad \le C (k-j)^m2^{(j-k)(n\delta _2+\xi )}\Vert b\Vert ^m_*\left\| a_j\right\| _{L^{q(\cdot )}} . \end{aligned}$$

(4.6)

Next we estimate ${\mathrm{J_{1}}}$ by considering two cases: $1<p_1<\infty$ and $0<p_1\le 1$.

Case 1 $1<p_1<\infty$. For this case, take $1/p_1+1/p'_1=1$. Then, by (4.6), $n\delta _2+\xi -\alpha >0$ and Hölder’s inequality, we find that

$$\begin{aligned} {\mathrm{J_{1}}}&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty }(k-j)^m 2^{(j-k)(n\delta _2+\xi )}|\lambda _j|\left\| a_j\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j| (k-j)^m 2^{(j-k)(n\delta _2+\xi -\alpha )}\right) ^{p_1} \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|^{p_1}2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}/2}\right) \nonumber \\&\quad \times \,\left( \sum ^{k-3}_{j=-\infty } (k-j)^{mp'_1}2^{(j-k)(n\delta _2+\xi -\alpha ){p'_1}/2}\right) ^{p_1/p'_1}\nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|^{p_1}2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}/2}\right) \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} \left( \sum ^\infty _{k=j+3} 2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}/2}\right) \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{j=-\infty } |\lambda _j|^{p_1}. \end{aligned}$$

(4.7)

Case 2 $0<p_1\le 1$. For this case, by (4.6) and $n\delta _2+\xi -\alpha >0$, we know that

$$\begin{aligned} {\mathrm{J_{1}}}&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty }(k-j)^m 2^{(j-k)(n\delta _2+\xi )}|\lambda _j|\left\| a_j\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j| (k-j)^m 2^{(j-k)(n\delta _2+\xi -\alpha )}\right) ^{p_1} \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{k=-\infty } \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|^{p_1}(k-j)^{mp_1} 2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}}\right) \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} \left( \sum ^\infty _{k=j+3} (k-j)^{mp_1} 2^{(j-k)(n\delta _2+\xi -\alpha ){p_1}}\right) \nonumber \\&\le C \Vert b\Vert ^{mp_1}_*\sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} . \end{aligned}$$

(4.8)

For ${\mathrm{J_{2}}}$, from $[b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }]$ is bounded on $L^{q(\cdot )}$, it follows that

$$\begin{aligned} {\mathrm{J_{2}}}&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^\infty _{j=k-2} |\lambda _j|\left\| a_j\right\| _{L^{q(\cdot )}}\right) ^{p_1} \nonumber \\&\le C \sum ^\infty _{k=-\infty }\left( \sum ^\infty _{j=k-2} |\lambda _j|2^{(k-j)\alpha }\right) ^{p_1}\nonumber \\&\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} . \end{aligned}$$

(4.9)

Combining (4.10), (4.7), (4.8) and (4.9), we finish the proof of Theorem 2.10. $\square$

In order to Theorem 2.12, we also need the following lemma.

Lemma 4.2

([1, Theorem 1.8]) Let $q_1(\cdot ),\,q_2(\cdot )\in {\mathcal {P}}$ be such that $q^+_1<n/\nu$ and $1/q_1(x)-1/q_2(x)=\nu /n$. If $q_2(\cdot )(n-\nu )/n\in {\mathcal {B}}$, then $\Vert I_\nu f\Vert _{q_2(\cdot )}\le C \Vert f\Vert _{q_1(\cdot )}$, where $I_\nu$ is the fractional integral operator with $0<\nu <n$.

Proof of Theorem 2.12

Similar to Theorem 2.8, we only prove the homogeneous case. For any $f\in H{\dot{K}}^{\alpha ,\,p_1}_{q_1(\cdot )}$, by Lemma 3.1, we know that $f=\sum _{j=-\infty }^{\infty }\lambda _ja_j$ in ${{\mathcal {S}}}'$, where $\Vert f\Vert _{H{\dot{K}}^{\alpha ,\,p_1}_{q_1(\cdot )}}\approx \inf \left( \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1}\right) ^{1/p_1}$ (the infimum is taken over above decompositions of f), and $a_j$ is a dyadic central $(\alpha ,\,q_1(\cdot ))$-atom with the support $B_j$. Noticing that $p_1\le p_2$, we deduce that

$$\begin{aligned} \left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\right\| ^{p_1}_{{\dot{K}}^{\alpha ,\,p_2}_{q_2(\cdot )}}&=\left\{ \sum ^\infty _{k=-\infty } 2^{k\alpha p_2}\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\chi _k\right\| ^{p_2}_{L^{q_2(\cdot )}}\right\} ^{p_1/p_2}\nonumber \\&\le \sum ^\infty _{k=-\infty } 2^{k\alpha p_1}\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](f)\chi _k\right\| ^{p_1}_{L^{q_2(\cdot )}} \nonumber \\&\le C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^{k-3}_{j=-\infty } |\lambda _j|\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)\chi _k\right\| _{L^{q_2(\cdot )}}\right) ^{p_1} \nonumber \\&\quad + \,C \sum ^\infty _{k=-\infty } 2^{k\alpha p_1} \left( \sum ^\infty _{j=k-2} |\lambda _j|\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)\chi _k\right\| _{L^{q_2(\cdot )}}\right) ^{p_1} \nonumber \\&=:{\mathrm{U_{1}+U_{2}}}. \end{aligned}$$

