1 Introduction

Let \({\mathbb{R}^{n}}\) (\(n\ge2\)) be the n-dimensional Euclidean space and \({{S}^{n-1}}\) be the unit sphere in \({\mathbb{R}^{n}}\) equipped with the induced Lebesgue measure \(d\sigma=d\sigma(\cdot)\). Suppose that \(\Omega\in L^{1}({{{S}^{n-1}}})\) satisfies the cancelation condition

$$ \int_{{{S}^{n-1}}}\Omega\bigl(y'\bigr)\,d \sigma\bigl(y'\bigr)=0. $$
(1.1)

For a suitable function ϕ and a measurable function h on \([0,\infty)\), we denote by \(T_{\Omega,\phi,h}\) the singular integral operator along the surface

$$\Gamma=\bigl\{ x=\phi\bigl(\vert y\vert \bigr)y':y\in{ \mathbb{R}^{n}}\bigr\} $$

defined as follows:

$$ T_{\Omega,h,\phi}f(x) =\mathrm{p.v.}\int_{{\mathbb{R}^{n}}} \frac{h(|y|)\Omega(y')}{|y|^{n}} f\bigl(x-\phi\bigl(\vert y\vert \bigr)y'\bigr) \,dy $$
(1.2)

for f in the Schwartz class \(\mathcal{S}({\mathbb{R}^{n}})\). If \(\phi=1\), then \(T_{\Omega,h,\phi}\) is the classical singular integral operator \(T_{\Omega,h}\), which is defined by

$$ T_{\Omega,h}f(x) =\mathrm{p.v.}\int_{{\mathbb{R}^{n}}} \frac{h(|y|)\Omega(y')}{|y|^{n}} f(x-y)\,dy. $$
(1.3)

When \(h\equiv1\), we denote simply \(T_{\Omega,h,\phi}\) and \(T_{\Omega,h}\) by \(T_{\Omega,\phi}\) and \(T_{\Omega}\), respectively.

The \(L^{p}\) boundedness of singular integrals along the surface has attracted the attention of many authors [1–3], etc. There are several papers concerning rough kernels associated to surfaces as above [4–6]. As one of them, we count the following one.

Theorem A

([5])

Let \(h\in\Delta_{\gamma}\) for some \(\gamma\ge2\), \(1< p<\infty\), \(\Omega\in H^{1}({{S}^{n-1}})\). Let ϕ be a nonnegative \(C^{1}\) function on \((0,\infty)\) satisfying

  1. (i)

    \(\phi(t)\) is strictly increasing and \(\phi(2t)\ge\lambda\phi(t)\) for all \(t>0\) and some \(\lambda>1\),

  2. (ii)

    \(\phi(t)\) satisfies a doubling condition \(\phi(2t)\le c \phi(t)\) for all \(t>0\) and some \(c>1\),

  3. (iii)

    \(\phi'(t)\ge C_{1}\phi(t)/t\) for all \(t>0\) and some \(C_{1}\).

Then \(T_{\Omega,h,\phi}\) is bounded on \(L^{p}({\mathbb{R}^{n}})\).

This is, in fact, stated in the more general setting, i.e., for a weighted case (Theorem 1 and Corollary 1 in [5]), but we state this as above for our purpose and for the sake of simplicity. We note here that condition (i) follows from (iii).

On the other hand, Triebel-Lizorkin space boundedness of rough singular integrals was also investigated by many authors, see [7, 8] and [9].

Before stating the following result, let us recall the definitions of some function spaces. First we give the definition of the Hardy space \(H^{1}({S}^{n-1})\):

$$\begin{aligned}& H^{1}\bigl({S}^{n-1}\bigr) \\& \quad = \biggl\{ \omega\in L^{1} \bigl({S}^{n-1}\bigr)\Bigm| \|f\|_{H^{1}({S}^{n-1})}=\biggl\Vert \sup _{0\le r< 1}\biggl\vert \int_{{S}^{n-1}}\omega \bigl(y'\bigr)P_{r(\cdot)}\bigl(y'\bigr)\, d \sigma\bigl(y'\bigr)\biggr\vert \biggr\Vert _{L^{1}({S}^{n-1})}< \infty \biggr\} , \end{aligned}$$

where \(P_{ry'}(x')\) denotes the Poisson kernel on \({S}^{n-1}\) defined by

$$P_{ry'}\bigl(x'\bigr)=\frac{1-r^{2}}{|ry'-x'|^{n}}, \quad 0\le r< 1 \text{ and } x',y'\in{S}^{n-1}. $$

For \(1\le\gamma\le\infty\), \(\Delta_{\gamma}(\mathbb{R}_{+})\) is the collection of all measurable functions \(h:[0,\infty)\to\mathbb{C}\) satisfying

$$\|h\|_{\Delta_{\gamma}}= \sup_{R>0} \biggl(\frac{1}{R} \int_{0}^{R}\bigl\vert h(t)\bigr\vert ^{\gamma}\, dt \biggr)^{1/\gamma}< \infty. $$

Note that

$$ L^{\infty}(\mathbb{R}_{+})=\Delta_{\infty}(\mathbb{R}_{+})\subset \Delta _{\beta}(\mathbb{R}_{+}) \subset\Delta_{\alpha}(\mathbb{R}_{+}) \quad \text{for }\alpha< \beta, $$

and all these inclusions are proper.

As a result of boundedness on Triebel-Lizorkin spaces, we cite the following one, which is somewhat different from our setting, but closely related.

