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. □