Rough singular integrals associated to polynomial curves

Abstract

In this paper, the authors establish the boundedness of singular integral operators associated to polynomial curves as well as the related maximal operators with rough kernels $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ and $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma >1$ on the Triebel–Lizorkin spaces. It should be pointed out that the bounds are independent of the coefficients of the polynomials in the definition of the operators. The main results of this paper not only improve and generalize essentially some known results but also complement some recent boundedness results.

Introduction

It is well known that the Triebel–Lizorkin spaces contain many important function spaces such as Lebesgue spaces, Hardy spaces, Sobolev spaces, and Lipschitz spaces. Over the last several years, a considerable amount of attention has been given to investigate the boundedness for singular integral operators with various rough kernels on the Triebel–Lizorkin spaces. Particularly, many scholars devoted to studying the bounds for singular integral operators with singularity along various sets under the rough kernels $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ and $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma >1$. For example, see [10] for the polynomial mappings, [29] for the homogeneous mappings, [27] for the surfaces to revolution. It is unknown whether the singular integral operators associated to polynomial curves under the rough kernels are bounded on the Triebel–Lizorkin spaces. The main purpose of this paper is to address the question. In addition, we establish the bounds for the related maximal singular integral operators on the Lebesgue and Triebel–Lizorkin spaces.

Before stating our main results, let us recall some pertinent definitions, notations, and backgrounds. Let $n\geq 2$ be an integer and let ${\mathrm{S}}^{n-1}$ denote the unit sphere in $\mathbb{R}^{n}$ equipped with the normalized Lebesgue measure dσ. Let $\Omega \in L^{1}({\mathrm{S}}^{n-1})$ be a homogeneous function of degree zero on $\mathbb{R}^{n}$ and satisfy

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

(1.1)

The singular integral operator $T_{h,\Omega }$ is defined as

$$\begin{aligned} T_{h,\Omega }f(x):={\mathrm{p.v.}} \int _{\mathbb{R}^{n}} \frac{\Omega (y/ \vert y \vert )h( \vert y \vert )}{ \vert y \vert ^{n}}f(x-y)\,dy, \end{aligned}$$

(1.2)

where $f\in \mathcal{S}(\mathbb{R}^{n})$ (the Schwartz class) and $h\in \Delta _{1}(\mathbb{R}_{+})$. For $\gamma >0$, the notation $\Delta _{\gamma }(\mathbb{R}_{+})$ denotes the set of all measurable functions h on $\mathbb{R}_{+}:=(0,\infty )$ satisfying

$$\begin{aligned} \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}=\sup_{R>0} \biggl( \frac{1}{R} \int _{0}^{R} \bigl\vert h(t) \bigr\vert ^{\gamma }\,dt \biggr)^{1/\gamma }< \infty. \end{aligned}$$

It is not difficult to see that $L^{\infty }(\mathbb{R}_{+})=\Delta _{\infty } (\mathbb{R}_{+}) \subsetneq \Delta _{\gamma _{2}}(\mathbb{R}_{+})\subsetneq \Delta _{ \gamma _{1}}(\mathbb{R}_{+})$ for $0<\gamma _{1}<\gamma _{2}<\infty$. For the sake of simplicity, we denote $T_{h,\Omega }=T_{\Omega }$ when $h\equiv 1$.

The theory of singular integral originated in Calderón and Zygmund’s work [4] in which they used the rotation method to establish the $L^{p}(\mathbb{R}^{n}) (1< p<\infty )$ of $T_{\Omega }$ if $\Omega \in L\log L({\mathrm{S}}^{n-1})$. Since then, more and more scholars have been devoted to studying the boundedness of singular integrals with various rough kernels. Particularly, Coifman and Weiss [12] proved that $T_{\Omega }$ is of type $(p, p)$ for $1< p<\infty$ if $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ (see also [15]). It was remarkable that $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ turned out to be the weakest size condition for the $L^{p}$ boundedness of $T_{\Omega }$ up to now. Later on, an active extension to the theory was due to Fefferman [23] who discovered that the Calderón–Zygmund rotation method is no longer available if $T_{h,\Omega }$ is also rough in the radial direction, for instance $h\in L^{\infty }(\mathbb{R}_{+})$, so that new methods must be addressed. More precisely, Fefferman [23] showed that $T_{h,\Omega }$ is of type $(p, p)$ for $1< p<\infty$ if $\Omega \in \mathrm{Lip}_{\alpha }({\mathrm{S}}^{n-1})$ for some $\alpha >0$ and $h\in L^{\infty }(\mathbb{R}_{+})$. Fefferman’s result was later improved by Namazi [32] by assuming $\Omega \in L^{q}({\mathrm{S}}^{n-1})$ for some $q>1$ instead of $\Omega \in \mathrm{Lip}_{\alpha }({\mathrm{S}}^{n-1})$. Meanwhile, Duoandikoetxea and Rubio de Francia [16] used the Littlewood–Paley theory to improve the results to the case $\Omega \in L^{q}({\mathrm{S}}^{n-1})$ for any $q>1$ and $h\in \Delta _{2}(\mathbb{R}_{+})$. The boundedness for rough singular integral operators on Tribel–Lizorkin spaces has also been studied extensively by many authors. In 2002, Chen, Fan, and Ying [5] first showed that $T_{\Omega }$ is bounded on $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$ if $\Omega \in L^{r}({\mathrm{S}}^{n-1})$ for some $r>1$. Later on, the result was extended and improved by many authors. For example, see [2, 6] for the case $\Omega \in \mathcal{F}_{\beta }({\mathrm{S}}^{n-1})$ (the Grafakos–Stefanov function class in [25]), [9, 10] for the case $\Omega \in H^{1}({\mathrm{S}}^{n-1})$.

For the operators $T_{\Omega }$ and $T_{h,\Omega }$, the singularities are along the diagonal $\{x=y\}$. However, many problems in analysis have led one to consider singular integral operators with singularity along more general sets. One of the principal motivations for the study of such operators is the requirements of several complex variables and large classes of “subelliptic” equations (see [37, 39]). So more and more scholars are devoted to studying the $L^{p}$ bounds for rough singular integral operators with singularity along various sets. For example, see [3, 22, 34] for polynomial mappings, [17, 19] for real-analytic submanifolds, [11, 28] for homogeneous mappings, [1, 18, 20, 26] for polynomial curves. Other interesting works can be found in [7, 8, 35, 36, 42], among others.

In this paper we focus on the singular integrals associated to polynomial curves with rough kernels. Let $h, \Omega$ be given as in (1.2) and P be a real polynomial on $\mathbb{R}$ satisfying $P(0)=0$. For a function $\varphi:\mathbb{R}_{+}\rightarrow \mathbb{R}$, we define the singular integral operator associated to polynomial compound curves $\{P(\varphi (|y|))y/|y|; y\in \mathbb{R}^{n}\}$ by

$$\begin{aligned} T_{h,\Omega,P,\varphi }f(x):={\mathrm{p.v.}} \int _{\mathbb{R}^{n}}f\bigl(x-P\bigl( \varphi \bigl( \vert y \vert \bigr)\bigr)y/ \vert y \vert \bigr)\frac{\Omega (y/ \vert y \vert )h( \vert y \vert )}{ \vert y \vert ^{n}}\,dy, \end{aligned}$$

(1.3)

where $f\in \mathcal{S}(\mathbb{R}^{n})$. When $\varphi (t)\equiv t$, we denote $T_{h,\Omega,P,\varphi }=T_{h,\Omega,P}$. Particularly, $T_{h,\Omega,P}=T_{h, \Omega }$ when $P(t)\equiv t$. In 1997, Fan and Pan [20] first established the $L^{2}$ boundedness for $T_{h,\Omega,P}$ if $h\in L^{\infty }(\mathbb{R}_{+})$ and $\Omega \in H^{1}({\mathrm{S}}^{n-1})$. Subsequently, Al-Hasan and Pan [1] improved the result by establishing the following.

Theorem A

([1])

Let $h\in L^{\infty }(\mathbb{R}_{+})$ and $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ satisfy (1.1). Then, for $1< p<\infty$, there exists a constant $C>0$ independent of $h, \Omega$ and the coefficients of P such that

$$\begin{aligned} \Vert T_{h,\Omega,P}f \Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert h \Vert _{L^{\infty }( \mathbb{R}_{+})} \Vert \Omega \Vert _{H^{1}({\mathrm{S}}^{n-1})} \Vert f \Vert _{L^{p}( \mathbb{R}^{n})}, \quad\forall f\in L^{p}\bigl(\mathbb{R}^{n} \bigr). \end{aligned}$$

Later on, the $L^{p}$ mapping properties for $T_{h,\Omega,P}$ have been investigated by many authors. For example, see [18] for the case $h\equiv 1$ and $\Omega \in \mathcal{F}_{\beta }({\mathrm{S}}^{n-1})$, [26] for the case $\Omega \in L\log L({\mathrm{S}}^{n-1})$.

Based on (2.4) and Theorem A, a natural question is the following.

Question 1.1

Is $T_{h,\Omega,P}$ bounded on $F_{\alpha }^{p,q}(\mathbb{R}^{n})$ if $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma \in (1,\infty ]$ and $\Omega \in H^{1}({\mathrm{S}}^{n-1})$?

Our investigation will not only address this question, but also deal with a more general class of operators. More specifically, we have the following result.

Theorem 1.1

Let P be a real polynomial on $\mathbb{R}$ satisfying $P(0)=0$ and $\varphi \in \mathfrak{F}_{1}$ or $\mathfrak{F}_{2}$. Here, $\mathfrak{F}_{1}$ (resp., $\mathfrak{F}_{2}$) is the set of all functions $\phi:\mathbb{R}_{+}\rightarrow \mathbb{R}$ satisfying the following condition $(a)$ (resp., $(b)$):

• (a) ϕ is an increasing $\mathcal{C}^{1}$ function such that $t\phi '(t)\geq C_{\phi }\phi (t)$ and $\phi (2t)\leq c_{\phi }\phi (t)$ for all $t>0$, where $C_{\phi }$ and $c_{\phi }$ are independent of t.

• (b) ϕ is a decreasing $\mathcal{C}^{1}$ function such that $t\phi '(t)\leq - C_{\phi }\phi (t)$ and $\phi (t)\leq c_{\phi }\phi (2t)$ for all $t>0$, where $C_{\phi }$ and $c_{\phi }$ are independent of t.

  Suppose that $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ satisfies (1.1) and $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma \in (1,\infty ]$. Then

  • (i) For $\alpha \in \mathbb{R}$ and $(1/p,1/q)\in \mathcal{R}_{\gamma }$, there exists a constant $C>0$ independent of $h, \gamma, \Omega$ and the coefficients of P such that

    $$\begin{aligned} \Vert T_{h,\Omega,P,\varphi }f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Vert \Omega \Vert _{H^{1}({ \mathrm{S}}^{n-1})} \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})}. \end{aligned}$$

    Here, $\mathcal{R}_{\gamma }$ is the interior of the convex hull of three squares $(\frac{1}{2},\frac{1}{2}+\frac{1}{\max \{2,\gamma '\}})^{2}$, $(\frac{1}{2} -\frac{1}{\max \{2,\gamma '\}},\frac{1}{2})^{2}$, and $(\frac{1}{2\gamma }, 1-\frac{1}{2\gamma })^{2}$.

  • (ii) For $\alpha >0$ and $(1/p,1/q)\in \mathcal{R}_{\gamma }$, there exists a constant $C>0$ independent of $h, \gamma, \Omega$ and the coefficients of P such that

    $$\begin{aligned} \Vert T_{h,\Omega,P,\varphi }f \Vert _{F_{\alpha }^{p,q}(\mathbb{R}^{n})}\leq C \gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Vert \Omega \Vert _{H^{1}({ \mathrm{S}}^{n-1})} \Vert f \Vert _{F_{\alpha }^{p,q}(\mathbb{R}^{n})}. \end{aligned}$$

Remark 1.1

There are some model examples in the class $\mathfrak{F}_{1}$ such as $t^{\alpha } (\alpha >0), t^{\alpha }(\ln (1+t))^{\beta } (\alpha, \beta >0), t\ln \ln (e+t)$, real-valued polynomials P on $\mathbb{R}$ with positive coefficients and $P(0)=0$, and so on. We now give examples in the class $\mathfrak{F}_{2}$ such as $t^{\delta }(\delta <0)$ and $t^{-1}\ln (1+1/t)$. It was pointed out in [26] that for $\varphi \in \mathfrak{F}_{1}$ (or $\mathfrak{F}_{2}$) there exists a constant $B_{\varphi }>1$ such that $\varphi (2t)\geq B_{\varphi }\varphi (t)$ (or $\varphi (t)\geq B_{\varphi }\varphi (2t)$).

Remark 1.2

(i) It is clear that $\mathcal{R}_{\gamma _{1}} \subsetneq \mathcal{R}_{\gamma _{2}}$ for $\gamma _{1}<\gamma _{2}$ and $\mathcal{R}_{\infty }=(0,1)\times (0,1)$. In view of (2.4), we see that Theorem 1.1 essentially improved and generalized Theorem A.

(ii) Our methods used to deal with Fourier transform estimates of some measures are different from those in the proof of Theorem A. In fact, the authors in [1] used the $TT^{*}$ method to prove Theorem A. However, the $TT^{*}$ method is not needed in the proof of Theorem 1.1.

(iii) Part (i) of Theorem 1.1 improved and generalized Theorem 1 in [9], in which the authors showed that $T_{h,\Omega }$ is bounded on $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$ for $\alpha \in \mathbb{R}$ and $1< p, q<\infty$, provided that $h\in L^{\infty }(\mathbb{R}_{+})$ and $\Omega \in H^{1}({\mathrm{S}}^{n-1})$.

(iv) Theorem 1.1 is new, even in the special case $h\equiv 1$ or $\alpha =0$, $q=2$, $\varphi (t)\equiv t$, or $P(t)\equiv t$.

The second motivation of this paper is concerned with the $L^{p}$ boundedness of maximal truncated singular integrals associated to polynomial curves. Let $h, \Omega, P, \varphi$ be given as in (1.3). The maximal truncated singular integral operator $T_{h,\Omega,P,\varphi }^{*}$ is defined by

$$\begin{aligned} T_{h,\Omega,P,\varphi }^{*}f(x):=\sup_{\epsilon >0} \biggl\vert \int _{ \vert y \vert >\epsilon }f\bigl(x-P\bigl(\varphi \bigl( \vert y \vert \bigr)\bigr)y/ \vert y \vert \bigr) \frac{\Omega (y/ \vert y \vert )h( \vert y \vert )}{ \vert y \vert ^{n}}\,dy \biggr\vert , \end{aligned}$$

(1.4)

where $f\in \mathcal{S}(\mathbb{R}^{n})$. The type of operator $T_{h,\Omega,P,\varphi }^{*}$ was first studied by Fan, Guo, and Pan [18] who proved that $T_{h,\Omega,P,\varphi }^{*}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for $(2\beta -1)/(2\beta -2)< p<2\beta -1$ if $h\equiv 1$, $\varphi (t)\equiv t$, and $\Omega \in \mathcal{F}_{\beta }({\mathrm{S}}^{n-1})$ for some $\beta >3/2$. Recently, Liu [26] proved that $T_{h,\Omega,P, \varphi }^{*}$ is of type $(p, p)$ for $1< p<\infty$, provided that $\varphi \in \mathfrak{F}_{1}$ or $\mathfrak{F}_{2}$, $\Omega \in L\log L({\mathrm{S}}^{n-1})$ and h satisfies certain radial condition.

Based on (2.1), (2.2) and the results related to $T_{h,\Omega,P, \varphi }^{*}$, a natural question is the following.

Question 1.2

Is $T_{h,\Omega,P,\varphi }^{*}$ bounded on $L^{p}(\mathbb{R}^{n})$ for some $p>1$ under the same conditions of Theorem 1.1?

This question can be addressed by the following.

Theorem 1.2

Let $P, \varphi$ be given as in Theorem 1.1. Suppose that $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ satisfies (1.1) and $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma \in (4/3,\infty ]$. Then there exists a constant $C>0$ independent of $h, \gamma, \Omega$ and the coefficients of P such that

$$\begin{aligned} \bigl\Vert T_{h,\Omega,P,\varphi }^{*}f \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Vert \Omega \Vert _{H^{1}({\mathrm{S}}^{n-1})} \Vert f \Vert _{L^{p}(\mathbb{R}^{n})}, \quad\forall f\in L^{p} \bigl(\mathbb{R}^{n}\bigr). \end{aligned}$$

Here, $p\in (\gamma ',\infty )$ if $\gamma \geq 2$ or $p\in (\gamma ', {2\gamma '}/{(\gamma '-2)})$ if $\gamma \in (4/3,2)$.