Theorem B

([9])

Let \(\Omega\in H^{1}({{S}^{n-1}})\) satisfy the cancelation condition (1.1) and \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\). Let \(P=(P_{1},P_{2},\ldots,P_{d})\) be real polynomials in y. Then, for the singular integral

$$ T_{\Omega,P,h}f(x) =\mathrm{p.v.}\int_{{\mathbb{R}^{n}}} \frac{h(|y|)\Omega(y')}{|y|^{n}} f\bigl(x-P(y)\bigr)\,dy $$
  1. (i)

    for \(\alpha\in\mathbb{R}\) and \(|\frac{1}{p}-\frac{1}{2}|<\min(\frac{1}{2},\frac{1}{\gamma'})\) and \(|\frac{1}{q}-\frac{1}{2}|<\min(\frac{1}{2},\frac{1}{\gamma'})\), there exists a constant \(C>0\) such that \(\|T_{\Omega,P,h} f\|_{\dot{F}_{p,q}^{\alpha}(\mathbb{R}^{d})} \le C \|f\|_{\dot{F}_{p,q}^{\alpha}(\mathbb{R}^{d})}\);

  2. (ii)

    for \(\alpha\in\mathbb{R}\) and \(|\frac{1}{p}-\frac{1}{2}|<\min(\frac{1}{2},\frac{1}{\gamma'})\) and \(1< q<\infty\), there exists a constant \(C>0\) such that \(\|T_{\Omega,P,h} f\|_{\dot{B}_{p,q}^{\alpha}(\mathbb{R}^{d})} \le C \|f\|_{\dot{B}_{p,q}^{\alpha}(\mathbb{R}^{d})}\).

Remark 1

We think that there is a gap in the proof of part (i) in the above theorem. Their proof works in the same region as in our Theorem 1.1 below.

Besides \(H^{1}({{S}^{n-1}})\), there is another class of kernels which leads to \(L^{p}\) and Triebel-Lizorkin space boundedness of singular integral operators \(T_{\Omega,h}\). It is closely related to the class \(\mathcal{F}_{\alpha}\) introduced by Grafakos and Stefanov [10]. We say \(\Omega\in W\mathcal{F}_{\beta}=W\mathcal{F}_{\beta}({{S}^{n-1}})\) if

$$\begin{aligned} \|\Omega\|_{W\mathcal{F}_{\beta}}&:= \sup_{\xi'\in{{S}^{n-1}}} \biggl( \int_{{S}^{n-1}}\int_{{S}^{n-1}}\bigl\vert \Omega \bigl(y'\bigr){\Omega\bigl(z'\bigr)}\bigr\vert \log^{\beta} \frac{2e}{|(y'-z')\cdot\xi'|}\, d\sigma\bigl(y'\bigr)\, d \sigma\bigl(z'\bigr) \biggr)^{\frac{1}{2}} \\ &< \infty. \end{aligned}$$
(1.4)

We note that \(\bigcup_{r>1}L^{r}({{S}^{n-1}})\subset W\mathcal{F}_{\beta_{2}}({{S}^{n-1}}) \subset W\mathcal{F}_{\beta_{1}}({{S}^{n-1}})\) for \(0<\beta_{1}<\beta _{2}<\infty\).

About the inclusion relation between \(\mathcal{F}_{\beta _{1}}({{S}^{n-1}})\) and \(W\mathcal{F}_{\beta_{2}}({{S}^{n-1}})\), the following is known: when \(n=2\), Lemma 1 in [11] shows \(\mathcal{F}_{\beta}(S^{1})\subset W\mathcal{F}_{\beta}(S^{1})\). It is also known that \(W\mathcal{F}_{2\alpha}(S^{1})\setminus (\mathcal{F}_{\alpha}(S^{1})\cup H^{1}(S^{1}) )\ne\emptyset\), cf. [12].

Theorem C

([12])

Let \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\). Suppose that \(\Omega\in W\mathcal{F}_{\beta}=W\mathcal{F}_{\beta}({{S}^{n-1}})\) for some \(\beta>\max(\gamma', 2)\), and it satisfies the cancelation condition (1.1). Then the singular integral operator \(T_{\Omega,h}\) is bounded on \(\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) if \(\alpha\in \mathbb{R}\), and \((1/p,1/q)\) belongs to the interior of the parallelogram \(P_{1}P_{2}P_{3}P_{4}\), where \(P_{1}=(\frac{\max(\gamma',2)}{2\beta},\frac{\max(\gamma ',2)}{2\beta})\), \(P_{2}=(\frac{1}{\gamma'}+ \frac{\max(\gamma',2)}{2\beta}(\frac{1}{\gamma}-\frac{1}{\gamma '}), \frac{\max(\gamma',2)}{2\beta})\), \(P_{3}=(1-\frac{\max(\gamma',2)}{2\beta}, 1-\frac{\max(\gamma ',2)}{2\beta})\), and \(P_{4}=(\frac{1}{\gamma}-\frac{\max(\gamma',2)}{2\beta}(\frac {1}{\gamma} -\frac{1}{\gamma'}), 1-\frac{\max(\gamma',2)}{2\beta})\).