(4.10)

For ${\mathrm{U_{1}}}$, repeating this process which is similar to the one of estimating $\mu ^{\rho ,\,*}_{\Omega ,\,\lambda }(a_j)$ of Lemma 3.6 and generalized Hölder’s inequality (Lemma 3.3), we know that, for any $x\in A_k$, $0<\varepsilon < \min \{1,\,n/m\}$, $b\in {\mathrm{Lip_\varepsilon }}$ and $m\in {{\mathbb {Z}}}_+$,

$$\begin{aligned} {[}b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)(x)&\le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}\frac{2^{j\xi }}{2^{k(n+\xi -m\varepsilon )}}\int _{B_j}\left| a_j(z)\right| \,{\mathrm{d}}z \nonumber \\&\le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}2^{-k(n-m\varepsilon )}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q_1(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q_1'(\cdot )}}. \end{aligned}$$

(4.11)

Take $\nu =m\varepsilon$, since

$$\begin{aligned} I_{m\varepsilon }\left( \chi _{B_k}\right) (x)\ge \int _{B_k}\frac{{\mathrm{d}}y}{|x-y|^{n-m\varepsilon }}\chi _{B_k}(x)\ge C2^{km\varepsilon }\chi _{B_k}(x), \end{aligned}$$

From this, Lemmas 4.2, 3.4 and Proposition 2.1, it follows that, for any $x\in A_k$, $0<\varepsilon < \min \{1,\,n/m\}$, $b\in {\mathrm{Lip_\varepsilon }}$ and $m\in {{\mathbb {Z}}}_+$,

$$\begin{aligned}&\left\| [b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }](a_j)\chi _k\right\| _{L^{q_2(\cdot )}} \nonumber \\&\quad \le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}2^{-k(n-m\varepsilon )}2^{(j-k)\xi } \left\| a_j\right\| _{L^{q_1(\cdot )}}\left\| \chi _{B_j}\right\| _{L^{q_1'(\cdot )}}\left\| \chi _k\right\| _{L^{q_2(\cdot )}} \nonumber \\&\quad \le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}2^{-k(n-m\varepsilon )}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q_1(\cdot )}} \left\| \chi _{B_j}\right\| _{L^{q_1'(\cdot )}}\left\| \chi _{B_k}\right\| _{L^{q_2(\cdot )}} \nonumber \\&\quad \le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}2^{-kn}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q_1(\cdot )}} \left\| \chi _{B_j}\right\| _{L^{q_1'(\cdot )}}\left\| I_{m\varepsilon }(\chi _{B_k})\right\| _{L^{q_2(\cdot )}} \nonumber \\&\quad \le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}2^{(j-k)\xi }\left\| a_j\right\| _{L^{q_1(\cdot )}}\frac{\left\| \chi _{B_j}\right\| _{L^{q'_1(\cdot )}}}{\left\| \chi _{B_k}\right\| _{L^{q'_1(\cdot )}}} \nonumber \\&\quad \le C \Vert b\Vert ^{m}_{\mathrm{Lip_\varepsilon }}2^{(j-k)(n\delta _2+\xi )}\left\| a_j\right\| _{L^{q_1(\cdot )}}. \end{aligned}$$

(4.12)

By (4.12), $n\delta _2+\xi -\alpha >0$ and repeating this process which is similar to the one of proving ${\mathrm{I_{1}}}$ of Theorem 2.8, we know that

$$\begin{aligned} {\mathrm{U_{1}}}\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1}. \end{aligned}$$

For ${\mathrm{U_{2}}}$, from $[b^m,\,\mu ^{\rho ,\, *}_{\Omega ,\, \lambda }]$ is bounded from $L^{q_1(\cdot )}$ to $L^{q_2(\cdot )}$ and repeating this process which is similar to the one of proving ${\mathrm{I_{2}}}$ of Theorem 2.8, it follows that

$$\begin{aligned} {\mathrm{U_{2}}}\le C \sum ^\infty _{j=-\infty } |\lambda _j|^{p_1} . \end{aligned}$$

Combining the estimates of ${\mathrm{U_1}}$ and ${\mathrm{U_2}}$, we obtain the desired inequality. This finishes the proof of Theorem 2.12. $\square$