Remark 1.3

Theorem 1.2 is new, even in the special case $h\equiv 1$ or $\varphi (t)\equiv t$. It is unknown whether the operator $T_{h,\Omega,P,\varphi }$ appearing in Theorem 1.2 is bounded on $L^{p}(\mathbb{R}^{n})$ for some $p>1$ if $\gamma \in (1,4/3]$, even in the special case $\varphi (t)=t$, which is very interesting.

The third motivation of this paper is concerned with the boundedness of maximal truncated singular integrals associated to polynomial curves on Triebel–Lizorkin spaces. The first work related to the boundedness for maximal singular integral operator on Triebel–Lizorkin spaces was due to Zhang and Chen [43], who showed that the maximal singular integral operator is bounded on $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$ and $F_{\alpha }^{p,q}(\mathbb{R}^{n})$ for $0<\alpha <1$ and $1< p, q<\infty$ by assuming that $\Omega \in H^{1}({\mathrm{S}}^{n-1})$. Recently, Liu, Xue, and Yabuta [30] established the boundedness for the maximal singular integral operators associated to polynomial mappings on Triebel–Lizorkin spaces under the conditions $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ with some $\gamma >1$ and $\Omega \in L\log L({\mathrm{S}}^{n-1})$. Very recently, the authors [31] obtained the boundedness for $T_{h,\Omega,P, \varphi }^{*}$ on Triebel–Lizorkin spaces, provided that $h\equiv 1$, $\Omega \in \mathcal{F}_{\beta }({\mathrm{S}}^{n-1})$ with some $\beta >3/2$ and $\varphi \in \mathfrak{F}_{3}$, where $\mathfrak{F}_{3}$ is the set of all functions ϕ satisfying the following conditions:

(a) ϕ is a positive increasing function on $(0,\infty )$ such that $t^{\delta }\phi '(t)$ is monotonic on $(0,\infty )$ for some $\delta \in \mathbb{R}$;

(b) There exist positive constants $C_{\phi }$ and $c_{\phi }$ such that $t\phi '(t)\geq C_{\phi }\phi (t)$ and $\phi (2t)\leq c_{\phi }\phi (t)$ for all $t>0$.

It is clear that $\mathfrak{F}_{3}\subsetneq \mathfrak{F}_{1}$. There are some model examples for the class $\mathfrak{F}_{3}$ such as $t^{\alpha } (\alpha >0)$, $t^{\beta }\ln (1+t) (\beta \geq 1)$, $t\ln \ln (e+t)$, real-valued polynomials P on $\mathbb{R}$ with positive coefficients and $P(0)=0$ and so on.

Based on the above, it is natural to ask the following question.

Question 1.3

Is $T_{h,\Omega,P,\varphi }^{*}$ defined in (1.4) bounded on the Triebel–Lizorkin spaces if $h\equiv 1$ and $\Omega \in H^{1}({\mathrm{S}}^{n-1})$?

Our next result will give a positive answer to Question 1.3.

Theorem 1.3

Let P be a real polynomial on $\mathbb{R}$ satisfying $P(0)=0$ and $\varphi \in \mathfrak{F}_{3}$. Suppose that $h\equiv 1$ and $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ satisfies (1.1). Then, for $0<\alpha <1$ and $1< p, q<\infty$, there exists a constant $C>0$ independent of Ω and the coefficients of P such that

$$\begin{aligned} &\bigl\Vert T_{h,\Omega,P,\varphi }^{*}f \bigr\Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \leq C \Vert \Omega \Vert _{H^{1}({\mathrm{S}}^{n-1})} \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}( \mathbb{R}^{n})},\quad \forall f\in \dot{F}_{\alpha }^{p,q}\bigl(\mathbb{R}^{n} \bigr);\\ &\bigl\Vert T_{h,\Omega,P,\varphi }^{*}f \bigr\Vert _{F_{\alpha }^{p,q}(\mathbb{R}^{n})} \leq C \Vert \Omega \Vert _{H^{1}({\mathrm{S}}^{n-1})} \Vert f \Vert _{F_{\alpha }^{p,q}( \mathbb{R}^{n})},\quad \forall f\in F_{\alpha }^{p,q}\bigl(\mathbb{R}^{n} \bigr). \end{aligned}$$

Moreover, both $T_{h,\Omega,P,\varphi }^{*}:F_{\alpha }^{p,q}(\mathbb{R}^{n}) \rightarrow \dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$ and $T_{h,\Omega,P, \varphi }^{*}:F_{\alpha }^{p,q}(\mathbb{R}^{n}) \rightarrow F_{\alpha }^{p,q} (\mathbb{R}^{n})$ are continuous.

Remark 1.4

The boundedness part in Theorem 1.3 implies [43, Theorem 1.2] when $P(t)=\varphi (t)\equiv t$. It should be pointed out that Theorem 1.3 is new, even in the special case $\varphi (t)\equiv t$.

The paper is organized as follows. In Sect. 2 we present some preliminary definitions and lemmas, which are the main ingredients of proving Theorems 1.1–1.3. The proofs of Theorems 1.1–1.3 will be given in Sect. 3. It should be pointed out that the main methods and ideas employed in this paper are a combination of ideas and arguments from [1, 21, 22, 27, 30, 41]. However, some new techniques are needed in the main proofs. The new ideas invented in our proofs are to define suitable measures and to estimate them suitably.

Throughout the paper, for any $p\in [1,\infty ]$, we denote $p'$ by the conjugate index of p, which satisfies ${1}/{p}+{1}/{p'}=1$. Here, we set $1'=\infty$ and $\infty '=1$. The letter C or c, sometimes with certain parameters, will stand for positive constants not necessarily the same one at each occurrence, but are independent of the essential variables. In what follows, we set $\mathfrak{R}_{n}=\{\zeta \in \mathbb{R}^{n}; 1/2<|\zeta |\leq 1\}$. Let $\triangle _{\zeta }(f)$ be the difference of f for an arbitrary function f defined on $\mathbb{R}^{n}$ and $\zeta \in \mathbb{R}^{n}$, i.e., $\triangle _{\zeta }(f)(x)=f_{\zeta }(x)-f(x)$, where $f_{\zeta }(x)=f(x+\zeta )$. For any $t\in \mathbb{R}$, we set $\exp (t)=e^{-2\pi it}$. We also use the conventions $\sum_{i\in \emptyset }a_{i}=0$ and $\prod_{i\in \emptyset }a_{i}=1$.

Preliminary definitions and lemmas

Preliminary definitions

In this subsection we give the definitions of several rough kernels and their relationships.

Definition 2.1

(Hardy spaces)

The Hardy space $H^{1}({\mathrm{S}}^{n-1})$ is the set of all $L^{1}({\mathrm{S}}^{n-1})$ functions which satisfy $\|f\|_{H^{1}({\mathrm{S}}^{n-1})}<\infty$, where

$$\begin{aligned} \Vert \Omega \Vert _{H^{1}({\mathrm{S}}^{n-1})}:= \int _{{\mathrm{S}}^{n-1}}\sup_{0\leq r< 1} \biggl\vert \int _{{\mathrm{S}}^{n-1}}\Omega (\theta ) \frac{1-r^{2}}{ \vert rw-\theta \vert ^{n}}\,d\sigma (\theta ) \biggr\vert \,d\sigma (w). \end{aligned}$$

Definition 2.2

($L(\log L)^{\alpha }({\mathrm{S}}^{n-1})$ class)

The class $L(\log L)^{\alpha }({\mathrm{S}}^{n-1})$ for $\alpha >0$ denotes the class of all measurable functions Ω on ${\mathrm{S}}^{n-1}$ which satisfy

$$\begin{aligned} \Vert \Omega \Vert _{L(\log L)^{\alpha }({\mathrm{S}}^{n-1})}:= \int _{{\mathrm{S}}^{n-1}} \bigl\vert \Omega (\theta ) \bigr\vert \log ^{\alpha }\bigl( \bigl\vert \Omega (\theta ) \bigr\vert +2\bigr)\,d \sigma (\theta )< \infty. \end{aligned}$$

Definition 2.3

(Grafakos–Stefanov class)

The Grafakos–Stefanov class $\mathcal{F}_{\beta }({\mathrm{S}}^{n-1})$ for $\beta >0$ denotes the set of all integrable functions over ${\mathrm{S}}^{n-1}$ which satisfy the condition

$$\begin{aligned} \sup_{u\in {\mathrm{S}}^{n-1}} \int _{{\mathrm{S}}^{n-1}} \bigl\vert \Omega (v) \bigr\vert \biggl(\log ^{+}\frac{1}{ \vert u\cdot v \vert } \biggr)^{\beta }\,d\sigma (v)< \infty. \end{aligned}$$

We remark that $\mathcal{F}_{\beta }({\mathrm{S}}^{n-1})$ was introduced by Grafakos and Stefanov [25] in the study of the $L^{p}$ boundedness of singular integral operator with rough kernels.

The following inclusion relations are known:

$$\begin{aligned} &L^{r}\bigl({\mathrm{S}}^{n-1}\bigr)\subsetneq L(\log L)^{\beta _{1}}\bigl({\mathrm{S}}^{n-1}\bigr) \subsetneq L(\log L)^{\beta _{2}}\bigl({\mathrm{S}}^{n-1}\bigr) \quad{\text{for }} r>1 {\text{ and }} 0< \beta _{2}< \beta _{1}; \\ &L(\log L)^{\beta }\bigl({\mathrm{S}}^{n-1}\bigr)\subsetneq H^{1}\bigl({\mathrm{S}}^{n-1}\bigr) \subsetneq L^{1} \bigl({\mathrm{S}}^{n-1}\bigr) \quad{\text{for }} \beta \geq 1; \end{aligned}$$

(2.1)

$$\begin{aligned} &L(\log L)^{\beta }\bigl({\mathrm{S}}^{n-1}\bigr)\nsubseteq H^{1}\bigl({\mathrm{S}}^{n-1}\bigr) \nsubseteq L(\log L)^{\beta }\bigl({\mathrm{S}}^{n-1}\bigr) \quad{\text{for }} 0< \beta < 1; \\ &{\mathcal{F}}_{\beta _{1}}\bigl({\mathrm{S}}^{n-1}\bigr)\subsetneq { \mathcal{F}}_{ \beta _{2}}\bigl({\mathrm{S}}^{n-1}\bigr),\quad 0< \beta _{2}< \beta _{1}; \\ &\bigcup_{q>1}L^{q}\bigl({ \mathrm{S}}^{n-1}\bigr)\subsetneq {\mathcal{F}}_{\beta }\bigl({ \mathrm{S}}^{n-1}\bigr),\quad \beta >0; \\ &\bigcap_{\beta >1}{\mathcal{F}}_{\beta }\bigl({ \mathrm{S}}^{n-1}\bigr)\nsubseteq H^{1}\bigl({ \mathrm{S}}^{n-1}\bigr)\nsubseteq \bigcup_{\beta >1}{ \mathcal{F}}_{\beta }\bigl({ \mathrm{S}}^{n-1}\bigr). \end{aligned}$$

(2.2)

Let us present the definitions of Triebel–Lizorkin spaces.

Definition 2.4

(Triebel–Lizorkin spaces)

Let $\mathcal{S}'(\mathbb{R}^{n})$ be the tempered distribution class on $\mathbb{R}^{n}$. For $\alpha \in \mathbb{R}$ and $0< p, q\le \infty (p\neq \infty )$, we define the homogeneous Triebel–Lizorkin spaces $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$ by

$$\begin{aligned} \dot{F}_{\alpha }^{p,q}\bigl(\mathbb{R}^{n}\bigr):= \biggl\{ f\in \mathcal{S}' \bigl( \mathbb{R}^{n}\bigr): \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} = \biggl\Vert \biggl(\sum _{i\in \mathbb{Z}}2^{-i\alpha q} \vert \Psi _{i}*f \vert ^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}< \infty \biggr\} , \end{aligned}$$

(2.3)

where $\widehat{\Psi _{i}}(\xi )=\phi (2^{i}\xi )$ for $i\in \mathbb{Z}$ and $\phi \in \mathcal{C}_{c}^{\infty }(\mathbb{R}^{n})$ satisfies the conditions: $0\leq \phi (x)\leq 1$; $\operatorname{supp}(\phi )\subset \{x: 1/2\leq |x|\leq 2\}$; $\phi (x)>c>0$ if $3/5\leq |x|\leq 5/3$. The inhomogeneous versions of Triebel–Lizorkin spaces are denoted by $F_{\alpha }^{p,q}(\mathbb{R}^{n})$ and are obtained by adding the term $\|\Theta *f\|_{L^{p}(\mathbb{R}^{n})}$ to the right-hand side of (2.3) with $\sum_{i\in \mathbb{Z}}$ replaced by $\sum_{i\geq 1}$, where $\Theta \in \mathcal{S}(\mathbb{R}^{n})$, $\operatorname{supp}(\hat{\Theta })\subset \{\xi: |\xi |\leq 2\}$, $\hat{\Theta }(x)>c>0$ if $|x|\leq 5/3$.

The following properties are well known (see [24, 40]):

$$\begin{aligned} &\dot{F}_{0}^{p,2}\bigl(\mathbb{R}^{n} \bigr)=L^{p}\bigl(\mathbb{R}^{n}\bigr) \quad{\text{for }} 1< p< \infty; \end{aligned}$$

(2.4)

$$\begin{aligned} &F_{\alpha }^{p,q}\bigl(\mathbb{R}^{n}\bigr) \sim \dot{F}_{\alpha }^{p,q}\bigl( \mathbb{R}^{n}\bigr) \cap L^{p}\bigl(\mathbb{R}^{n}\bigr)\quad {\text{and}} \\ & \Vert f \Vert _{F_{\alpha }^{p,q}(\mathbb{R}^{n})}\sim \Vert f \Vert _{\dot{F}_{\alpha }^{p,q} (\mathbb{R}^{n})}+ \Vert f \Vert _{L^{p}(\mathbb{R}^{n})}\quad {\text{for }} \alpha >0,1< p, q< \infty. \end{aligned}$$

(2.5)

Our next definition is concerned with the $H^{1}({\mathrm{S}}^{n-1})$ atom.

Definition 2.5

($H^{1}({\mathrm{S}}^{n-1})$ atom)

A function $a:{\mathrm{S}}^{n-1}\rightarrow \mathbb{C}$ is a $(1,\infty )$ atom if there exist $\vartheta \in {\mathrm{S}}^{n-1}$ and $\varrho \in (0,1]$ such that

$$\begin{aligned} &\operatorname{supp}(a)\subset {\mathrm{S}}^{n-1}\cap B(\vartheta,\varrho ),\quad {\text{where }} B(\vartheta,\varrho )=\bigl\{ y\in \mathbb{R}^{n}: \vert y-\vartheta \vert < \varrho \bigr\} ; \end{aligned}$$

(2.6)

$$\begin{aligned} &\Vert a \Vert _{L^{\infty }({\mathrm{S}}^{n-1})}\leq \varrho ^{-n+1}; \end{aligned}$$

(2.7)

$$\begin{aligned} &\int _{{\mathrm{S}}^{n-1}}a(y)\,d\sigma (y)=0. \end{aligned}$$

(2.8)

Preliminary lemmas

We start now the following atomic decomposition of $H^{1}({\mathrm{S}}^{n-1})$.

Lemma 2.1

([13, 14])

Let $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ satisfy (1.1). Then there exist a sequence of complex numbers $\{c_{j}\}_{j\geq 1}$ and a sequence of $(1,\infty )$ atoms $\{\Omega _{j}\}_{j\geq 1}$ such that

$$\begin{aligned} \Omega =\sum_{j=1}^{\infty }c_{j} \Omega _{j}, \quad \Vert \Omega \Vert _{H^{1}({\mathrm{S}}^{n-1})}\approx \sum _{j=1}^{\infty } \vert c_{j} \vert . \end{aligned}$$

In order to deal with certain estimates for Fourier transforms of some measures, we need the following properties for $(1,\infty )$ atom.