Let us recall the definitions of the homogeneous Triebel-Lizorkin spaces \(\dot{F}_{p,q}^{\alpha}=\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) and the homogeneous Besov spaces \(\dot{B}_{p,q}^{\alpha}=\dot{B}_{p,q}^{\alpha}({\mathbb{R}^{n}})\). For \(0< p,q\leq\infty\) (\(p\neq\infty\)) and \(\alpha\in\mathbb{R}\), \(\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) is defined by

$$ \dot{F}_{p,q}^{\alpha}\bigl({\mathbb{R}^{n}} \bigr) = \biggl\{ f\in\mathcal{S}'\bigl({\mathbb{R}^{n}} \bigr):\|f\|_{\dot {F}_{p,q}^{\alpha}} =\biggl\Vert \biggl(\sum _{k\in\mathbb{Z}} 2^{k\alpha q}\vert \Psi _{k}*f\vert ^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}}< \infty \biggr\} $$
(1.5)

and \(\dot{B}_{p}^{\alpha,q}({\mathbb{R}^{n}})\) is defined by

$$ \dot{B}_{p,q}^{\alpha}\bigl({\mathbb{R}^{n}} \bigr)= \biggl\{ f\in\mathcal {S}'\bigl({\mathbb{R}^{n}} \bigr): \|f\|_{\dot{B}_{p,q}^{\alpha}}= \biggl(\sum_{k\in\mathbb{Z}} 2^{k\alpha q}\Vert \Psi_{k}*f\Vert _{L^{p}}^{q} \biggr)^{1/q}< \infty \biggr\} , $$
(1.6)

where \(\mathcal{S}'({\mathbb{R}^{n}})\) denotes the tempered distribution class on \({\mathbb{R}^{n}}\), \(\widehat{\Psi}_{k}(\xi)=\Phi(2^{-k}\xi)\) for \(k\in\mathbb{Z}\) and \(\Phi\in C_{c}^{\infty}({\mathbb{R}}^{n})\) is a radial function satisfying the following conditions:

$$ \begin{aligned} (\mathrm{i})&\quad 0\leq\Phi\leq1; \\ (\mathrm{ii})&\quad \operatorname{supp}\Phi\subset\bigl\{ \xi:1/2\leq \vert \xi \vert \leq2\bigr\} ; \\ (\mathrm{iii})&\quad \Phi>c>0 \quad \text{if } 3/5\leq|\xi|\leq5/3; \\ (\mathrm{iv})&\quad \sum_{j\in\mathbb{Z}}\Phi \bigl(2^{-j}\xi\bigr)=1 \quad (\xi \ne0). \end{aligned} $$
(1.7)

The inhomogeneous versions of Triebel-Lizorkin space and Besov space, which are denoted by \(F_{p,q}^{\alpha}({\mathbb{R}^{n}})\) and \(B_{p,q}^{\alpha}({\mathbb{R}^{n}})\) respectively, are obtained by adding the term \(\|\Phi_{0} * f\|_{p}\) to the right-hand side of (1.5) or (1.6) with \(\sum_{k\in\mathbb{Z}}\) replaced by \(\sum_{k=0}^{\infty}\), where \(\Phi_{0} \in\mathcal{S}({\mathbb{R}^{n}})\), \(\operatorname{supp}\widehat{\Phi}_{0}\subset\{\xi:|\xi|\leq2\}\), and \(\widehat{\Phi}_{0}(\xi)>c>0\) if \(|\xi|\leq5/3\).

The following properties of the Triebel-Lizorkin space and the Besov space are well known. Let \(1< p,q<\infty\), \(\alpha\in\mathbb{R}\), and \(1/p+1/p'=1\), \(1/q+1/q'=1\):

$$ \begin{aligned} (\mathrm{a}) &\quad \dot{F}_{2,2}^{0}=\dot{B}_{2,2}^{0}=L^{2}, \qquad \dot{F}_{p,2}^{0}=L^{p}\quad \text{and} \\ &\quad \dot{F}_{p,p}^{\alpha} =\dot{B}_{p,p}^{\alpha} \quad \text{for } 1< p<\infty \quad \text{and}\quad \dot{F}_{\infty,2}^{0}= \mathrm{BMO}; \\ (\mathrm{b}) &\quad F_{p,q}^{\alpha}\sim\dot{F}_{p,q}^{\alpha} \cap L^{p} \quad \text{and} \quad \|f\|_{F_{p,q}^{\alpha}} \sim\|f \|_{\dot{F}_{p,q}^{\alpha}}+\|f\|_{L^{p}} \quad (\alpha>0); \\ (\mathrm{c}) &\quad B_{p,q}^{\alpha}\sim\dot{B}_{p,q}^{\alpha} \cap L^{p} \quad \text{and} \quad \|f\|_{B_{p,q}^{\alpha}} \sim\|f \|_{\dot{B}_{p,q}^{\alpha}}+\|f\|_{L^{p}}\quad (\alpha>0); \\ (\mathrm{d}) &\quad\bigl(\dot{F}_{p,q}^{\alpha}\bigr)^{*}= \dot{F}_{p',q'}^{-\alpha } \quad \text{and}\quad \bigl(F_{p,q}^{\alpha} \bigr)^{*}=F_{p',q'}^{-\alpha}; \\ (\mathrm{e})&\quad \bigl(\dot{B}_{p,q}^{\alpha}\bigr)^{*}= \dot{B}_{p',q'}^{-\alpha } \quad \text{and} \quad \bigl(B_{p,q}^{\alpha}\bigr)^{*}=B_{p',q'}^{-\alpha}; \\ (\mathrm{f}) &\quad \bigl(\dot{F}_{p,q_{1}}^{\alpha_{1}}, \dot{F}_{p,q_{2}}^{\alpha_{2}} \bigr)_{\theta,q}= \dot{B}_{p,q}^{\alpha}\\ &\quad \quad \bigl(\alpha_{1}\ne\alpha_{2}, 0<p<\infty, 0<q,q_{1},q_{2}\le\infty, \alpha=(1-\theta) \alpha_{1}+\theta\alpha_{2}, 0<\theta<1\bigr). \end{aligned} $$
(1.8)

See [13] and [14] for more properties of \(\dot{F}_{p,q}^{\alpha}\) and \(\dot {B}_{p,q}^{\alpha}\). See Triebel [14], p.64 and p.244, for (f).