Lemma 2.2

([21])

Let $\zeta =(\zeta _{1},\ldots,\zeta _{n})\neq (0,\ldots,0)$ and $\zeta '= \zeta /|\zeta |=(\zeta _{1}',\ldots,\zeta _{n}')$. Suppose that $n\geq 3$ and $b(\cdot )$ is a $(1,\infty )$ atom on ${\mathrm{S}}^{n-1}$ supported in ${\mathrm{S}}^{n-1}\cap B(\zeta ',\varrho )$, where $\varrho \in (0,1]$. Let

$$\begin{aligned} &F_{b}(s)=\bigl(1-s^{2}\bigr)^{{(n-3)}/{2}}\chi _{(-1,1)}(s) \int _{{\mathrm{S}}^{n-2}}b\bigl(s,\bigl(1-s^{2} \bigr)^{{1}/{2}} \tilde{y}\bigr)\,d\sigma (\tilde{y}),\\ &G_{b}(s)=\bigl(1-s^{2}\bigr)^{{(n-3)}/{2}}\chi _{(-1,1)}(s) \int _{{\mathrm{S}}^{n-2}} \bigl\vert b\bigl(s,\bigl(1-s^{2} \bigr)^{{1}/{2}} \tilde{y}\bigr) \bigr\vert \,d\sigma ( \tilde{y}). \end{aligned}$$

Then there exists a positive constant C, independent of b, such that

$$\begin{aligned} &\operatorname{supp}(F_{b})\subset \bigl(\zeta _{1}'-2r \bigl(\zeta '\bigr),\zeta _{1}'+2r\bigl( \zeta '\bigr)\bigr),\\ & \operatorname{supp}(G_{b})\subset \bigl(\zeta _{1}'-2r\bigl(\zeta '\bigr), \zeta _{1}'+2r\bigl(\zeta '\bigr)\bigr);\\ &\Vert F_{b} \Vert _{L^{\infty }(\mathbb{R})}\leq C \bigl\vert r\bigl( \zeta '\bigr) \bigr\vert ^{-1}, \qquad \Vert G_{b} \Vert _{L^{\infty }(\mathbb{R})}\leq C \bigl\vert r\bigl(\zeta '\bigr) \bigr\vert ^{-1};\\ &\int _{\mathbb{R}}F_{b}(s)\,ds=0, \end{aligned}$$

where $r(\zeta ')=|\zeta |^{-1}|A_{\varrho }(\zeta )|$ and $A_{\varrho }(\zeta )= (\varrho ^{2}\zeta _{1},\varrho \zeta _{2}, \ldots,\varrho \zeta _{n})$.

Lemma 2.3

([21])

Let $\zeta =(\zeta _{1},\zeta _{2})\neq (0,0)$ and $\zeta '=\zeta /|\zeta |=(\zeta _{1}', \zeta _{2}')$. Suppose that $n=2$ and $b(\cdot )$ is a $(1,\infty )$ atom on ${\mathrm{S}}^{1}$ supported in ${\mathrm{S}}^{1}\cap B(\zeta ',\varrho )$, where $\varrho \in (0,1]$. Let

$$\begin{aligned} &F_{b}(s)=\bigl(1-s^{2}\bigr)^{-{1}/{2}}\chi _{(-1,1)}(s) \bigl(b\bigl(s,\bigl(1-s^{2}\bigr)^{{1}/{2}} \bigr)+b\bigl(s,-\bigl(1-s^{2}\bigr)^{{1}/{2}}\bigr) \bigr),\\ &G_{b}(s)=\bigl(1-s^{2}\bigr)^{-{1}/{2}}\chi _{(-1,1)}(s) \bigl( \bigl\vert b\bigl(s,\bigl(1-s^{2} \bigr)^{{1}/{2}}\bigr) \bigr\vert + \bigl\vert b\bigl(s,- \bigl(1-s^{2}\bigr)^{{1}/{2}}\bigr) \bigr\vert \bigr). \end{aligned}$$

Then there exists a positive constant C, independent of b, such that

$$\begin{aligned} &\operatorname{supp}(F_{b})\subset \bigl(\zeta _{1}'-2r \bigl(\zeta '\bigr),\zeta _{1}'+2r\bigl( \zeta '\bigr)\bigr),\qquad \operatorname{supp}(G_{b})\subset \bigl(\zeta _{1}'-2r\bigl(\zeta '\bigr), \zeta _{1}'+2r\bigl(\zeta '\bigr)\bigr);\\ &\int _{\mathbb{R}}F_{b}(s)\,ds=0;\\ &\Vert F_{b} \Vert _{L^{q}(\mathbb{R})}\leq C \bigl\vert r\bigl( \zeta '\bigr) \bigr\vert ^{-1+{1}/{q}}, \qquad \Vert G_{b} \Vert _{L^{q}(\mathbb{R})}\leq C \bigl\vert r\bigl(\zeta '\bigr) \bigr\vert ^{-1+{1}/{q}}, \end{aligned}$$

for some $q\in (1,2)$, where $r(\zeta ')=|\zeta |^{-1}|A_{\varrho }(\zeta )|$ and $A_{\varrho }(\zeta )=(\varrho ^{2}\zeta _{1},\varrho \zeta _{2})$.

The following oscillatory estimates are useful for our proofs.

Lemma 2.4

([33, Corollary, p. 186])

Let $l\in \mathbb{N}\setminus \{0\}$, $\{\mu _{i}\}_{i=1}^{l}\subset \mathbb{R}$, and $\{d_{i}\}_{i=1}^{l}$ be distinct positive real numbers. Let $\psi \in \mathcal{C}^{1}([0,1])$. Then there exists $C>0$ independent of $\{\mu _{j}\}_{j=1}^{l}$ such that

$$\begin{aligned} \biggl\vert \int _{\delta }^{\tau }\exp \bigl(\mu _{1}t^{d_{1}}+ \cdots +\mu _{l}t^{d_{l}}\bigr) \psi (t)\,dt \biggr\vert \leq C \vert \mu _{1} \vert ^{-\epsilon } \biggl( \bigl\vert \psi ( \tau ) \bigr\vert + \int _{ \delta }^{\tau } \bigl\vert \psi '(t) \bigr\vert \,dt \biggr) \end{aligned}$$

holds for $0\leq \delta <\tau \leq 1$ and $\epsilon =\min \{{1}/{d_{1}},{1}/{l}\}$.

Lemma 2.5

([31])

Let $\Phi (t)=t^{\alpha _{1}}+\mu _{2}t^{\alpha _{2}}+\cdots +\mu _{n}t^{ \alpha _{n}}$, where $\{\mu _{i}\}_{i=2}^{n}$ are real parameters, and $\{\alpha _{i}\}_{i=1}^{n}$ are distinct positive (not necessarily integer) exponents. Suppose that $\varphi \in \mathfrak{F}_{3}$ and $t^{\delta }\varphi '(t)$ is monotonic on $(0,\infty )$ for some $\delta \in \mathbb{R}$. Then, for any $r>0$ and $\lambda \neq 0$,

$$\begin{aligned} \biggl\vert \int _{r/2}^{r}\exp \bigl(\lambda \Phi \bigl(\varphi (t)\bigr)\bigr)\frac{dt}{t} \biggr\vert \leq C \bigl\vert \lambda \varphi (r)^{\alpha _{1}} \bigr\vert ^{-\epsilon }, \end{aligned}$$

with $\epsilon =\min \{1/\alpha _{1},1/n\}$. Here, $C>0$ is independent of $\{\mu _{i}\}_{i=2}^{n}$, but may depend on φ and δ.

We end this section by presenting a well-known result.

Lemma 2.6

([38, pp. 476–478])

Let $\mathcal{P}=(P_{1},\ldots,P_{d})$ with each $P_{i}$ being a real polynomial defined on $\mathbb{R}^{n}$. Then the maximal operator $M_{\mathcal{P}}$ defined by

$$\begin{aligned} M_{\mathcal{P}}f(x)=\sup_{r>0}\frac{1}{r^{n}} \biggl\vert \int _{ \vert t \vert \leq r}f\bigl(x-\mathcal{P}(t)\bigr)\,dt \biggr\vert \end{aligned}$$

satisfies

$$\begin{aligned} \Vert M_{\mathcal{P}}f \Vert _{L^{p}(\mathbb{R}^{d})}\leq C_{p} \Vert f \Vert _{L^{p}( \mathbb{R}^{d})},\quad \forall 1< p< \infty {\textit{ and }} f\in L^{p} \bigl( \mathbb{R}^{d}\bigr). \end{aligned}$$

Here, $C_{p}>0$ is independent of the coefficients of $\{P_{i}\}_{i=1}^{d}$ and f.

Proofs of Theorems 1.1–1.3

In this section we prove Theorems 1.1–1.3. In Sect. 3.1 we present some notation and lemmas, which are the main ingredients of proving Theorems 1.1–1.3. The proofs of Theorems 1.1–1.3 will be given in Sect. 3.2.

Some notation and lemmas

In what follows, let $N\in \mathbb{N}\setminus \{0\}$ and $P(t)=\sum_{i=1}^{N}a_{i}t^{i}$ with $a_{N}\neq 0$. Then there exist $0< l_{1}< l_{2}<\cdots <l_{\Lambda }=N$ such that $P(t)=\sum_{i=1}^{\Lambda }a_{l_{i}}t^{l_{i}}$ with $a_{l_{i}}\neq 0$ for all $1\leq i\leq \Lambda$. Set

$$\begin{aligned} P_{0}(t)=0,\qquad P_{s}(t)=\sum_{i=1}^{s}a_{l_{s}}t^{l_{s}},\quad 1\leq s\leq \Lambda. \end{aligned}$$

(3.1)

It is clear that $P(t)=P_{\Lambda }(t)$ and $l_{s}\geq s$ for $1\leq s\leq \Lambda$.

Let $h, \Omega$ be given as in (1.2). For $0\leq s\leq \Lambda$, $y, \xi \in \mathbb{R}^{n}$, a vector $\theta \in {\mathrm{S}}^{n-1}$, and a function $\varphi:[0,\infty )\rightarrow \mathbb{R}$, we set

$$\begin{aligned} \Gamma _{s,\theta }(y,\xi )=\sum_{i=s+1}^{\Lambda }a_{l_{i}} \varphi \bigl( \vert y \vert \bigr)^{l_{i}}\theta \cdot \xi. \end{aligned}$$

Define the measures $\{\sigma _{h,\Omega,k,\theta,s}\}_{k\in \mathbb{Z}}$ and $\{|\sigma _{h,\Omega,k,\theta,s}|\}_{k\in \mathbb{Z}}$ by

$$\begin{aligned} &\widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi )= \int _{2^{k\gamma '}< \vert y \vert \leq 2^{(k+1)\gamma '}}\exp \bigl(P_{s}\bigl(\varphi \bigl( \vert y \vert \bigr)\bigr)y'\cdot \xi +\Gamma _{s, \theta }(y, \xi )\bigr)\frac{\Omega (y/ \vert y \vert )h( \vert y \vert )}{ \vert y \vert ^{n}}\,dy,\\ &\widehat{ \vert \sigma _{h,\Omega,k,\theta,s} \vert }(\xi )= \int _{2^{k\gamma '}< \vert y \vert \leq 2^{(k+1)\gamma '}}\exp \bigl(P_{s}\bigl(\varphi \bigl( \vert y \vert \bigr)\bigr)y'\cdot \xi +\Gamma _{s, \theta }(y, \xi )\bigr)\frac{ \vert \Omega (y/ \vert y \vert )h( \vert y \vert ) \vert }{ \vert y \vert ^{n}}\,dy, \end{aligned}$$

where $P_{s}$ is given as in (3.1). Note that $\Gamma _{s,\theta }(y,\xi )$ is independent of ${y}/{|y|}$. In view of (1.1), it is easy to see that

$$\begin{aligned} \sigma _{h,\Omega,k,\theta,0}(\xi )=0,\quad \forall k\in \mathbb{Z}, \xi \in \mathbb{R}^{n}. \end{aligned}$$

(3.2)

We have the following estimates.

Lemma 3.1

Let $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma \in (1,\infty ]$ and Ω be a $(1,\infty )$ atom satisfying (2.6)–(2.8) with $0<\varrho \leq 1$ and $\vartheta =\theta =(1,0,\ldots,0)\in {\mathrm{S}}^{n-1}$. Assume that $\varphi \in \mathfrak{F}_{1}$ or $\varphi \in \mathfrak{F}_{2}$. Then, for $1\leq s\leq \Lambda$ and $\xi =(\xi _{1},\ldots,\xi _{n})\neq (0,\ldots,0)$, there exists a constant $C>0$ independent of $h, \Omega, \gamma, \xi$ and $\{a_{l_{s}}\}_{s=1}^{\Lambda }$ such that

$$\begin{aligned} \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi )- \widehat{\sigma _{h,\Omega,k,\theta,s-1}}(\xi ) \bigr\vert \leq C\gamma ' \Vert h \Vert _{ \Delta _{\gamma }(\mathbb{R}_{+})}\min \bigl\{ 1,\varphi \bigl(2^{(k+1)\gamma '} \bigr)^{l_{s}} \bigl\vert L_{s}( \xi ) \bigr\vert \bigr\} , \end{aligned}$$

(3.3)

where

$$\begin{aligned} L_{s}(\xi )=\bigl(a_{l_{s}}\varrho ^{2}\xi _{1},a_{l_{s}}\varrho \xi _{2}, \ldots,a_{l_{s}} \varrho \xi _{n}\bigr). \end{aligned}$$

(3.4)

Proof

We only prove (3.3) for the case $\varphi \in \mathfrak{F}_{1}$ since another case $\varphi \in \mathfrak{F}_{2}$ is analogous. Fix $1\leq s\leq \Lambda$ and $\xi '=\xi /|\xi |=(\xi _{1}',\ldots, \xi _{n}')$. Let $\mathcal{O}$ be the rotation such that $\mathcal{O}(\xi ')=\vartheta$ and $\mathcal{O}^{-1}$ denote the inverse of $\mathcal{O}$. Then $\mathcal{O}^{2}(\xi ')=(\xi _{1}',\eta _{2}',\ldots,\eta _{n}')$. Let $Q_{n-1}$ be a rotation in $\mathbb{R}^{n-1}$ such that $Q_{n-1}(\xi _{2}',\ldots,\xi _{n}')=(\eta _{2}',\ldots,\eta _{n}')$ and R be a transformation by $R(z_{1},z_{2},\ldots,z_{n})=(z_{1},Q_{n-1}(z_{2},\ldots,z_{n}))$. Then, for any $y'=(u,y_{2}',\ldots,y_{n}')\in {\mathrm{S}}^{n-1}$, we have $\vartheta \cdot R(y')=\vartheta \cdot y'=u$ and $\Omega (\mathcal{O}^{-1}R(y'))$ is a $(1,\infty )$ atom with supported in ${\mathrm{S}}^{n-1}\cap B(\xi ',\varrho )$. By some changes of variables, we have

$$\begin{aligned} &\widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi ) \\ &\quad= \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \exp \Biggl( \sum _{i=s+1}^{\Lambda }a_{l_{i}}\varphi (t)^{l_{i}} \xi \cdot \theta \Biggr) \int _{{\mathrm{S}}^{n-1}}\Omega \bigl(y'\bigr)\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}}\xi \cdot y' \Biggr)\,d\sigma \bigl(y'\bigr)h(t)\frac{dt}{t} \\ &\quad= \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}}\exp \Biggl( \sum _{i=s+1}^{\Lambda }a_{l_{i}}\varphi (t)^{l_{i}} \vert \xi \vert \xi _{1}' \Biggr) \\ & \qquad{}\times \int _{{\mathrm{S}}^{n-1}}A\bigl(y'\bigr)\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi (t)^{l_{i}} \vert \xi \vert \xi '\cdot \mathcal{O}^{-1}R\bigl(y'\bigr) \Biggr)\,d\sigma \bigl(y'\bigr)h(t)\frac{dt}{t} \\ &\quad= \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}}\exp \Biggl( \sum _{i=s+1}^{\Lambda }a_{l_{i}}\varphi (t)^{l_{i}} \vert \xi \vert \xi _{1}' \Biggr) \int _{\mathbb{R}}F_{A}(u)\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,duh(t)\frac{dt}{t}, \end{aligned}$$

(3.5)

where $A(y')=\Omega (\mathcal{O}^{-1}R(y'))$ and $F_{A}$ is defined as in Lemma 2.2 (in case $n>2$) or Lemma 2.3 (in case $n=2$). Notice that $A(\cdot )$ is a $(1,\infty )$ atom with supported in $B(\xi ',\varrho )$. Invoking Lemmas 2.2 and 2.3, one finds that

$$\begin{aligned} &\operatorname{supp}(F_{A})\subset \bigl(\xi _{1}'-2r \bigl(\xi '\bigr),\xi _{1}'+2r\bigl(\xi '\bigr)\bigr). \end{aligned}$$

(3.6)

$$\begin{aligned} &\Vert F_{A} \Vert _{L^{\infty }(\mathbb{R})}\leq C \bigl\vert r\bigl(\xi '\bigr) \bigr\vert ^{-1}, \quad{\text{if }} n\geq 3; \end{aligned}$$

(3.7)

$$\begin{aligned} &\Vert F_{A} \Vert _{L^{q}(\mathbb{R})}\leq C \bigl\vert r\bigl(\xi '\bigr) \bigr\vert ^{-1+1/q},\quad {\text{if }} n=2 \end{aligned}$$

(3.8)

for some $q\in (1,2)$. Here, $r(\xi ')=|\xi |^{-1}L_{\varrho }(\xi )$, where $A_{\varrho }(\xi )=(\varrho ^{2}\xi _{1},\varrho \xi _{2},\ldots, \varrho \xi _{n})$ for $n\geq 3$ and $A_{\varrho }(\xi )=(\varrho ^{2}\xi _{1},\varrho \xi _{2})$ for $n=2$.

In view of (3.5) and (3.6),

$$\begin{aligned} & \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi )- \widehat{ \sigma _{h,\Omega,k,\theta,s-1}}(\xi ) \bigr\vert \\ &\quad= \Biggl\vert \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}}\exp \Biggl(\sum _{i=s+1}^{\Lambda }a_{l_{i}}\varphi (t)^{l_{i}} \vert \xi \vert \xi _{1}' \Biggr) \int _{\mathbb{R}}F_{A}(u)\exp \Biggl(\sum _{i=1}^{s-1}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr) \\ &\qquad{} \times \bigl(\exp \bigl(a_{l_{s}}\varphi (t)^{l_{s}} \vert \xi \vert \xi _{1}'\bigr)-\exp \bigl(a_{l_{s}}\varphi (t)^{l_{s}} \vert \xi \vert u\bigr)\bigr)\,duh(t) \frac{dt}{t} \Biggr\vert \\ &\quad\leq \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \int _{ \mathbb{R}} \bigl\vert F_{A}(u) \bigr\vert \min \bigl\{ 2,2\pi \varphi \bigl(2^{(k+1)\gamma '}\bigr)^{l_{s}} \vert a_{l_{s}} \xi \vert \bigl\vert \xi _{1}'-u \bigr\vert \bigr\} \,du \bigl\vert h(t) \bigr\vert \frac{dt}{t} \\ &\quad\leq \min \bigl\{ 2,4\pi \vert a_{l_{s}}\xi \vert r\bigl(\xi '\bigr)\varphi \bigl(2^{(k+1) \gamma '}\bigr)^{l_{s}}\bigr\} \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \bigl\vert h(t) \bigr\vert \frac{dt}{t} \int _{\mathbb{R}} \bigl\vert F_{A}(u) \bigr\vert \,du. \end{aligned}$$

(3.9)

From (3.7) and (3.8), one sees that there exists $C>0$ independent of $h, \Omega, \gamma$ such that

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert F_{A}(u) \bigr\vert \,du \leq C. \end{aligned}$$

(3.10)

Moreover, by Hölder’s inequality, one has

$$\begin{aligned} \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \bigl\vert h(t) \bigr\vert \frac{dt}{t}& = \sum_{i=0}^{[\gamma ']} \int _{2^{k \gamma '+i}}^{2^{k\gamma '+i+1}} \bigl\vert h(t) \bigr\vert \frac{dt}{t} \\ & \leq \sum_{i=0}^{[ \gamma ']} \biggl( \int _{2^{k\gamma '+i}}^{2^{k\gamma '+i+1}} \bigl\vert h(t) \bigr\vert ^{\gamma }\frac{dt}{t} \biggr)^{1/\gamma } \biggl( \int _{2^{k\gamma '+i}}^{2^{k \gamma '+i+1}}\frac{dt}{t} \biggr)^{1/\gamma '} \\ & \leq 2^{1/\gamma }\bigl(\bigl[\gamma '\bigr]+1\bigr) \Vert h \Vert _{ \Delta _{\gamma }(\mathbb{R}_{+})}(\ln 2)^{1/\gamma '}\leq 4\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}. \end{aligned}$$

(3.11)

Here, $[x]=\max \{k\in \mathbb{Z}:k\leq x\}$ for $x\in \mathbb{R}$. Finally, it follows from (3.9)–(3.11) that

$$\begin{aligned} \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi )- \widehat{\sigma _{h,\Omega,k,\theta,s-1}}(\xi ) \bigr\vert \leq C\gamma ' \Vert h \Vert _{ \Delta _{\gamma }(\mathbb{R}_{+})}\min \bigl\{ 1,\varphi \bigl(2^{(k+1)\gamma '} \bigr)^{l_{s}} \bigl\vert L_{s}( \xi ) \bigr\vert \bigr\} , \end{aligned}$$

where $C>0$ is independent of $h, \Omega, \gamma$. This proves (3.3) and completes the proof. □

Lemma 3.2

Let $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma \in (1,\infty ]$ and Ω be a $(1,\infty )$ atom satisfying (2.6)–(2.8) with $0<\varrho \leq 1$ and $\vartheta =\theta =(1,0,\ldots,0)\in {\mathrm{S}}^{n-1}$. Assume that $\varphi \in \mathfrak{F}_{1}$ or $\varphi \in \mathfrak{F}_{2}$. Then, for $1\leq s\leq \Lambda$ and $\xi =(\xi _{1},\ldots,\xi _{n})\neq (0,\ldots,0)$, there exist $\delta >0$ and $C>0$ independent of $h, \Omega, \gamma, \xi$, and $\{a_{l_{s}}\}_{s=1}^{\Lambda }$ such that

$$\begin{aligned} \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi ) \bigr\vert \leq C\gamma ' \Vert h \Vert _{ \Delta _{\gamma }(\mathbb{R}_{+})}\min \bigl\{ 1,\bigl(\varphi \bigl(2^{k\gamma '}\bigr)^{l_{s}} \bigl\vert L_{s}( \xi ) \bigr\vert \bigr)^{-1/(2l_{s}\gamma '\delta )}\bigr\} , \end{aligned}$$

(3.12)

where $L_{s}(\xi )$ is given as (3.4) and $\delta =1$ if $n\geq 3$ and $\delta >2$ if $n=2$.

Proof

We only prove (3.12) for the case $\varphi \in \mathfrak{F}_{1}$ since another case is analogous. By (3.5) and Hölder’s inequality, we have

$$\begin{aligned} & \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi ) \bigr\vert \\ &\quad\leq \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \Biggl\vert \int _{\mathbb{R}}F_{A}(u)\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert \bigl\vert h(t) \bigr\vert \frac{dt}{t} \\ &\quad\leq \biggl( \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \bigl\vert h(t) \bigr\vert ^{\gamma }\frac{dt}{t} \biggr)^{1/\gamma } \Biggl( \int _{2^{k\gamma '}}^{2^{(k+1) \gamma '}} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{\gamma '}\frac{dt}{t} \Biggr)^{1/ \gamma '} \\ &\quad\leq \biggl( \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \bigl\vert h(t) \bigr\vert ^{\gamma }\frac{dt}{t} \biggr)^{1/\gamma } \Vert F_{A} \Vert _{L^{1}(\mathbb{R})}^{ \max \{1-2/\gamma ',0\}}\bigl(\gamma ' \bigr)^{\max \{1/\gamma '-1/2,0\}} \\ &\qquad{} \times \Biggl( \int _{2^{k\gamma '}}^{2^{(k+1) \gamma '}} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \Biggr)^{\min \{1/ \gamma ',1/2\}}. \end{aligned}$$

(3.13)

Notice that

$$\begin{aligned} & \biggl( \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \bigl\vert h(t) \bigr\vert ^{\gamma }\frac{dt}{t} \biggr)^{1/\gamma } \\ &\quad\leq \Biggl(\sum_{i=0}^{[\gamma ']} \int _{2^{k \gamma '+i}}^{2^{k\gamma '+i+1}} \bigl\vert h(t) \bigr\vert ^{\gamma }\frac{dt}{t} \Biggr)^{1/ \gamma } \leq \bigl(2\bigl(\bigl[ \gamma '\bigr]+1\bigr) \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}^{\gamma } \bigr)^{1/\gamma }\leq \bigl(4\gamma '\bigr)^{1/\gamma } \Vert h \Vert _{\Delta _{\gamma }( \mathbb{R}_{+})}. \end{aligned}$$

This together with (3.10) and (3.13) implies

$$\begin{aligned} & \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi ) \bigr\vert \\ &\quad\leq \bigl(4\gamma '\bigr)^{ \max \{1/2,1/\gamma \}} \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \\ &\qquad{} \times \Biggl( \int _{2^{k\gamma '}}^{2^{(k+1) \gamma '}} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \Biggr)^{\min \{1/ \gamma ',1/2\}}. \end{aligned}$$

(3.14)

By some changes of variables and the properties for φ, we have

$$\begin{aligned} & \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \Biggl\vert \int _{ \mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \\ &\quad\leq \sum_{\mu =0}^{[\gamma ']} \int _{2^{k \gamma '+\mu }}^{2^{k\gamma '+\mu +1}} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \\ &\quad\leq \sum_{\mu =0}^{[\gamma ']} \int _{ \varphi (2^{k\gamma '+\mu })}^{\varphi (2^{k\gamma '+\mu +1})} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}t^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2} \frac{dt}{\varphi ^{-1}(t)\varphi '(\varphi ^{-1}(t))} \\ &\quad\leq \frac{1}{C_{\varphi }}\sum_{\mu =0}^{[ \gamma ']} \int _{\varphi (2^{k\gamma '+\mu })}^{\varphi (2^{k\gamma '+ \mu +1})} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}t^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \\ &\quad= \frac{1}{C_{\varphi }}\sum_{\mu =0}^{[\gamma ']} \int _{ \frac{\varphi (2^{k\gamma '+\mu +1})}{\varphi (2^{k\gamma '+\mu })}}^{1} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi \bigl(2^{k\gamma '+\mu +1}\bigr)^{l_{i}}t^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2} \frac{dt}{t} \\ &\quad\leq \frac{1}{C_{\varphi }}\sum_{\mu =0}^{[ \gamma ']} \int _{c_{\varphi }^{-1}}^{1} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi \bigl(2^{k\gamma '+\mu +1}\bigr)^{l_{i}}t^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \\ &\quad\leq \frac{1}{C_{\varphi }}\sum_{\mu =0}^{[ \gamma ']} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert F_{A}(u) \overline{F_{A}(v)} \bigr\vert \\ &\qquad{}\times \Biggl\vert \int _{c_{\varphi }^{-1}}^{1}\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi \bigl(2^{k\gamma '+\mu +1}\bigr)^{l_{i}}t^{l_{i}} \vert \xi \vert (u-v) \Biggr)\frac{dt}{t} \Biggr\vert \,du\,dv. \end{aligned}$$

(3.15)

Fix $\mu \in \{0,1,\ldots,[\gamma ']\}$, we get by Lemma 2.4 that

$$\begin{aligned} & \Biggl\vert \int _{c_{\varphi }^{-1}}^{1}\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi \bigl(2^{k\gamma '+\mu +1}\bigr)^{l_{i}}t^{l_{i}} \vert \xi \vert (u-v) \Biggr)\frac{dt}{t} \Biggr\vert \\ &\quad\leq C\min \bigl\{ 1,\bigl( \vert a_{l_{s}}\xi \vert \varphi \bigl(2^{k\gamma '+\mu +1}\bigr)^{l_{s}} \vert u-v \vert \bigr)^{-1/l_{s}} \bigr\} \\ &\quad\leq C\bigl( \vert a_{l_{s}}\xi \vert \varphi \bigl(2^{k\gamma '+i+1}\bigr)^{l_{i}} \vert u-v \vert \bigr)^{-1/(l_{s} \delta )}, \end{aligned}$$

(3.16)

where $\delta =1$ if $n\geq 3$ and $\delta =q'$ if $n=2$. Here, q is given as in the proof of Lemma 3.1. Here, the constant $C>0$ is independent of $u, v, \xi, \mu, k$, and $\{a_{l_{i}}\}_{i=1}^{s}$. In view of (3.15) with (3.16),

$$\begin{aligned} & \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \Biggl\vert \int _{ \mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}} \varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \\ &\quad\leq C\gamma '\bigl(\varphi \bigl(2^{k\gamma '} \bigr)^{l_{s}} \vert a_{l_{s}} \xi \vert \bigr)^{-1/(l_{s}\delta )} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert F_{A}(u) \overline{F_{A}(v)} \bigr\vert \vert u-v \vert ^{-1/(l_{s}\delta )} \,du\,dv. \end{aligned}$$

(3.17)

Define the function $b(u)=r(\xi ')F_{A}(r(\xi ')u+\xi _{1}')$. In view of (3.6)–(3.8) we see that $\operatorname{supp}(b)\subset (-2,2)$ and $\|b\|_{L^{\infty }(\mathbb{R})}\leq C$ for $n\geq 3$ and $\|b\|_{L^{q}(\mathbb{R})}\leq C$ for $n=2$. By some changes of variables,

$$\begin{aligned} & \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert F_{A}(u) \overline{F_{A}(v)} \bigr\vert \vert u-v \vert ^{-1/(l_{s}\delta )} \,du\,dv \\ &\quad= \bigl\vert r\bigl(\xi '\bigr) \bigr\vert ^{-1/(l_{s}\delta )} \int _{-2}^{2} \int _{-2}^{2} \bigl\vert b(u) \overline{b(v)} \bigr\vert \vert u-v \vert ^{-1/(l_{s}\delta )}\,du\,dv. \end{aligned}$$

(3.18)

When $n\geq 3$, by the fact $\|b\|_{L^{\infty }(\mathbb{R})}\leq C$ and $\delta =1$, we get

$$\begin{aligned} \int _{-2}^{2} \int _{-2}^{2} \bigl\vert b(u)\overline{b(v)} \bigr\vert \vert u-v \vert ^{-1/l_{s}}\,du\,dv \leq C \int _{-2}^{2} \int _{-2}^{2} \vert u-v \vert ^{-1/l_{s}} \,du\,dv\leq C. \end{aligned}$$

When $n=2$, by the fact $\|b\|_{L^{q}(\mathbb{R})}\leq C$ and Hölder’s inequality,

$$\begin{aligned} &\int _{-2}^{2} \int _{-2}^{2} \bigl\vert b(u)\overline{b(v)} \bigr\vert \vert u-v \vert ^{-1/(l_{s}q')}\,du\,dv\\ &\quad \leq C \Vert b \Vert _{L^{q}(\mathbb{R}^{n})}^{2} \biggl( \int _{-2}^{2} \int _{-2}^{2} \vert u-v \vert ^{-1/l_{s}} \,du\,dv \biggr)^{1/q'}\leq C. \end{aligned}$$

Therefore, we get from (3.18) that

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert F_{A}(u) \overline{F_{A}(v)} \bigr\vert \vert u-v \vert ^{-1/(sl_{s} \delta )} \,du\,dv\leq C \bigl\vert r\bigl(\xi '\bigr) \bigr\vert ^{-1/(l_{s}\delta )}. \end{aligned}$$

(3.19)

It follows from (3.19) and (3.17) that

$$\begin{aligned} &\int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \Biggl\vert \int _{\mathbb{R}}F_{A}(u) \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi (t)^{l_{i}} \vert \xi \vert u \Biggr)\,du \Biggr\vert ^{2}\frac{dt}{t} \\ &\quad\leq C\gamma '\bigl(\varphi \bigl(2^{k\gamma '} \bigr)^{l_{s}} \bigl\vert L_{s}( \xi ) \bigr\vert \bigr)^{-1/(l_{s}\delta )}, \end{aligned}$$

(3.20)

where $C>0$ is independent of $h, \Omega, \gamma, \varrho, \xi, k$ and $\{a_{l_{i}}\}_{i=1}^{s}$. In view of (3.20) and (3.14),

$$\begin{aligned} \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi ) \bigr\vert \leq C\gamma ' \Vert h \Vert _{ \Delta _{\gamma }(\mathbb{R}_{+})}\bigl(\varphi \bigl(2^{k\gamma '}\bigr)^{l_{s}} \bigl\vert L_{s}( \xi ) \bigr\vert \bigr)^{-\min \{1/\gamma ',1/2\}/(l_{s}\delta )}, \end{aligned}$$