Now we can state our first result.

Theorem 1.1

Let ϕ be a positive increasing function on \((0,\infty)\) satisfying

$$ \phi(2t)\le c_{1}\phi(t) \quad (t>0) \textit{ for some }c_{1}>1 $$
(1.9)

and

$$ \varphi(t)=\phi(t)/\bigl(t\phi'(t)\bigr)\in L^{\infty}(0,\infty). $$
(1.10)

Let \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\). Suppose \(\Omega\in H^{1}({{S}^{n-1}})\) satisfying the cancelation condition (1.1). Then

  1. (i)

    \(T_{\Omega,h,\phi}\) is bounded on \(\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) for \(\alpha\in\mathbb{R}\) and p, q with \((\frac{1}{p},\frac{1}{q})\) belonging to the interior of the octagon \(P_{1}P_{2}R_{2}P_{3}P_{4}P_{5}R_{4}P_{6}\) (hexagon \(P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}\) in the case \(1<\gamma\le2\)), where \(P_{1}=(\frac{1}{2}-\frac{1}{\max\{2,\gamma'\}}, \frac{1}{2}-\frac{1}{\max\{2,\gamma'\}})\), \(P_{2}=(\frac{1}{2},\frac{1}{2}-\frac{1}{\max\{2,\gamma'\}})\), \(P_{3}=(\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}},\frac{1}{2})\), \(P_{4}=(\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}}, \frac{1}{2}+\frac{1}{\max\{2,\gamma'\}})\), \(P_{5}=(\frac{1}{2},\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}})\), \(P_{6}=(\frac{1}{2}-\frac{1}{\max\{2,\gamma'\}},\frac{1}{2})\), \(R_{2}=(1-\frac{1}{2\gamma}, \frac{1}{2\gamma})\), and \(R_{4}=(\frac{1}{2\gamma},1-\frac{1}{2\gamma})\);

  2. (ii)

    \(T_{\Omega,h,\phi}\) is bounded on \(\dot{B}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) for \(\alpha\in \mathbb{R}\) and p, q satisfying \(|\frac{1}{2}-\frac{1}{p}|<\min\{\frac{1}{2},\frac{1}{\gamma '}\}\) and \(1< q<\infty\).

See Figures 1 and 2 for the conclusion (i) of Theorem 1.1.

Figure 1
figure 1

( \(\pmb{1<\gamma<2}\) ).

Full size image
Figure 2
figure 2

( \(\pmb{2\leq\gamma\leq\infty}\) ).

Full size image

Example 1

As typical examples of ϕ satisfying conditions (1.9) and (1.10), we list the following three: \(t^{\alpha}\log ^{\beta}(1+t)\) (\(\alpha>0\), \(\beta\ge0\)), \((2t^{2}-2t+1)t^{1+\alpha}\) (\(\alpha\ge0\)), and \(\phi(t)=2t^{2}+t\) (\(0< t<\frac{\pi}{2}\)), \(\phi(t)=2t^{2}+t\sin t\) (\(t\ge\frac{\pi}{2}\)). Note that linear combinations with positive coefficients of functions ϕ’s satisfying the above two conditions also satisfy them, cf. [15].

We shall state our following result, which relates to two function spaces \(L(\log L)({{S}^{n-1}})\) and the block spaces \(B_{q}^{(0,0)}({{S}^{n-1}})\). Let \(L(\log L)^{\alpha}({{S}^{n-1}})\) (for \(\alpha>0\)) denote the class of all measurable functions Ω on \({{S}^{n-1}}\) which satisfy

$$ \|\Omega\|_{L(\log L)^{\alpha}({{S}^{n-1}})}=\int_{{{S}^{n-1}}}\bigl\vert \Omega \bigl(y'\bigr)\bigr\vert \log^{\alpha}\bigl(2+\bigl\vert \Omega\bigl(y'\bigr)\bigr\vert \bigr)\,d\sigma \bigl(y'\bigr)< \infty. $$

Denote by \(L(\log L)({{S}^{n-1}})\) for \(L(\log L)^{1}({{S}^{n-1}})\). A well-known fact is \(L(\log L)({{S}^{n-1}})\subset H^{1}({{S}^{n-1}})\).

Next, we turn to the block space \(B_{q}^{(0,v)}({{S}^{n-1}})\). A q-block on \({{S}^{n-1}}\) is an \(L^{q}({{S}^{n-1}})\) (\(1< q\le\infty \)) function b which satisfies

$$ \begin{aligned} (\mathrm{i})&\quad \operatorname{supp}b \subset I; \\ (\mathrm{ii})&\quad \|b\|_{q}\le|I|^{-1/q'}, \end{aligned} $$
(1.11)

where \(|I|=\sigma(I)\), and \(I=B(x_{0}',\theta_{0})\cap{{S}^{n-1}}\) is a cap on \({{S}^{n-1}}\) for some \(x_{0}'\in{{S}^{n-1}}\) and \(\theta_{0}\in(0,1]\). For \(1< q\le\infty\) and \(v>-1\), the block space \(B_{q}^{(0,v)}({{S}^{n-1}})\) is defined by

$$ B_{q}^{(0,v)}\bigl({{S}^{n-1}}\bigr)= \Biggl\{ \Omega\in L^{1}\bigl({{S}^{n-1}}\bigr); \Omega =\sum _{j=1}^{\infty} \lambda_{j} b_{j}, M_{q}^{(0,v)}\bigl(\{\lambda_{j}\} \bigr)< \infty \Biggr\} , $$
(1.12)

where \(\lambda_{j}\in\mathbb{C}\) and \(b_{j}\) is a q-block supported on a cap \(I_{j}\) on \({{S}^{n-1}}\), and

$$ M_{q}^{(0,v)}\bigl(\{\lambda_{j}\} \bigr)=\sum_{j=1}^{\infty}|\lambda_{j}| \bigl\{ 1+\log^{(v+1)} \bigl(|I_{j}|^{-1} \bigr) \bigr\} . $$
(1.13)