(3.21)

where $C>0$ is independent of $h, \Omega, \gamma, \varrho, \xi, k$, and $\{a_{l_{i}}\}_{i=1}^{s}$. On the other hand, we get by (3.5), (3.10), and (3.11) that

$$\begin{aligned} \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi ) \bigr\vert \leq \Vert F_{A} \Vert _{L^{1}( \mathbb{R})} \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \bigl\vert h(t) \bigr\vert \frac{dt}{t}\leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}. \end{aligned}$$

(3.22)

Then (3.12) follows from (3.21) and (3.22). □

Lemma 3.3

Let $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some $\gamma \in (1,\infty ]$ and Ω be a $(1,\infty )$ atom satisfying (2.1)–(2.3) with $0<\varrho \leq 1$ and $\vartheta =\theta =(1, 0,\ldots,0)\in {\mathrm{S}}^{n-1}$. Let $\varphi \in \mathfrak{F}_{1}$ or $\varphi \in \mathfrak{F}_{2}$. Then, for $\gamma '< p<\infty$, there exists a constant $C>0$ independent of $h, \Omega, \gamma, \xi, \theta$, and $\{a_{l_{s}}\}_{s=1}^{\Lambda }$ such that

$$\begin{aligned} \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert \vert \sigma _{h,\Omega,k,\theta,0} \vert *f \bigr\vert \Bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C \gamma ' \Vert h \Vert _{\Delta _{\gamma }( \mathbb{R}_{+})} \Vert f \Vert _{L^{p}(\mathbb{R}^{n})}, \quad\forall f\in L^{p}\bigl( \mathbb{R}^{n} \bigr). \end{aligned}$$

(3.23)

Proof

We only consider the case $\varphi \in \mathfrak{F}_{1}$ since another one can be obtained similarly. Fix $k\in \mathbb{Z}$, by a change of variables,

$$\begin{aligned} \vert \sigma _{h,\Omega,k,\theta,0} \vert *f(x)&= \int _{2^{k \gamma '}< \vert y \vert \leq 2^{(k+1)\gamma '}}f \Biggl(x-\sum_{j=1}^{\Lambda }a_{l_{j}} \varphi \bigl( \vert y \vert \bigr)^{l_{j}}\theta \Biggr) \frac{ \vert \Omega (y)h( \vert y \vert ) \vert }{ \vert y \vert ^{n}}\,dy \\ & = \int _{2^{k\gamma '}}^{2^{(k+1) \gamma '}}f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}\varphi (t)^{l_{j}} \theta \Biggr) \bigl\vert h(t) \bigr\vert \frac{dt}{t} \Vert \Omega \Vert _{L^{1}({\mathrm{S}}^{n-1})}. \end{aligned}$$

It is clear that $\|\Omega \|_{L^{1}({\mathrm{S}}^{n-1})}\leq C$. By Hölder’s inequality and a change of variables, one has

$$\begin{aligned} & \bigl\vert \vert \sigma _{h,\Omega,k,\theta,0} \vert *f(x) \bigr\vert \\ &\quad\leq C \int _{2^{k\gamma '}}^{2^{(k+1)\gamma '}} \Biggl\vert f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}\varphi (t)^{l_{j}} \theta \Biggr) \Biggr\vert \bigl\vert h(t) \bigr\vert \frac{dt}{t} \\ &\quad\leq C\sum_{i=0}^{[\gamma ']} \int _{2^{k \gamma '+i}}^{2^{k\gamma '+i+1}} \Biggl\vert f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}\varphi (t)^{l_{j}}\theta \Biggr) \Biggr\vert \bigl\vert h(t) \bigr\vert \frac{dt}{t} \\ &\quad\leq C \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}\sum_{i=0}^{[\gamma ']} \Biggl( \int _{2^{k\gamma '+i}}^{2^{k\gamma '+i+1}} \Biggl\vert f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}\varphi (t)^{l_{j}} \theta \Biggr) \Biggr\vert ^{\gamma '}\frac{dt}{t} \Biggr)^{1/\gamma '} \\ &\quad= C \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}\sum_{i=0}^{[ \gamma ']} \Biggl( \int _{\varphi (2^{k\gamma '+i})}^{\varphi (2^{k \gamma '+i+1})} \Biggl\vert f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}t^{l_{j}} \theta \Biggr) \Biggr\vert ^{\gamma '} \frac{dt}{\varphi ^{-1}(t)\varphi '(\varphi ^{-1}(t))} \Biggr)^{1/ \gamma '} \\ &\quad\leq C \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})}\sum_{i=0}^{[\gamma ']} \Biggl( \int _{\varphi (2^{k\gamma '+i})}^{ \varphi (2^{k\gamma '+i+1})} \Biggl\vert f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}t^{l_{j}} \theta \Biggr) \Biggr\vert ^{\gamma '}\frac{dt}{t} \Biggr)^{1/\gamma '} \\ &\quad\leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Biggl(\sup _{r>0}\frac{1}{r} \int _{ \vert t \vert \leq r} \Biggl\vert f \Biggl(x- \sum _{j=1}^{\Lambda }a_{l_{j}}t^{l_{j}}\theta \Biggr) \Biggr\vert ^{ \gamma '}\,dt \Biggr)^{1/\gamma '}. \end{aligned}$$

It follows that

$$\begin{aligned} \sup_{k\in \mathbb{Z}} \bigl\vert \vert \sigma _{h,\Omega,k,\theta,0} \vert *f(x) \bigr\vert \leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Biggl(\sup_{r>0}\frac{1}{r} \int _{ \vert t \vert \leq r} \Biggl\vert f \Biggl(x-\sum _{j=1}^{\Lambda }a_{l_{j}}t^{l_{j}}\theta \Biggr) \Biggr\vert ^{\gamma '}\,dt \Biggr)^{1/ \gamma '}. \end{aligned}$$

This together with Lemma 2.6 yields (3.23). □

The following result is the main ingredient of proving Theorem 1.2.

Lemma 3.4

Let $A>0$, $\Lambda \in \mathbb{N}\setminus \{0\}$ and $\{\sigma _{k,s}:0\leq s\leq \Lambda {\textit{ and }}k\in \mathbb{Z}\}$ be a family of uniformly bounded Borel measures on $\mathbb{R}^{n}$ with $\sigma _{k,0}(\xi )=0$ for every $k\in \mathbb{Z}$ and $\xi \in \mathbb{R}^{n}$. For $1\leq s\leq \Lambda$, let $\eta _{s}>1$, $v\geq 1$, $\delta _{s}, \beta _{s}>0$, $\{a_{k,s,v}\}$ be a sequence of positive numbers, $\ell _{s}\in \mathbb{N}\setminus \{0\}$ and $L_{s}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell _{s}}$ be a linear transformation. Suppose that there exists a constant $C>0$ independent of A such that the following are satisfied for $k\in \mathbb{Z}, \xi \in \mathbb{R}^{n}$ and $s\in \{1,\ldots,\Lambda \}$:

• (a) $|\widehat{\sigma _{k,s}}(\xi )|\leq CA\min \{1,|a_{k,s,v}L_{s}(\xi )|^{- \delta _{s}/v}\}$;

• (b) $|\widehat{\sigma _{k,s}}(\xi )-\widehat{\sigma _{k,s-1}}(\xi ) \leq CA|a_{k,s,v}L_{s}( \xi )|^{\beta _{s}/v}$;

• (c) $\inf_{k\in \mathbb{Z}}\frac{a_{k+1,s,v}}{a_{k,s,v}}\geq \eta _{s}^{v}$ or $\inf_{k\in \mathbb{Z}}\frac{a_{k,s,v}}{a_{k+1,s,v}}\geq \eta _{s}^{v}$;

• (d) For some $q\in (1,\infty )$, it holds that

  $$\begin{aligned} \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert \vert \sigma _{k,s} \vert *f \bigr\vert \Bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq CA \Vert f \Vert _{L^{q}(\mathbb{R}^{n})}, \quad\forall f\in L^{q}\bigl( \mathbb{R}^{n}\bigr). \end{aligned}$$

  Then there exists a constant $C>0$ such that

  $$\begin{aligned} \Biggl\Vert \sup_{k\in \mathbb{Z}} \Biggl\vert \sum _{j=k}^{\infty }\sigma _{j,\Lambda }*f \Biggr\vert \Biggr\Vert _{L^{p}(\mathbb{R}^{n})}\leq CA \Vert f \Vert _{L^{p}( \mathbb{R}^{n})}, \quad\forall f\in L^{p}\bigl(\mathbb{R}^{n}\bigr), \end{aligned}$$

  (3.24)

  where $p=2$ if $q=2$, $p\in (q,2]$ if $q\in (1,2)$, and $p\in [2,\min \{q, \frac{2q}{q-1}\})$ if $q>2$. Here, $C>0$ is independent of $A, v, \{L_{s}\}_{s=1}^{\Lambda }, f$, but may depend on $p, n, \Lambda, \{\ell _{s}\}_{s=1}^{\Lambda }, \{\beta _{s}\}_{s=1}^{ \Lambda }$ and $\{\delta _{s}\}_{s=1}^{\Lambda }, \{\eta _{s}\}_{s=1}^{\Lambda }$.

Proof

We shall adopt the method following from [22] to prove this lemma. For simplicity, we only consider the case $\inf_{k\in \mathbb{Z}} \frac{a_{k+1,s,v}}{a_{k,s,v}}\geq \eta _{s}^{v}$, since another one can be proved similarly. For $s\in \{1,\ldots,\Lambda \}$, we set $r_{s}=\operatorname{rank}(L_{s})$ and let $\pi _{r_{s}}^{n}(\xi )=(\xi _{1},\ldots,\xi _{r_{s}})$ be the projection from $\mathbb{R}^{n}$ to $\mathbb{R}^{r_{s}}$. Invoking [22, Lemma 6.1], there exist two nonsingular linear transformations $H_{s}:\mathbb{R}^{r_{s}}\rightarrow \mathbb{R}^{r_{s}}$ and $G_{s}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ such that

$$\begin{aligned} \bigl\vert H_{s}\pi _{r_{s}}^{n}G_{s}( \xi ) \bigr\vert \leq \bigl\vert L_{s}(\xi ) \bigr\vert \leq \ell _{s} \bigl\vert H_{s} \pi _{r_{s}}^{n}G_{s}( \xi ) \bigr\vert . \end{aligned}$$

(3.25)

Let $\phi \in \mathcal{C}_{0}^{\infty }(\mathbb{R})$ be such that $\operatorname{supp}(\phi )=\{|t|\leq 1\}$ and $\phi (t)\equiv 1$ for $|t|<1/2$. For $s\in \{1,\ldots,\Lambda \}$, we define a sequence of measures $\{\mu _{k,s}\}_{k\in \mathbb{Z}}$ on $\mathbb{R}^{n}$ by

$$\begin{aligned} \widehat{\mu _{k,s}}(\xi )=\widehat{\sigma _{k,s}}(\xi ) \prod_{j=s+1}^{\Lambda }\phi \bigl( \bigl\vert a_{k,j,v}H_{j}\pi _{r_{j}}^{n}G_{j}( \xi ) \bigr\vert \bigr)- \widehat{\sigma _{k,s-1}}(\xi )\prod _{j=s}^{\Lambda }\phi \bigl( \bigl\vert a_{k,j,v}H_{j} \pi _{r_{j}}^{n}G_{j}( \xi ) \bigr\vert \bigr). \end{aligned}$$

(3.26)

It is not difficult to see that

$$\begin{aligned} \sigma _{k,\Lambda }=\sum_{s=1}^{\Lambda }\mu _{k,s}. \end{aligned}$$

(3.27)

In view of (3.27) we write

$$\begin{aligned} \sup_{k\in \mathbb{Z}} \Biggl\vert \sum_{j=k}^{\infty } \sigma _{j,\Lambda }*f \Biggr\vert \leq \sum_{s=1}^{\Lambda } \sup_{k\in \mathbb{Z}} \Biggl\vert \sum_{j=k}^{\infty } \mu _{j,s}*f \Biggr\vert =:\sum_{s=1}^{\Lambda }T_{s}^{*}(f). \end{aligned}$$

(3.28)

Therefore, for (3.24), it suffices to show that

$$\begin{aligned} \bigl\Vert T_{s}^{*}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})} \leq C_{p}A \Vert f \Vert _{L^{p}( \mathbb{R}^{n})} \end{aligned}$$

(3.29)

for all $1\leq s\leq \Lambda$, where $p=2$ if $q=2$, and $p\in (q,2]$ if $q\in (1,2)$, and $p\in [2,\min \{q,\frac{2q}{q-1}\})$ if $q>2$. Here, $C_{p}>0$ is independent of $A, v, L_{s}, f$, but may depend on $p, n, \Lambda, \ell _{s}, \beta _{s}, \delta _{s}, \eta _{s}$.

Let $\psi \in \mathcal{S}(\mathbb{R})$ be such that $\psi (\xi )\equiv 1$ when $|\xi |<1$ and $\psi (\xi )\equiv 0$ when $|\xi |>\eta _{s}$. Define the function $\Phi _{k}$ by $\widehat{\Phi _{k}}(\xi )=\psi (a_{k,s,v}|H_{s}\pi _{r_{s}}^{n} G_{s}( \xi )|)$. Write

$$\begin{aligned} \sum_{j=k}^{\infty }\mu _{j,s}*f&= \Phi _{k}*T_{s}(f)+( \delta -\Phi _{k})*\sum _{j=k}^{\infty }\mu _{j,s}*f-\Phi _{k}* \sum_{j=-\infty }^{k-1}\mu _{j,s}*f\\ &=:I_{k,1}(f)+I_{k,2}(f)+I_{k,3}(f), \end{aligned}$$

where δ is the Dirac delta function and

$$\begin{aligned} T_{s}(f)=\sum_{k\in \mathbb{Z}}\mu _{k,s}*f. \end{aligned}$$

(3.30)

It follows that

$$\begin{aligned} T_{s}^{*}(f)\leq \sup_{k\in \mathbb{Z}} \bigl\vert I_{k,1}(f) \bigr\vert +\sup_{k\in \mathbb{Z}} \bigl\vert I_{k,2}(f) \bigr\vert +\sup_{k\in \mathbb{Z}} \bigl\vert I_{k,3}(f) \bigr\vert . \end{aligned}$$

(3.31)

We first prove that

$$\begin{aligned} \bigl\Vert T_{s}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C_{p}A \Vert f \Vert _{L^{p}( \mathbb{R}^{n})} \end{aligned}$$

(3.32)

for $p\in (\frac{2q}{q+1},\frac{2q}{q-1})$ and $s\in \{1,\ldots,\Lambda \}$, where $C_{p}>0$ is independent of $A, v, L_{s}, f$, but may depend on $p, n, \Lambda, \ell _{s}, \beta _{s}, \delta _{s}, \eta _{s}$. In view of assumptions (a) and (b) and (3.26),

$$\begin{aligned} &\bigl\vert \widehat{\mu _{k,s}}(\xi ) \bigr\vert \leq CA\bigl( \bigl\vert a_{k,s,v} L_{s}(\xi ) \bigr\vert ^{\beta _{s}/v}+ \bigl\vert a_{k,s,v} L_{s}(\xi ) \bigr\vert ^{1/v}\bigr); \end{aligned}$$

(3.33)

$$\begin{aligned} &\bigl\vert \widehat{\mu _{k,s}}(\xi ) \bigr\vert \leq CA\min \bigl\{ 1, \bigl\vert a_{k,s,v} L_{s}(\xi ) \bigr\vert ^{- \delta _{s}/v}+ \bigl\vert a_{k,s,v}L_{s}(\xi ) \bigr\vert ^{-1/v}\bigr\} . \end{aligned}$$

(3.34)

Let $\{\Psi _{k,s}\}_{k\in \mathbb{Z}}$ be a sequence of nonnegative functions in $\mathcal{C}_{0}^{\infty }(\mathbb{R})$ such that

$$\begin{aligned} &\mathop{\operatorname{supp}}(\Psi _{k,s})\subset \bigl[a_{k+1,s,v}^{-1},a_{k-1,s,v}^{-1} \bigr], \qquad\sum_{k\in \mathbb{Z}}\Psi _{k,s}^{2}(t)=1,\\ &\biggl\vert \biggl(\frac{d}{dt} \biggr)^{j}\Psi _{k,s}(t) \biggr\vert \leq C_{j} \vert t \vert ^{-j} (j=1,2,\ldots ) \quad{\text{for all }} t>0 {\text{ and }} j\in \mathbb{N}, \end{aligned}$$

where $C_{j}$ are independent of $s, v$, and k. Define the Fourier multiplier operator $S_{j,s}$ by

$$\begin{aligned} \widehat{S_{j,s}f}(\xi )=\Psi _{j,s}\bigl( \bigl\vert H_{s}\pi _{r_{s}}^{n}G_{s}(\xi ) \bigr\vert \bigr) \hat{f}(\xi ) \quad{\text{for }} j\in \mathbb{Z}. \end{aligned}$$

(3.35)

Thus, the operator $T_{s}$ can be decomposed as

$$\begin{aligned} T_{s}(f)=\sum_{k\in \mathbb{Z}}\mu _{k,s}* \sum_{j \in \mathbb{Z}}S_{j+k,s}S_{j+k,s}f =\sum _{j\in \mathbb{Z}} \sum_{k\in \mathbb{Z}}S_{j+k,s}( \mu _{k,s}*S_{j+k,s}f) =: \sum_{j\in \mathbb{Z}}T_{s,j}(f). \end{aligned}$$

(3.36)

By the Littlewood–Paley theory, Plancherel’s theorem, and assumption (c), we then use (3.33) and (3.34) to obtain

$$\begin{aligned} \bigl\Vert T_{s,j}(f) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} &\leq C \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \vert \mu _{k,s}*S_{j+k,s}f \vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{2}(\mathbb{R}^{n})} \\ & \leq C \biggl(\sum_{k\in \mathbb{Z}} \int _{a_{j+k+1,s,v}^{-1}\leq \vert H_{s}\pi _{r_{s}}^{n}G_{s}( \xi ) \vert \leq a_{j+k-1,s,v}^{-1}} \bigl\vert \widehat{\mu _{k,s}}(\xi ) \bigr\vert ^{2} \bigl\vert \hat{f}(\xi ) \bigr\vert ^{2}\,d\xi \biggr)^{1/2} \\ & \leq CA\eta _{s}^{-c \vert j \vert } \Vert f \Vert _{L^{2}(\mathbb{R}^{n})} \end{aligned}$$

(3.37)

for some $c>0$, where $C>0$ is independent of $A, v, L_{s}, f$, but may depend on $\ell _{s}, \beta _{s}, \delta _{s}, \eta _{s}$.

On the other hand, by our assumption (d), (3.26) and a well-known result on maximal functions (see [22]), there exists a constant $C>0$ independent of $A, v, L_{s}$ such that

$$\begin{aligned} \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert \vert \mu _{k,s} \vert *f \bigr\vert \Bigr\Vert _{L^{q}( \mathbb{R}^{n})} \leq CA \Vert f \Vert _{L^{q}(\mathbb{R}^{n})}, \quad\forall f\in L^{q}\bigl( \mathbb{R}^{n}\bigr) \end{aligned}$$

(3.38)

for any $1\leq s\leq \Lambda$. Using (3.38) and the lemma in [16, pp. 544],

$$\begin{aligned} \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \vert \mu _{k,s}*g_{k} \vert ^{2} \biggr)^{{1}/{2}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C_{p}A \biggl\Vert \biggl(\sum _{k \in \mathbb{Z}} \vert g_{k} \vert ^{2} \biggr)^{{1}/{2}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})} \end{aligned}$$

(3.39)

for $|1/p-1/2|=1/(2q)$ and arbitrary functions $\{g_{k}\}_{k}\in L^{p}(\ell ^{2}, \mathbb{R}^{n})$. Here, $C_{p}>0$ is independent of $A, v, L_{s}$. Combining (3.39) with the Littlewood–Paley theory implies

$$\begin{aligned} \bigl\Vert T_{s,j}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})} &\leq C \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \vert \mu _{k,s}*S_{j+k,s}f \vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{p}(\mathbb{R}^{n})} \\ & \leq CA \biggl\Vert \biggl(\sum_{k \in \mathbb{Z}} \vert S_{j+k,s}f \vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{p}(\mathbb{R}^{n})} \leq CA \Vert f \Vert _{L^{p}(\mathbb{R}^{n})}, \end{aligned}$$

(3.40)

where $|1/p-1/2|=1/(2q)$ and $C>0$ is independent of $A, v, L_{s}$. By interpolation between (3.37) and (3.40), we have that, for any $p\in (\frac{2q}{q+1},\frac{2q}{q-1})$ and some $c'>0$,

$$\begin{aligned} \bigl\Vert T_{s,j}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq CA\eta _{s}^{-c' \vert j \vert } \Vert f \Vert _{L^{p}( \mathbb{R}^{n})}. \end{aligned}$$

(3.41)

Inequality (3.41) together with (3.36) and Minkowski’s inequality implies (3.32).

By (3.32) and a well-known result on maximal functions (see [22]), we have that, for all $p\in (\frac{2q}{q+1},\frac{2q}{q-1})$,

$$\begin{aligned} \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert I_{k,1}(f) \bigr\vert \Bigr\Vert _{L^{p}( \mathbb{R}^{n})} \leq C \bigl\Vert T_{s}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C_{p}A \Vert f \Vert _{L^{p}(\mathbb{R}^{n})}, \end{aligned}$$

(3.42)

where $C_{p}>0$ is independent of $A, v, L_{s}, f$, but may depend on $p, n, \ell _{s}, \beta _{s}, \delta _{s}, \eta _{s}$.

We now estimate $\|\sup_{k\in \mathbb{Z}}|I_{k,2}(f)|\|_{L^{p}(\mathbb{R}^{n})}$. Write

$$\begin{aligned} \sup_{k\in \mathbb{Z}} \bigl\vert I_{k,2}(f) \bigr\vert \leq \sum_{j=0}^{ \infty }\sup _{k\in \mathbb{Z}} \bigl\vert (\delta -\Phi _{k})*\mu _{j+k,s}*f \bigr\vert =:\sum_{j=0}^{\infty }I_{j}(f). \end{aligned}$$

(3.43)

An application of (3.38) shows that

$$\begin{aligned} \bigl\Vert I_{j}(f) \bigr\Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert \vert \mu _{j+k,s} \vert * \vert f \vert \bigr\vert \Bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq CA \Vert f \Vert _{L^{q}(\mathbb{R}^{n})}. \end{aligned}$$

(3.44)

In view of Plancherel’s theorem, (3.25), and (3.33), we have that, for some $c>0$,

$$\begin{aligned} \bigl\Vert I_{j}(f) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}^{2} &\leq \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \bigl\vert ( \delta -\Phi _{k})*\mu _{j+k,s}*f \bigr\vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{2}(\mathbb{R}^{n})}^{2} \\ & \leq \sum_{k\in \mathbb{Z}} \int _{\{a_{k,s,v} \vert H_{s}\pi _{r_{s}}^{n}G_{s}(\xi ) \vert \geq 1 \}} \bigl\vert \widehat{\mu _{j+k,s}}(\xi ) \bigr\vert ^{2} \bigl\vert \hat{f}(\xi ) \bigr\vert ^{2}\,d\xi \\ & \leq \sum_{k\in \mathbb{Z}}\sum_{i=-\infty }^{k} \int _{\{a_{i,s,v}^{-1} \leq \vert L_{s}( \xi ) \vert < a_{i-1,s,v}^{-1}\}} \bigl\vert \widehat{\mu _{j+k,s}}(\xi ) \bigr\vert ^{2} \bigl\vert \hat{f}( \xi ) \bigr\vert ^{2}\,d\xi \\ & \leq C\sum_{k\in \mathbb{Z}}\sum _{i=-\infty }^{k}A^{2}\eta _{s}^{-c(j+k-i)} \int _{\{a_{i,s,v}^{-1} \leq \vert L_{s}(\xi ) \vert < a_{i-1,s,v}^{-1}\}} \bigl\vert \hat{f}(\xi ) \bigr\vert ^{2}\,d\xi \\ & \leq CA^{2}\eta _{s}^{-jc} \sum _{i=0}^{\infty }\eta _{s}^{-ic} \Vert f \Vert _{L^{2}(\mathbb{R}^{n})}^{2} \\ & \leq CA^{2}\eta _{s}^{-jc} \Vert f \Vert _{L^{2}( \mathbb{R}^{n})}^{2}. \end{aligned}$$

It follows that

$$\begin{aligned} \bigl\Vert I_{j}(f) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\leq CA\eta _{s}^{-jc/2} \Vert f \Vert _{L^{2}( \mathbb{R}^{n})}. \end{aligned}$$

(3.45)

An interpolation between (3.44) and (3.45) gives that

$$\begin{aligned} \bigl\Vert I_{j}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq CA\eta _{s}^{-\tau j} \Vert f \Vert _{L^{p}( \mathbb{R}^{n})} \end{aligned}$$

for some $\tau >0$ and $p\in [2,q]$ if $q>2$ or $p\in (q,2]$ if $q\in (1,2)$ or $p=2$ if $q=2$. Combining this with (3.43) leads to

$$\begin{aligned} \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert I_{k,2}(f) \bigr\vert \Bigr\Vert _{L^{p}( \mathbb{R}^{n})}\leq CA \Vert f \Vert _{L^{p}(\mathbb{R}^{n})} \end{aligned}$$

(3.46)

for $p\in [2,q]$ if $q>2$ or $p\in (q,2]$ if $q\in (1,2)$ or $p=2$ if $q=2$.

It remains to estimate $\|\sup_{k\in \mathbb{Z}}|I_{k,3}(f)|\|_{L^{p}(\mathbb{R}^{n})}$. Write

$$\begin{aligned} \sup_{k\in \mathbb{Z}} \bigl\vert I_{k,3}(f) \bigr\vert = \sup_{k\in \mathbb{Z}} \Biggl\vert \sum_{j=1}^{\infty } \Phi _{k}*\mu _{k-j,s}*f \Biggr\vert \leq \sum _{j=1}^{\infty }\sup_{k\in \mathbb{Z}} \vert \Phi _{k}*\mu _{k-j,s}*f \vert =:\sum _{j=1}^{\infty }J_{j}(f). \end{aligned}$$

(3.47)

In view of (3.38), one can get

$$\begin{aligned} \bigl\Vert J_{j}(f) \bigr\Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert \vert \mu _{j-k,s} \vert * \vert f \vert \bigr\vert \Bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq CA \Vert f \Vert _{L^{q}(\mathbb{R}^{n})}. \end{aligned}$$

(3.48)

In view of Plancherel’s theorem, we use (3.33) and (3.25) to get

$$\begin{aligned} & \bigl\Vert J_{j}(f) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &\quad\leq \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \vert \Phi _{k}*\mu _{k-j,s}*f \vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &\quad\leq \biggl(\sum_{k\in \mathbb{Z}} \int _{\{a_{k,s,v} \vert H_{s} \pi _{r_{s}}^{n}G_{s}(\xi ) \vert \leq \eta _{s}\}} \bigl\vert \widehat{\mu _{k-j,s}}( \xi ) \bigr\vert ^{2} \bigl\vert \hat{f}(\xi ) \bigr\vert ^{2}\,d\xi \biggr)^{1/2} \\ &\quad\leq C \biggl( \int _{\mathbb{R}^{n}}\sum_{k\in \mathbb{Z}} \bigl\vert \widehat{\mu _{k-j,s}}(\xi ) \bigr\vert ^{2}\chi _{\{a_{k,s,v} \vert L_{s}( \xi ) \vert \leq \ell _{s}\eta _{s}\}} \bigl\vert \hat{f}(\xi ) \bigr\vert ^{2}\,d\xi \biggr)^{1/2} \\ &\quad\leq CA\bigl(\eta _{s}^{-\beta _{s}j}+\eta _{s}^{-j} \bigr) \Vert f \Vert _{L^{2}( \mathbb{R}^{n})} \\ &\qquad{} \times \biggl(\sup_{\xi \in \mathbb{R}^{n}} \sum_{k\in \mathbb{Z}} \bigl( \bigl\vert a_{k,s,v}L_{s}(\xi ) \bigr\vert ^{2\beta _{s}/v}+ \bigl\vert a_{k,s,v} L_{s}(\xi ) \bigr\vert ^{2/v}\bigr)\chi _{\{a_{k,s,v} \vert L_{s}(\xi ) \vert \leq \ell _{s}\eta _{s} \}} \biggr)^{1/2} \\ &\quad\leq CA\bigl(\eta _{s}^{-\beta _{s}j}+\eta _{s}^{-j} \bigr) \Vert f \Vert _{L^{2}( \mathbb{R}^{n})}, \end{aligned}$$

(3.49)

where in the last inequality of (3.49) we have used the properties of lacunary sequence and the fact that $\ell _{s}\eta _{s}>1$, $v\geq 1$. Here, $C>0$ is independent of $A, v, L_{s}$, but may depend on $n, \ell _{s}, \beta _{s}, \delta _{s}, \eta _{s}$. An interpolation between (3.48) and (3.49) leads to

$$\begin{aligned} \bigl\Vert J_{j}(f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq CA\bigl(\eta _{s}^{-\theta \beta _{s}j}+ \eta _{s}^{-\theta j}\bigr) \Vert f \Vert _{L^{2}(\mathbb{R}^{n})} \end{aligned}$$

(3.50)

for some $\theta >0$, where $p\in [2,q]$ if $q>2$ or $p\in (q,2]$ if $q\in (1,2)$ or $p=2$ if $q=2$. By (3.47), (3.50), and Minkowski’s inequality,

$$\begin{aligned} \Bigl\Vert \sup_{k\in \mathbb{Z}} \bigl\vert I_{k,3}(f) \bigr\vert \Bigr\Vert _{L^{p}( \mathbb{R}^{n})}\leq CA \Vert f \Vert _{L^{p}(\mathbb{R}^{n})} \end{aligned}$$

(3.51)

for $p\in [2,q]$ if $q>2$ or $p\in (q,2]$ if $q\in (1,2)$ or $p=2$ if $q=2$. Then (3.29) follows from (3.31), (3.42), (3.46), and (3.51). This finishes the proof of Lemma 3.4. □

The following result is the main ingredient of proving Theorem 1.1.

Lemma 3.5

Let $\Lambda, v\in \mathbb{N}\setminus \{0\}$. For $1\leq s\leq \Lambda$, let $\{a_{k,s,v}\}_{k\in \mathbb{Z}}$ be a lacunary sequence of positive numbers. For $1\leq s\leq \Lambda$, let $\delta _{s}>0, \eta _{s}>1, \ell _{s}\in \mathbb{N}\setminus \{0 \}$, and $L_{s}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell _{s}}$ be linear transformations. Let $\{\sigma _{s,k}: 0\leq s\leq \Lambda{\textit{ and }}k\in \mathbb{Z}\}$ be a family of measures on $\mathbb{R}^{n}$ with $\sigma _{0,k}=0$ for every $k\in \mathbb{Z}$. Suppose that there exist $p_{0}, q_{0}>1$ satisfying $(p_{0},q_{0})\neq (2,2)$ and $c, A>0$ independent of v and $\{L_{s}\}_{s=1}^{\Lambda }$ such that the following conditions are satisfied for any $1\leq s\leq \Lambda$, $k\in \mathbb{Z}$, $\xi \in \mathbb{R}^{n}$, and $\{g_{k,j}\}\in L^{p_{0}}(\mathbb{R}^{n},\ell ^{q_{0}}(\ell ^{2}))$:

• (a) $|\widehat{\sigma _{s,k}}(\xi )|\leq cA\min \{1,|a_{k,s,v}L_{s}(\xi )|^{-{ \delta _{s}}/{v}}\}$;

• (b) $|\widehat{\sigma _{s,k}}(\xi )-\widehat{\sigma _{s-1,k}}(\xi ) | \leq cA|a_{k,s,v}L_{s}(\xi )|^{{1}/{v}}$;

• (c) $\inf_{k\in \mathbb{Z}}\frac{a_{k+1,s,v}}{a_{k,s,v}}\geq \eta _{s}^{v}$ or $\inf_{k\in \mathbb{Z}}\frac{a_{k,s,v}}{a_{k+1,s,v}}\geq \eta _{s}^{v}$;

• (d)
  $$\begin{aligned} &\biggl\Vert \biggl(\sum_{j\in \mathbb{Z}} \biggl(\sum _{k\in \mathbb{Z}} \vert \sigma _{s,k}*g_{k,j} \vert ^{2} \biggr)^{q_{0}/2} \biggr)^{1/q_{0}} \biggr\Vert _{L^{p_{0}}(\mathbb{R}^{n})}\\ &\quad\leq cA \biggl\Vert \biggl(\sum _{j \in \mathbb{Z}} \biggl(\sum_{k\in \mathbb{Z}} \vert g_{k,j} \vert ^{2} \biggr)^{q_{0}/2} \biggr)^{1/q_{0}} \biggr\Vert _{L^{p_{0}}(\mathbb{R}^{n})}. \end{aligned}$$

  Then, for $\alpha \in \mathbb{R}$ and $({1}/{p},{1}/{q})\in B_{1}B_{2} \setminus \{({1}/{p_{0}},{1}/{q_{0}}),({1}/{2},{1}/{2}) \}$, there exists a constant $C>0$ independent of v and $\{L_{s}\}_{s=1}^{\Lambda }$ such that

  $$\begin{aligned} \biggl\Vert \sum_{k\in \mathbb{Z}}\sigma _{\Lambda,k}*f \biggr\Vert _{ \dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \leq CA \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}( \mathbb{R}^{n})}, \end{aligned}$$

  where $B_{1}=({1}/{2},{1}/{2})$, $B_{2}=({1}/{p_{0}},{1}/{q_{0}})$ and $B_{1}B_{2}$ is the line segment from $B_{1}$ to $B_{2}$.