For \(\Omega\in B_{q}^{(0,v)}({{S}^{n-1}})\), denote

$$\|\Omega\|_{B_{q}^{(0,v)}({{S}^{n-1}})}=\inf \Biggl\{ M_{q}^{(0,v)}\bigl(\{ \lambda_{j}\}\bigr); \Omega=\sum_{j=1}^{\infty} \lambda_{j} b_{j}, b_{j} \text{ is a } q \text{-block} \Biggr\} . $$

Then \(\|\cdot\|_{B_{q}^{(0,v)}({{S}^{n-1}})}\) is a norm on the space \(B_{q}^{(0,v)}({{S}^{n-1}})\), and \((B_{q}^{(0,v)} ({{S}^{n-1}}), \|\cdot\|_{B_{q}^{(0,v)}({{S}^{n-1}})} )\) is a Banach space.

Historically, the block spaces in \(\mathbb{R}^{n}\) originated in the work of Taibleson and Weiss on the convergence of the Fourier series in connection with the developments of the real Hardy spaces. The block spaces on \({{S}^{n-1}}\) were introduced by Jiang and Lu [16] in studying the homogeneous singular integral operators. For further information about the theory of spaces generated by blocks and its applications to harmonic analysis, see the book [17] and survey article [18]. The following inclusion relations are known:

$$ \begin{aligned} (\mathrm{a}) &\quad B_{q}^{(0,v_{1})}\bigl({{S}^{n-1}}\bigr)\subset B_{q}^{(0,v_{2})}\bigl({{S}^{n-1}}\bigr)\quad \text{if }v_{1}> v_{2}>-1; \\ (\mathrm{b}) &\quad B_{q_{1}}^{(0,v)}\bigl({{S}^{n-1}} \bigr)\subset B_{q_{2}}^{(0,v)}\bigl({{S}^{n-1}}\bigr) \quad \text{if } 1< q_{2}<q_{1}\text{ for any }v>-1; \\ (\mathrm{c}) &\quad \bigcup_{p>1}L^{p} \bigl({{S}^{n-1}}\bigr)\subset B_{q}^{(0,v)} \bigl({{S}^{n-1}}\bigr) \quad \text{for any }q>1, v>-1; \\ (\mathrm{d}) &\quad \bigcup_{q>1}B_{q}^{(0,v)} \bigl({{S}^{n-1}}\bigr)\not\subset \bigcup_{q>1}L^{q} \bigl({{S}^{n-1}}\bigr)\quad \text{for any }v>-1; \\ (\mathrm{e}) &\quad B_{q}^{(0,v)}\bigl(S^{n-1}\bigr) \subset H^{1}\bigl(S^{n-1}\bigr) +L(\log L)^{1+v} \bigl(S^{n-1}\bigr) \quad \text{for any }q>1, v>-1; \\ (\mathrm{f}) &\quad \bigcup_{q>1} B_{q}^{(0,0)}\bigl({{S}^{n-1}}\bigr)\subset H^{1}\bigl({{S}^{n-1}}\bigr). \end{aligned} $$
(1.14)

The following theorem shows that if Ω belongs to \(L\log L({{S}^{n-1}})\) or block spaces, then we can get better results than Theorem 1.1.

Theorem 1.2

Let ϕ be a positive increasing function on \((0,\infty)\) satisfying the same condition as in Theorem  1.1. Let \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\), and \(\Omega \in L^{1}({{S}^{n-1}})\) satisfy the cancelation condition (1.1). Then if \(\Omega\in L(\log L)({{S}^{n-1}})\cup (\bigcup_{1< q<\infty} B_{q}^{(0,0)}({{S}^{n-1}}) )\), then

  1. (i)

    \(T_{\Omega,h,\phi}\) is bounded on \(\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) for \(\alpha\in\mathbb{R}\) and p, q with \((\frac{1}{p},\frac{1}{q})\) belonging to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)), where \(Q_{1}=(0,0)\), \(Q_{2}=(\frac{1}{\gamma'},0)\), \(Q_{3}=(1,1)\), \(Q_{4}=(\frac{1}{\gamma},1)\), \(P_{3}=(\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}},\frac{1}{2})\), \(P_{6}=(\frac{1}{2}1\frac{1}{\max\{2,\gamma'\}},\frac{1}{2})\), \(R_{2}=(1-\frac{1}{2\gamma}, \frac{1}{2\gamma})\), and \(R_{4}=(\frac{1}{2\gamma},1-\frac{1}{2\gamma})\);

  2. (ii)

    \(T_{\Omega,h,\phi}\) is bounded on \(\dot{B}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) for \(\alpha\in \mathbb{R}\) and \(1< p,q<\infty\).

See Figures 3 and 4 for the conclusion of Theorem 1.2 for the cases \(1<\gamma<2\) and \(2\le\gamma<\infty\), respectively.

Figure 3
figure 3

( \(\pmb{1<\gamma<2}\) ).

Full size image
Figure 4
figure 4

( \(\pmb{2\leq\gamma\leq\infty}\) ).