Proof

Assume that $\inf_{k\in \mathbb{Z}}\frac{a_{k+1,s,v}}{a_{k,s,v}}\geq \eta _{s}^{v}$ for all $1\leq s\leq \Lambda$, the corresponding result has been proved in [27, Lemma 2.5]. Similar arguments will give the corresponding result for the case $\inf_{k\in \mathbb{Z}}\frac{a_{k,s,v}}{a_{k+1,s,v}} \geq \eta _{s}^{v}$. The details are omitted. □

In order to prove Theorem 1.3, we need the following characterization of the Triebel–Lizorkin spaces.

Lemma 3.6

([41])

Let $0<\alpha <1$, $1< p<\infty$, $1< q\leq \infty$, and $1\leq r<\min \{p,q\}$. Then

$$\begin{aligned} \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})}\approx \biggl\Vert \biggl( \sum _{k\in \mathbb{Z}}2^{kq\alpha } \biggl( \int _{\mathfrak{R}_{n}} \bigl\vert \triangle _{2^{-k}\zeta }(f) \bigr\vert ^{r}\,d\zeta \biggr)^{q/r} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}. \end{aligned}$$

Our main ingredient of proving Theorem 1.3 is the following boundedness criterion.

Lemma 3.7

Let $v\geq 1$, $\Lambda \in \mathbb{N} \setminus \{0\}$, and $\{\sigma _{k,s}:k\in \mathbb{Z}, 1\leq s\leq \Lambda \}$ be a family of Borel measures on $\mathbb{R}^{n}$ with $\sigma _{k,0}=0$ for all $k\in \mathbb{Z}$. Let $|\sigma _{k,s}|$ be the total variation of $\sigma _{k,s}$. Let $\{a_{k,s,v}\}_{k\in \mathbb{Z}}$ be a lacunary sequence of positive numbers. For $1\leq s\leq \Lambda$, let $\eta _{s}>1, \beta _{s}, \gamma _{s}>0$, $M_{s}\in \mathbb{N}\setminus \{0\}$, and $L_{s}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{M_{s}}$ be linear transformations. Suppose that there exist $C, A>0$ independent of v such that, for $1\leq s\leq \Lambda$, $k\in \mathbb{Z}$, and $\xi \in \mathbb{R}^{n}$, the following conditions are satisfied:

• (a) $\max \{|\widehat{\sigma _{k,s}}(\xi ) -\widehat{\sigma _{k,s-1}}( \xi )|,|\widehat{|\sigma _{k,s}|}(\xi ) -\widehat{|\sigma _{k,s-1}|}( \xi )|\}\leq CA|a_{k,s,v}L_{s}(\xi )|^{1/v}$;

• (b) $\max \{|\widehat{\sigma _{k,s}}(\xi )|,|\widehat{|\sigma _{k,s}|}( \xi )|\} \leq CA\min \{1,|a_{k,s,v}L_{s}(\xi )|^{-\beta _{s}/v}\}$;

• (c) There exists $\vartheta \in \mathbb{R}^{n}$ such that $\sup_{k\in \mathbb{Z}}||\sigma _{k,0}|*f(x)|\leq CA|f(x+\vartheta )|$ for any $x\in \mathbb{R}^{n}$;

• (d) $\inf_{k\in \mathbb{Z}}\frac{a_{k+1,s,v}}{a_{k,s,v}}\geq \eta _{s}^{v}$ or $\inf_{k\in \mathbb{Z}}\frac{a_{k,s,v}}{a_{k+1,s,v}}\geq \eta _{s}^{v}$;

• (e) There exist $p_{0}, q_{0}>1$ satisfying $(p_{0},q_{0})\neq (2,2)$, $1< r_{0}<\min \{p_{0},q_{0}\}$, and $2\le u<\infty$ such that

  $$\begin{aligned} & \biggl\Vert \biggl(\sum_{l\in \mathbb{Z}} \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \bigl\vert \vert \sigma _{k,s} \vert * g_{l, \zeta,k} \bigr\vert ^{u} \biggr)^{1/u} \biggr\Vert _{L^{r_{0}}(\mathfrak{R}_{n})}^{q_{0}} \biggr)^{1/q_{0}} \biggr\Vert _{L^{p_{0}}(\mathbb{R}^{n})} \\ &\quad\leq CA \biggl\Vert \biggl(\sum_{l\in \mathbb{Z}} \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \vert g_{l,\zeta,k} \vert ^{u} \biggr)^{1/u} \biggr\Vert _{L^{r_{0}}(\mathfrak{R}_{n})}^{q_{0}} \biggr)^{1/q_{0}} \biggr\Vert _{L^{p_{0}}( \mathbb{R}^{n})}. \end{aligned}$$

  Then, for $\alpha \in (0,1)$ and $({1}/{p},{1}/{q})\in P_{1}P_{2}\setminus \{({1}/{p_{0}},{1}/{q_{0}}) \}$, there exists a constant $C>0$ independent of A and v such that

  $$\begin{aligned} &\biggl\Vert \biggl(\sum_{l\in \mathbb{Z}}2^{lq\alpha } \biggl( \int _{ \mathfrak{R}_{n}}\sup_{k\in \mathbb{Z}} \bigl\vert \vert \sigma _{k,s} \vert * \vert \triangle _{2^{-l}\zeta }f \vert \bigr\vert \,d\zeta \biggr)^{q} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}\leq CA \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}( \mathbb{R}^{n})}, \\ &\quad\forall 1\leq s\leq \Lambda. \end{aligned}$$

  Here, $P_{1}P_{2}$ denotes the line segment from $P_{1}$ to $P_{2}$ with $P_{1}=({1}/{2}, {1}/{2})$ and $P_{2}=({1}/{p_{0}},{1}/{q_{0}})$.

• (g) Suppose also that the following inequality holds for $1\leq s\leq \Lambda$:

  $$\begin{aligned} &\biggl\Vert \biggl(\sum_{j\in \mathbb{Z}} \biggl(\sum _{k\in \mathbb{Z}} \vert \sigma _{k,s}*g_{k,j} \vert ^{2} \biggr)^{q_{0}/2} \biggr)^{1/q_{0}} \biggr\Vert _{L^{p_{0}}(\mathbb{R}^{n})}\\ &\quad \leq CA \biggl\Vert \biggl(\sum _{j \in \mathbb{Z}} \biggl(\sum_{k\in \mathbb{Z}} \vert g_{k,j} \vert ^{2} \biggr)^{q_{0}/2} \biggr)^{1/q_{0}} \biggr\Vert _{L^{p_{0}}(\mathbb{R}^{n})}. \end{aligned}$$

  Then, for $\alpha \in (0,1)$ and $({1}/{p},{1}/{q})\in P_{1}P_{2}\setminus \{({1}/{p_{0}},{1}/{q_{0}}),({1}/{2},{1}/{2}) \}$, there exists a constant $C>0$ independent of A and v such that

  $$\begin{aligned} \Biggl\Vert \Biggl(\sum_{l\in \mathbb{Z}}2^{lq\alpha } \Biggl( \int _{ \mathfrak{R}_{n}}\sup_{k\in \mathbb{Z}} \Biggl\vert \sum _{j=k}^{ \infty }\sigma _{j,\Lambda }* \triangle _{2^{-l}\zeta }f \Biggr\vert \,d\zeta \Biggr)^{q} \Biggr)^{{1}/{q}} \Biggr\Vert _{L^{p}(\mathbb{R}^{n})}\leq CA \Vert f \Vert _{\dot{F}_{ \alpha }^{p,q}(\mathbb{R}^{n})}. \end{aligned}$$

Proof

The lemma can be proved by the arguments similar to those used in deriving [30, Lemma 2.9]. We omit the details. □

Proofs of Theorems 1.1–1.3

Proof of Theorem 1.1

Let $h, \Omega, P, \varphi$ be given as in Theorem 1.1. Invoking Lemma 2.1, there exist a sequence of complex numbers $\{c_{j}\}_{j=1}^{\infty }$ and a sequence of $(1,\infty )$ atoms $\{\Omega _{j}\}_{j=1}^{\infty }$ such that $\Omega =\sum_{j=1}^{\infty }c_{j}\Omega _{j}$ and $\|\Omega \|_{H^{1}({\mathrm{S}}^{n-1})} \approx \sum_{j=1}^{\infty }|c_{j}|$. By the definition of $T_{h,\Omega,P,\varphi }$, one has

$$\begin{aligned} T_{h,\Omega,P,\varphi }f=\sum_{j=1}^{\infty }c_{j}T_{h,\Omega _{j},P, \varphi }f. \end{aligned}$$

(3.52)

In view of (3.52) and the definition of $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$, we have that, for $1< p, q<\infty$ and $\alpha \in \mathbb{R}$,

$$\begin{aligned} \Vert T_{h,\Omega,P,\varphi }f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \leq \sum _{j=1}^{\infty } \vert c_{j} \vert \Vert T_{h,\Omega _{j},P,\varphi }f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})}. \end{aligned}$$

Therefore, to prove Theorem 1.1, it suffices to prove that there exists $C>0$ is independent of $h, \gamma, \Omega$ and the coefficients of P such that

$$\begin{aligned} \Vert T_{h,\Omega,P,\varphi }f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})}, \end{aligned}$$

(3.53)

holds for any $(1,\infty )$ atom Ω and $\alpha \in \mathbb{R}$ and $(p,q)\in \mathcal{R}_{\gamma }$.

Given a $(1,\infty )$ atom Ω satisfying (2.6)–(2.8) with $0<\varrho \leq 1$ and $\vartheta \in {\mathrm{S}}^{n-1}$. Without loss of generality we may assume that $\vartheta =\theta =(1,0,\ldots,0)$. By the definition of $\sigma _{h,\Omega,k,\theta,\Lambda }$, we have

$$\begin{aligned} T_{h,\Omega,P,\varphi }f=\sum_{k\in \mathbb{Z}}\sigma _{h, \Omega,k,\theta,\Lambda }*f. \end{aligned}$$

(3.54)

Note that if $\varphi \in \mathfrak{F}_{1}$ or $\varphi \in \mathfrak{F}_{2}$, there exist $C_{1}, C_{2}>0$ depending only on φ such that

$$\begin{aligned} C_{1}\leq \frac{\varphi (2t)}{\varphi (t)}\leq C_{2}, \quad\forall t>0. \end{aligned}$$

(3.55)

In view of (3.55) and Lemma 3.1,

$$\begin{aligned} \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s}}(\xi )- \widehat{\sigma _{h,\Omega,k,\theta,s-1}}(\xi ) \bigr\vert \leq C\gamma ' \Vert h \Vert _{ \Delta _{\gamma }(\mathbb{R}_{+})}\min \bigl\{ 1,\bigl(\varphi \bigl(2^{k\gamma '} \bigr)^{l_{s}} \bigl\vert L_{s}( \xi ) \bigr\vert \bigr)^{1/\gamma '}\bigr\} . \end{aligned}$$

(3.56)

By the properties of φ and applying the arguments similar to those used in deriving [27, Lemma 2.4], one obtains that there exists $C>0$ independent of $h, \Omega, \gamma$ and $\{a_{l_{i}}\}_{i=1}^{\Lambda }$ such that

$$\begin{aligned} &\biggl\Vert \biggl(\sum_{j\in \mathbb{Z}} \biggl(\sum _{k\in \mathbb{Z}} \vert \sigma _{h,\Omega,k,\theta,s}*g_{k,j} \vert ^{2} \biggr)^{q/2} \biggr)^{1/q} \biggr\Vert _{L^{p}(\mathbb{R}^{n})} \\ &\quad\leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \biggl\Vert \biggl(\sum_{j\in \mathbb{Z}} \biggl(\sum_{k\in \mathbb{Z}} \vert g_{k,j} \vert ^{2} \biggr)^{q/2} \biggr)^{1/q} \biggr\Vert _{L^{p}(\mathbb{R}^{n})} \end{aligned}$$

(3.57)

for all $1\leq s\leq \Lambda$ and $(1/p,1/q)\in \mathcal{R}_{\gamma }$. Then (3.53) follows from (3.2), (3.54), (3.56), (3.57), and Lemmas 3.2 and 3.5. □

Proof of Theorem 1.2

Let $h, \Omega, P, \varphi$ be given as in Theorem 1.2. By Lemma 2.1, there exist a sequence of complex numbers $\{c_{j}\}_{j=1}^{\infty }$ and a sequence of $(1,\infty )$ atoms $\{\Omega _{j}\}_{j=1}^{\infty }$ such that $\Omega =\sum_{j=1}^{\infty }c_{j}\Omega _{j}$ and $\|\Omega \|_{H^{1}({\mathrm{S}}^{n-1})} \approx \sum_{j=1}^{\infty }|c_{j}|$. In view of the definition of $T_{h,\Omega,P, \varphi }^{*}$,

$$\begin{aligned} T_{h,\Omega,P,\varphi }^{*}f\leq \sum_{j=1}^{\infty } \vert c_{j} \vert T_{h, \Omega _{j},P,\varphi }^{*}f. \end{aligned}$$

(3.58)

In view of (3.58), to prove Theorem 1.2, it suffices to show that there exists $C>0$ independent of $h, \gamma, \Omega$ and the coefficients of P such that

$$\begin{aligned} \bigl\Vert T_{h,\Omega,P,\varphi }^{*}f \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C\gamma ' \Vert h \Vert _{\Delta _{\gamma }(\mathbb{R}_{+})} \Vert f \Vert _{L^{p}(\mathbb{R}^{n})} \end{aligned}$$

(3.59)

holds for any $(1,\infty )$ atom Ω and $p\in (\gamma ',\infty )$ if $\gamma \geq 2$ or $p\in (\gamma ',\frac{2\gamma '}{\gamma '-2})$ if $\gamma \in (4/3,2)$. Let Ω be a $(1,\infty )$ atom satisfying (2.6)–(2.8) with $0<\varrho \leq 1$ and $\vartheta \in {\mathrm{S}}^{n-1}$. Without loss of generality, we may assume that $\vartheta =\theta =(1,0,\ldots,0)$. Let $\{\sigma _{h,\Omega,k,\theta,s}\}_{s=0}^{\Lambda }$ be given as in the proof of Theorem 1.1. By a simple argument following from the proof of [18, Theorem 2], one has

$$\begin{aligned} T_{h,\Omega,P,\varphi }^{*}f\leq \sup_{k\in \mathbb{Z}} \bigl\vert \vert \sigma _{h,\Omega,k,\theta,\Lambda } \vert *f \bigr\vert +\sup _{k\in \mathbb{Z}} \Biggl\vert \sum_{j=k}^{\infty } \sigma _{h,\Omega,j, \theta,\Lambda }*f \Biggr\vert . \end{aligned}$$

(3.60)

By (3.2), (3.54), (3.56), (3.60), and Lemmas 3.2–3.4, we have (3.59) for $p\in (\gamma ',\infty )$ if $\gamma \geq 2$ or $p\in (\gamma ', \frac{2\gamma '}{\gamma '-2})$ if $\gamma \in (4/3,2)$. □

Proof of Theorem 1.3

Let $h, \Omega, P, \varphi$ be given as in Theorem 1.3. Notice that

$$\begin{aligned} \bigl\vert \triangle _{\zeta }\bigl(T_{h,\Omega,P,\varphi }^{*}f \bigr) (x) \bigr\vert & = \bigl\vert T_{h,\Omega,P, \varphi }^{*}f(x+\zeta )-T_{h,\Omega,P,\varphi }^{*}f(x) \bigr\vert \\ & = \bigl\vert T_{h,\Omega,P,\varphi }^{*}f_{\zeta }(x)-T_{h,\Omega,P,\varphi }^{*}f(x) \bigr\vert \leq T_{h,\Omega,P,\varphi }^{*}\bigl( \triangle _{\zeta }(f)\bigr) (x),\quad \forall x, \zeta \in \mathbb{R}^{n}. \end{aligned}$$