Full size image

As a corresponding result to Theorem C, we have the following theorem.

Theorem 1.3

Let ϕ be a positive increasing function on \((0,\infty)\) satisfying the same condition as in Theorem  1.1. Let \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\). Suppose \(\Omega\in W\mathcal{F}_{\beta}=W\mathcal{F}_{\beta}({{S}^{n-1}})\) for some \(\beta>\max(\gamma', 2)\), and it satisfies the cancelation condition (1.1). Then

  1. (i)

    the singular integral operator \(T_{\Omega,h,\phi}\) is bounded on \(\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\), if \(\alpha\in \mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{R}_{2}\mathcal{P}_{3}\mathcal{Q}_{3}\mathcal{Q}_{4} \mathcal{R}_{4}\mathcal{P}_{6}\) (hexagon \(\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{P}_{3}\mathcal{Q}_{3}\mathcal{Q}_{4} \mathcal{P}_{6}\) in the case \(1<\gamma\le2\)), where \(\mathcal{Q}_{1}= (\frac{\max(\gamma',2)}{2\beta}, \frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{Q}_{2}= (\frac{1}{\gamma'}+ \frac{\max(\gamma',2)}{\beta}(\frac{1}{2}-\frac{1}{\gamma'}), \frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{P}_{3}= (\frac{1}{2}+\frac{1}{\max(\gamma',2)}-\frac {1}{\beta}, \frac{1}{2} )\), \(\mathcal{Q}_{3}= (1-\frac{\max(\gamma',2)}{2\beta}, 1-\frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{Q}_{4} = (\frac{1}{\gamma}-\frac{\max(\gamma',2)}{\beta}(\frac {1}{\gamma} -\frac{1}{2}), 1-\frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{P}_{6}= (\frac{1}{2}-\frac{1}{\max(\gamma',2)}+\frac {1}{\beta}, \frac{1}{2} )\), \(\mathcal{R}_{2}= (1-\frac{1}{2\gamma}-\frac{\max(\gamma ',2)}{2\beta\gamma'}, \frac{1}{2\gamma}+\frac{\max(\gamma',2)}{2\beta\gamma'} )\), and \(\mathcal{R}_{4}= (\frac{1}{2\gamma}+\frac{\max(\gamma ',2)}{2\beta\gamma'}, 1-\frac{1}{2\gamma}-\frac{\max(\gamma',2)}{2\beta\gamma'} )\);

  2. (ii)

    \(T_{\Omega,h,\phi}\) is bounded on \(\dot{B}_{p,q}^{\alpha}({\mathbb{R}^{n}})\), if \(\alpha\in\mathbb{R}\), \(\frac{\max(\gamma',2)}{2\beta}< p<1-\frac{\max(\gamma ',2)}{2\beta}\) and \(1< q<\infty\).

This improves Theorem C sufficiently. See Figures 3 and 4 for the conclusion (i) of Theorem 1.3.

The proofs of Theorems 1.1 and 1.3 will be given in Sections 2 and 3, respectively, and the proof of Theorem 1.2 will be given in Section 4. The letter C will denote a positive constant that may vary at each occurrence but is independent of the essential variables.

2 Proof of Theorem 1.1

2.1 Some lemmas

In [19], the following atom-decomposition of \(H^{1}(S^{n-1})\) was given. If \(\Omega\in H^{1}(S^{n-1})\) satisfying (1.1), then

$$ \Omega=\sum_{j=1}^{\infty}\lambda_{j} a_{j}, $$
(2.1)

where \(\sum_{j=1}^{\infty}|\lambda_{j}|\le C\|\Omega\|_{H^{1}(S^{n-1})}\) and each \(a_{j}\) is a regular \(H^{1}(S^{n-1})\) atom. A function a on \({S}^{n-1}\) is called regular ∞-atom in \(H^{1}(S^{n-1})\) if there exist \(\zeta\in{S}^{n-1}\) and \(\rho\in(0,2]\) such that

  1. (i)

    \(\operatorname{supp}(a)\subset{S}^{n-1}\cap B(\zeta,\rho)\), where \(B(\zeta,\rho)=\{y\in{R}^{n}: |y-\zeta|<\rho\}\);

  2. (ii)

    \(\|a\|_{L^{\infty}}\le\rho^{-n+1}\);

  3. (iii)

    \(\int_{{S}^{n-1}}a(y)\, d\sigma(y)=0\).

Let a be a regular ∞-atom. When \(n\ge3\), set

$$ E_{a}\bigl(s,\xi'\bigr)= \bigl(1-s^{2}\bigr)^{\frac{n-3}{2}}\chi_{(-1,1)}(s)\int _{S^{n-2}} a \bigl(s, \sqrt{1-s^{2}} \tilde{y} \bigr)\,d \sigma (\tilde{y}), $$
(2.2)

and when \(n=2\), set

$$ e_{a}\bigl(s,\xi'\bigr)= \frac{1}{\sqrt{1-s^{2}}}\chi_{(-1,1)}(s) \bigl[ a \bigl(s, \sqrt{1-s^{2}} \bigr) +a \bigl(s, -\sqrt{1-s^{2}} \bigr) \bigr]. $$
(2.3)

Next we prepare two lemmas, whose proofs can be found in Fan and Pan [4].