This together with Lemma 3.6 and (3.52) implies that, for $\alpha \in (0,1)$ and $1< p, q<\infty$,

$$\begin{aligned} &\bigl\Vert T_{h,\Omega,P,\varphi }^{*}f \bigr\Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \\ &\quad\leq C \biggl\Vert \biggl(\sum_{l\in \mathbb{Z}}2^{lq \alpha } \biggl( \int _{\mathfrak{R}_{n}} \bigl\vert \triangle _{2^{-l}\zeta } \bigl(T_{h, \Omega,P,\varphi }^{*}f\bigr) \bigr\vert \,d\zeta \biggr)^{q} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}( \mathbb{R}^{n})} \\ & \quad\leq C \biggl\Vert \biggl( \sum_{l\in \mathbb{Z}}2^{lq\alpha } \biggl( \int _{\mathfrak{R}_{n}} \bigl\vert T_{h, \Omega,P,\varphi }^{*}\bigl( \triangle _{2^{-l}\zeta }(f)\bigr) \bigr\vert \,d\zeta \biggr)^{q} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})} \\ &\quad \leq C\sum_{j=1}^{\infty } \vert c_{j} \vert \biggl\Vert \biggl(\sum_{l\in \mathbb{Z}}2^{lq\alpha } \biggl( \int _{\mathfrak{R}_{n}} \bigl\vert T_{h,\Omega _{j},P,\varphi }^{*}\bigl( \triangle _{2^{-l}\zeta }(f)\bigr) \bigr\vert \,d\zeta \biggr)^{q} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}( \mathbb{R}^{n})}. \end{aligned}$$

(3.61)

Therefore, to establish the bounds for $T_{h,\Omega,P,\varphi }^{*}$ on $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$, it suffices to show that

$$\begin{aligned} \biggl\Vert \biggl(\sum_{l\in \mathbb{Z}}2^{lq\alpha } \biggl( \int _{ \mathfrak{R}_{n}} \bigl\vert T_{h,\Omega,P,\varphi }^{*}\bigl( \triangle _{2^{-l}\zeta }(f)\bigr) \bigr\vert \,d \zeta \biggr)^{q} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})} \end{aligned}$$

(3.62)

holds for any $(1,\infty )$ atom Ω and $\alpha \in (0,1)$ and $1< p, q<\infty$. Here, $C>0$ is independent of Ω and the coefficients of P.

In what follows, let Ω be a $(1,\infty )$ atom satisfying (2.6)–(2.8) with $0<\varrho \leq 1$ and $\vartheta \in {\mathrm{S}}^{n-1}$. Without loss of generality, we may assume that $\vartheta =\theta =(1, 0,\ldots,0)$. Let $P, \{P_{s}\}_{s=0}^{\Lambda }, \{\Gamma _{s, \theta }\}_{s=0}^{\Lambda }$, $\{L_{s}\}_{s=1}^{\Lambda }$, and $\{\sigma _{h, \Omega,k,\theta,s}\}_{s=0}^{\Lambda }$ be given as in the proof of Theorem 1.1. We define the measures $\{\nu _{k,s}\}_{0}^{2\Lambda }$ and $\{|\nu _{k,s}|\}_{0}^{2\Lambda }$ by

$$\begin{aligned} &\widehat{\nu _{k,s}}(\xi )= \int _{2^{k}< \vert y \vert \leq 2^{k+1}}\exp \Biggl( \sum_{i=1}^{s}a_{l_{i}} \varphi \bigl( \vert y \vert \bigr)^{l_{i}}\xi \cdot \theta \Biggr) \frac{\Omega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, \quad 0\leq s\leq \Lambda,\\ &\nu _{k,s}(\xi )=\sigma _{h,\Omega,k,\theta,s-\Lambda }, \quad\Lambda +1\leq s\leq 2 \Lambda,\\ &\widehat{ \vert \nu _{k,s} \vert }(\xi )= \int _{2^{k}< \vert y \vert \leq 2^{k+1}}\exp \Biggl( \sum_{i=1}^{s}a_{l_{i}} \varphi \bigl( \vert y \vert \bigr)^{l_{i}}\xi \cdot \theta \Biggr) \frac{ \vert \Omega (y/ \vert y \vert ) \vert }{ \vert y \vert ^{n}}\,dy,\quad 0\leq s\leq \Lambda,\\ &\vert \nu _{k,s} \vert (\xi )= \vert \sigma _{h,\Omega,k,\theta,s-\Lambda } \vert ,\quad \Lambda +1\leq s\leq 2\Lambda. \end{aligned}$$

Let $\xi =(\xi _{1},\ldots,\xi _{n})$. By (1.1) and a change of variable, one has

$$\begin{aligned} \widehat{\nu _{k,s}}(\xi )=0,\quad \forall 0\leq s\leq \Lambda. \end{aligned}$$

(3.63)

Invoking Lemma 2.5, one finds

$$\begin{aligned} \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi ) \bigr\vert = \Biggl\vert \int _{2^{k}}^{2^{(k+1)}} \exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi (t)^{l_{i}} \xi _{1} \Biggr)\frac{dt}{t} \Biggr\vert \Vert \Omega \Vert _{L^{1}({\mathrm{S}}^{n-1})} \leq C\bigl(\varphi \bigl(2^{k+1}\bigr)^{l_{s}} \vert a_{l_{s}}\xi _{1} \vert \bigr)^{-1/l_{s}}. \end{aligned}$$

Combining this with the trivial estimate $|\widehat{|\nu _{k,s}|}(\xi )|\leq C$ yields that

$$\begin{aligned} \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi ) \bigr\vert \leq C\min \bigl\{ 1,\bigl(\varphi \bigl(2^{k}\bigr)^{l_{s}} \vert a_{l_{s}} \xi _{1} \vert \bigr)^{-1/l_{s}}\bigr\} ,\quad 1\leq s\leq \Lambda. \end{aligned}$$

(3.64)

By the definition of $|\nu _{k,s}|$ and the arguments similar to those used to derive (3.12),

$$\begin{aligned} \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi ) \bigr\vert & = \bigl\vert \widehat{ \vert \sigma _{h,\Omega,k,s-\Lambda } \vert }( \xi ) \bigr\vert \\ & \leq C\min \bigl\{ 1,\bigl(\varphi \bigl(2^{k}\bigr)^{l_{s-\Lambda }} \bigl\vert L_{s- \Lambda }(\xi ) \bigr\vert \bigr)^{-1/(2(s-\Lambda )l_{s-\Lambda }\delta )}\bigr\} , \quad{\text{for }} \Lambda +1\leq s\leq 2\Lambda. \end{aligned}$$

(3.65)

We get from (3.12) that

$$\begin{aligned} \bigl\vert \widehat{\nu _{k,s}}(\xi ) \bigr\vert & = \bigl\vert \widehat{\sigma _{h,\Omega,k,s-\Lambda }}(\xi ) \bigr\vert \\ & \leq C\min \bigl\{ 1,\bigl(\varphi \bigl(2^{k}\bigr)^{l_{s-\Lambda }} \bigl\vert L_{s- \Lambda }(\xi ) \bigr\vert \bigr)^{-1/(2(s-\Lambda )l_{s-\Lambda }\delta )}\bigr\} , \quad{\text{for }} \Lambda +1\leq s\leq 2\Lambda. \end{aligned}$$

(3.66)

One can easily check that

$$\begin{aligned} & \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi )-\widehat{ \vert \nu _{k,s-1} \vert }(\xi ) \bigr\vert \\ &\quad= \Biggl\vert \int _{2^{k}}^{2^{(k+1)}} \Biggl(\exp \Biggl(\sum _{i=1}^{s}a_{l_{i}}\varphi (t)^{l_{i}} \xi _{1} \Biggr)-\exp \Biggl(\sum_{i=1}^{s-1}a_{l_{i}} \varphi (t)^{l_{i}}\xi _{1} \Biggr) \Biggr)\frac{dt}{t} \Biggr\vert \Vert \Omega \Vert _{L^{1}({\mathrm{S}}^{n-1})} \\ &\quad\leq C\varphi \bigl(2^{k+1}\bigr)^{l_{s}} \vert a_{l_{s}}\xi _{1} \vert \leq C\varphi \bigl(2^{k} \bigr)^{l_{s}} \vert a_{l_{s}} \xi _{1} \vert , \quad{\text{for }} 1\leq s\leq \Lambda. \end{aligned}$$

(3.67)

Arguments similar to (3.56) show that

$$\begin{aligned} \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi )-\widehat{ \vert \nu _{k,s-1} \vert }(\xi ) \bigr\vert &= \bigl\vert \widehat{ \vert \sigma _{h,\Omega,k,\theta,s-\Lambda } \vert }(\xi )- \widehat{ \vert \sigma _{h,\Omega,k,\theta,s-\Lambda -1} \vert }(\xi ) \bigr\vert \\ & \leq C\min \bigl\{ 1,\varphi \bigl(2^{k}\bigr)^{l_{s- \Lambda }} \bigl\vert L_{s-\Lambda }(\xi ) \bigr\vert \bigr\} \quad{\text{for }} \Lambda +1 \leq s \leq 2\Lambda. \end{aligned}$$

(3.68)

In view of (3.56),

$$\begin{aligned} \bigl\vert \widehat{\nu _{k,s}}(\xi )-\widehat{\nu _{k,s-1}}(\xi ) \bigr\vert &= \bigl\vert \widehat{\sigma _{h,\Omega,k,\theta,s-\Lambda }}(\xi )- \widehat{\sigma _{h,\Omega,k,\theta,s-\Lambda -1}}(\xi ) \bigr\vert \\ & \leq C\min \bigl\{ 1,\varphi \bigl(2^{k}\bigr)^{l_{s- \Lambda }} \bigl\vert L_{s-\Lambda }(\xi ) \bigr\vert \bigr\} \quad{\text{for }} \Lambda +1 \leq s \leq 2\Lambda. \end{aligned}$$

(3.69)

We now define linear transformations $I_{s}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ for $1\leq s\leq 2\Lambda$ by

$$\begin{aligned} I_{s}(\xi )= \textstyle\begin{cases} a_{l_{s}}\xi _{1}& {\text{if }} 1\leq s\leq \Lambda; \\ L_{s-\Lambda }(\xi )& {\text{if }} \Lambda +1\leq s\leq 2\Lambda. \end{cases}\displaystyle \end{aligned}$$

We also set

$$\begin{aligned} \gamma _{s}= \textstyle\begin{cases} l_{s}& {\text{if }} 1\leq s\leq \Lambda; \\ l_{s-\Lambda }& {\text{if }} \Lambda +1\leq s\leq 2\Lambda \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned} \beta _{s}= \textstyle\begin{cases} \frac{1}{sl_{s}}& {\text{if }} 1\leq s\leq \Lambda; \\ \frac{1}{2(s-\Lambda )l_{s-\Lambda }\delta }& {\text{if }} \Lambda +1 \leq s\leq 2\Lambda. \end{cases}\displaystyle \end{aligned}$$

It follows from (3.63)–(3.69) that

$$\begin{aligned} &\max \bigl\{ \bigl\vert \widehat{\nu _{k,s}}(\xi )-\widehat{\nu _{k,s-1}}(\xi ) \bigr\vert , \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi ) -\widehat{ \vert \nu _{k,s-1} \vert }(\xi ) \bigr\vert \bigr\} \leq C \bigl\vert 2^{k \gamma _{s}}I_{s}( \xi ) \bigr\vert ,\quad 1\leq s\leq 2\Lambda; \end{aligned}$$

(3.70)

$$\begin{aligned} &\max \bigl\{ \bigl\vert \widehat{\nu _{k,s}}(\xi ) \bigr\vert , \bigl\vert \widehat{ \vert \nu _{k,s} \vert }(\xi ) \bigr\vert \bigr\} \leq CA\min \bigl\{ 1, \bigl\vert 2^{k\gamma _{s}}I_{s}(\xi ) \bigr\vert ^{-\beta _{s}}\bigr\} ,\quad 1 \leq s\leq 2\Lambda. \end{aligned}$$

(3.71)

It is not difficult to see that

$$\begin{aligned} \sup_{k\in \mathbb{Z}} \bigl\vert \vert \nu _{k,0} \vert *f(x) \bigr\vert \leq C \vert f \vert (x). \end{aligned}$$

(3.72)

From (3.60) we see that

$$\begin{aligned} T_{h,\Omega,P,\varphi }^{*}f\leq \sup_{k\in \mathbb{Z}} \bigl\vert \vert \nu _{k,2\Lambda } \vert *f \bigr\vert +\sup_{k\in \mathbb{Z}} \Biggl\vert \sum_{i=k}^{\infty }\nu _{i,2\Lambda }*f \Biggr\vert . \end{aligned}$$

(3.73)

Using Lemmas 2.4 and 2.5 in [31], we obtain that, for any $1\leq s\leq 2\Lambda$ and $1< p, q, r<\infty$,

$$\begin{aligned} &\biggl\Vert \biggl(\sum_{i\in \mathbb{Z}} \biggl(\sum _{k\in \mathbb{Z}} \vert \nu _{k,s}*g_{k,i} \vert ^{2} \biggr)^{q/2} \biggr)^{1/q} \biggr\Vert _{L^{p}( \mathbb{R}^{n})} \leq C \biggl\Vert \biggl(\sum _{i\in \mathbb{Z}} \biggl(\sum_{k\in \mathbb{Z}} \vert g_{k,i} \vert ^{2} \biggr)^{q/2} \biggr)^{1/q} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}; \end{aligned}$$

(3.74)

$$\begin{aligned} & \biggl\Vert \biggl(\sum_{i\in \mathbb{Z}} \biggl\Vert \biggl( \sum_{k\in \mathbb{Z}} \bigl\vert \vert \nu _{k,s} \vert *g_{i,\zeta,k} \bigr\vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{r}(\mathfrak{R}_{n})}^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}( \mathbb{R}^{n})} \\ &\quad\leq C \biggl\Vert \biggl(\sum_{i\in \mathbb{Z}} \biggl\Vert \biggl(\sum_{k\in \mathbb{Z}} \vert g_{i,\zeta,k} \vert ^{2} \biggr)^{1/2} \biggr\Vert _{L^{r}(\mathfrak{R}_{n})}^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}( \mathbb{R}^{n})}. \end{aligned}$$

(3.75)

By (3.63), (3.70)–(3.72), (3.74), (3.75) and invoking Lemma 3.7, we have that, for $\alpha \in (0,1)$, $1< p, q<\infty$, and $1\leq s\leq 2\Lambda$,

$$\begin{aligned} &\biggl\Vert \biggl(\sum_{l\in \mathbb{Z}}2^{lq\alpha } \biggl( \int _{ \mathfrak{R}_{n}}\sup_{k\in \mathbb{Z}} \bigl\vert \vert \nu _{k,s} \vert * \vert \triangle _{2^{-l}\zeta }f \vert \bigr\vert \,d\zeta \biggr)^{q} \biggr)^{{1}/{q}} \biggr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{\dot{F}_{\alpha }^{p,q}( \mathbb{R}^{n})}, \end{aligned}$$

(3.76)

$$\begin{aligned} &\Biggl\Vert \Biggl(\sum_{l\in \mathbb{Z}}2^{lq\alpha } \Biggl( \int _{ \mathfrak{R}_{n}}\sup_{k\in \mathbb{Z}} \Biggl\vert \sum _{i=k}^{ \infty }\nu _{i,2\Lambda }*\triangle _{2^{-l}\zeta }f \Biggr\vert \,d\zeta \Biggr)^{q} \Biggr)^{{1}/{q}} \Biggr\Vert _{L^{p}(\mathbb{R}^{n})} \leq C \Vert f \Vert _{\dot{F}_{ \alpha }^{p,q}(\mathbb{R}^{n})}. \end{aligned}$$

(3.77)

Then (3.62) follows from (3.73), (3.76), and (3.77). Furthermore, the boundedness for $T_{h,\Omega,P,\varphi }^{*}$ on $F_{\alpha }^{p,q}(\mathbb{R}^{n})$ follows from the boundedness for $T_{h,\Omega,P,\varphi }^{*}$ on $\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{n})$, (2.4), (2.5), and Theorem 1.2. By (3.61), (3.62) and the arguments similar to those used in deriving the continuity part of [31, Theorem 1.1], we can get the continuity part in Theorem 1.3. This completes the proof of Theorem 1.3. □