Lemma 2.1

Let Ω be a regular ∞-atom in \(H^{1}(S^{n-1})\) (\(n\ge3\)). Then there exists a constant \(c>0\), independent of Ω, such that \(c E_{\Omega}(s,\xi')\) is an ∞-atom in \(H^{1}(\mathbb{R})\). That is, \(c E_{\Omega}(s,\xi')\) satisfies

$$ \begin{aligned} &\|c E_{\Omega}\|_{L^{\infty}}\le\frac{1}{4r(\xi')}, \qquad \operatorname {supp}E_{\Omega}\subset \bigl(\xi_{1}'-2r \bigl(\xi'\bigr), \xi_{1}'+2r\bigl( \xi'\bigr) \bigr)\quad \textit {and} \\ &\int_{\mathbb{R}}E_{\Omega}\bigl(s,\xi'\bigr)\,ds=0, \end{aligned} $$
(2.4)

where \(r(\xi')=|\xi|^{-1}|A_{\tau}\xi|\) and \(A_{\tau}(\xi)=(\tau^{2}\xi _{1},\tau\xi_{2},\ldots, \tau\xi_{n})\).

Lemma 2.2

Let Ω be a regular ∞-atom in \(H^{1}(S^{1})\). Then, for \(1< q<2\), there exists a constant \(c>0\), independent of Ω, such that \(c e_{\Omega}(s,\xi')\) is a q-atom in \(H^{1}(\mathbb{R})\), the center of whose support is \(\xi_{1}'\) and the radius \(r(\xi')=|\xi|^{-1} (\tau^{4}\xi_{1}^{2}+\tau^{2}\xi_{2}^{2} )^{1/2}\).

For \(\Omega\in L^{1}({{S}^{n-1}})\), \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\), and a suitable function ϕ on \(\mathbb{R}_{+}\), we define the maximal functions \(M_{\Omega,h,\phi}\) by

$$ M_{\Omega,h,\phi} f(x)= \sup_{k\in\mathbb{Z}} \frac{1}{2^{kn}}\int_{2^{k-1}< |y|\le2^{k}} \bigl\vert \Omega \bigl(y'\bigr)h\bigl(\vert y\vert \bigr)f\bigl(x-\phi\bigl( \vert y\vert \bigr)y'\bigr)\bigr\vert \,dy. $$
(2.5)

Let ϕ be a positive increasing function on \((0,\infty)\) satisfying \(\phi(2t)\le c_{1}\phi(t)\) (\(t>0\)) for some \(c_{1}>1\), and \(\varphi(t)=\phi(t)/(t\phi'(t))\in L^{\infty}(0,\infty)\). Then, as in the proof of Lemma 2.3 in [20], p.246, we have

$$\begin{aligned} M_{\Omega,h,\phi} f(x) \le&\|h\|_{\Delta_{\gamma}}\bigl(\Vert \Omega \Vert _{L^{1}({{S}^{n-1}})}\bigr)^{\frac{1}{\gamma}} \\ &{}\times \biggl(\int _{{{S}^{n-1}}}\bigl\vert \Omega\bigl(y'\bigr)\bigr\vert M_{y'} \bigl(|f|^{\gamma'}\bigr) (x)\, d\sigma \bigl(y'\bigr) \biggr)^{\frac{1}{\gamma'}}, \end{aligned}$$
(2.6)

where \(M_{y'}g\) is the directional Hardy-Littlewood maximal function of g defined by

$$ M_{y'}g(x)=\sup_{r>0}\frac{1}{2r}\int _{|t|< r}\bigl\vert g\bigl(x-ty'\bigr)\bigr\vert \,dt. $$

For this directional maximal function \(M_{y'}\), we know that for \(1< p,q<\infty\),

$$\begin{aligned}& \biggl(\int_{{\mathbb{R}^{n}}} \biggl[ \biggl(\sum _{j\in\mathbb{Z}} \bigl(M_{y'}(f_{j}) (x) \bigr)^{q} \biggr)^{\frac{1}{q}} \biggr]^{p}\, dx \biggr)^{\frac{1}{p}} \\& \quad \le C_{p,q} \biggl(\int_{{\mathbb{R}^{n}}} \biggl[ \biggl(\sum_{j\in \mathbb{Z}} \bigl\vert f_{j}(x)\bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr]^{p}\, dx \biggr)^{\frac{1}{p}}. \end{aligned}$$
(2.7)

This is just (2.7) in the proof of Lemma 2.3 of [8], p.496. From (2.6) and (2.7), we get the following lemma.

Lemma 2.3

Let ϕ be a positive increasing function on \((0,\infty)\) satisfying \(\phi(2t)\le c_{1}\phi(t)\) (\(t>0\)) for some \(c_{1}>1\), and \(\varphi(t)=\phi(t)/(t\phi'(t))\in L^{\infty}(0,\infty)\). Let \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\). For \(\gamma'< p,q<\infty\), we have

$$ \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}}|M_{\Omega,h,\phi}f_{j}|^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \le C\biggl\Vert \biggl( \sum_{j\in\mathbb{Z}}|f_{j}|^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})}. $$
(2.8)

Proof

Let \(\{g_{j}\}_{j\in\mathbb{Z}}\) be a sequence of functions satisfying \(\|(\sum_{j\in\mathbb{Z}}|g_{j}|^{q'})^{1/{q'}}\|_{L^{p'}({\mathbb {R}^{n}})}\le1\). Then, noting \(p,q>\gamma'\) and using (2.6), the duality, and Minkowski’s inequality, we see that

$$\begin{aligned}& \biggl\vert \int_{{\mathbb{R}^{n}}}\sum _{j\in\mathbb{Z}}M_{\Omega ,h,\phi}f_{j}(x) g_{j}(x) \,dx\biggr\vert \\& \quad \le C\int_{{\mathbb{R}^{n}}}\sum _{j\in\mathbb{Z}} \biggl(\int_{{{S}^{n-1}}}\bigl\vert \Omega\bigl(y'\bigr)\bigr\vert M_{y'} \bigl(|f_{j}|^{\gamma'}\bigr) (x)\, d\sigma\bigl(y' \bigr) \biggr)^{\frac{1}{\gamma'}}\bigl\vert g_{j}(x)\bigr\vert \,dx \\& \quad \le C \biggl(\int_{{\mathbb{R}^{n}}} \biggl(\sum _{j\in\mathbb{Z}} \biggl(\int_{{{S}^{n-1}}}\bigl\vert \Omega\bigl(y'\bigr)\bigr\vert M_{y'} \bigl(|f_{j}|^{\gamma'}\bigr) (x)\, d\sigma\bigl(y' \bigr) \biggr)^{\frac{q}{\gamma'}} \biggr)^{\frac{p}{q}}\, dx \biggr)^{\frac{1}{p}} \\& \qquad {}\times \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}}\bigl\vert g_{j}(x)\bigr\vert ^{q'} \biggr)^{\frac{1}{q'}} \biggr\Vert _{L^{p'}({\mathbb{R}^{n}})} \\& \quad \le C \biggl\{ \int_{{{S}^{n-1}}}\bigl\vert \Omega \bigl(y'\bigr)\bigr\vert \biggl(\int_{{\mathbb{R}^{n}}} \biggl[ \biggl(\sum_{j\in\mathbb{Z}} \bigl(M_{y'} \bigl(|f_{j}|^{\gamma'}\bigr) (x) \bigr)^{\frac{q}{\gamma'}} \biggr)^{\frac{\gamma'}{q}} \biggr]^{\frac{p}{\gamma'}}\, dx \biggr)^{\frac{\gamma'}{p}} \,d \sigma\bigl(y'\bigr) \biggr\} ^{\frac{1}{\gamma'}}. \end{aligned}$$

Hence by (2.7) we have

$$\begin{aligned} \biggl\vert \int_{{\mathbb{R}^{n}}}\sum_{j\in\mathbb{Z}}M_{\Omega ,h,\phi}f_{j}(x) g_{j}(x)\,dx\biggr\vert &\le C \biggl\{ \int_{{{S}^{n-1}}} \bigl\vert \Omega\bigl(y'\bigr)\bigr\vert \biggl(\int _{{\mathbb{R}^{n}}} \biggl(\sum_{j\in\mathbb{Z}} \bigl\vert f_{j}(x)\bigr\vert ^{q} \biggr)^{\frac{p}{q}}\, dx \biggr)^{\frac{\gamma'}{p}} \,d\sigma\bigl(y'\bigr) \biggr\} ^{\frac{1}{\gamma'}} \\ &\le C\biggl\Vert \biggl(\sum_{j\in\mathbb{Z}}|f_{j}|^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})}, \end{aligned}$$

which implies our (2.8). □

Now, for \(\Omega\in L^{1}({{S}^{n-1}})\), we define the measures \(\sigma _{\Omega,h,\phi,k}\) on \({\mathbb{R}^{n}}\) and the maximal operator \(\sigma_{\Omega,h,\phi }^{*}f(x)\) by

$$\begin{aligned}& \int_{\mathbb{R}^{n}}f(x)\,d\sigma_{\Omega,h,\phi,k}(x) =\int _{\mathbb{R}^{n}}f\bigl(\phi\bigl(|x|\bigr)x'\bigr)\frac{\Omega(x')h(|x|)}{|x|^{n}} \chi_{2^{k-1}< |x|\le2^{k}}(x)\,dx, \end{aligned}$$
(2.9)
$$\begin{aligned}& \sigma_{\Omega,h,\phi}^{*}f(x) =\sup_{k\in\mathbb{Z}}\bigl\vert \vert \sigma_{\Omega,h,\phi ,k}\vert *f(x)\bigr\vert , \end{aligned}$$
(2.10)

where \(|\sigma_{\Omega,h,\phi,k}|\) is defined in the same way as \(\sigma_{\Omega,h,\phi,k}\), but with Ω replaced by \(|\Omega |\) and h by \(|h|\).

Then we have the following lemma.

Lemma 2.4

Let ϕ be a positive increasing function on \((0,\infty)\) satisfying \(\phi(2t)\le c_{1}\phi(t)\) (\(t>0\)) for some \(c_{1}>1\), and \(\varphi(t)=\phi(t)/(t\phi'(t))\in L^{\infty}(0,\infty)\). Let \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\), \(\Omega\in L^{1}({{S}^{n-1}})\). Then:

  1. (i)

    If \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(P_{1}P_{2}R_{2}P_{3}P_{4}P_{5}R_{4}P_{6}\), there exists \(C>0\) such that

    $$\begin{aligned}& \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl(\sum_{k\in\mathbb{Z}}\vert \sigma_{\Omega,h,\phi ,k}*g_{k,j} \vert ^{2} \biggr)^{\frac{q}{2}} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\& \quad \le C\|h\|_{\Delta_{\gamma}}\|\Omega\|_{L^{1}({{S}^{n-1}})} \biggl\Vert \biggl( \sum_{j\in\mathbb{Z}} \biggl(\sum_{k\in\mathbb{Z}}|g_{k,j}|^{2} \biggr)^{\frac{q}{2}} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})}, \end{aligned}$$
    (2.11)

    where \(P_{1}=(\frac{1}{2}-\frac{1}{\max\{2,\gamma'\}}, \frac{1}{2}-\frac{1}{\max\{2,\gamma'\}})\), \(P_{2}=(\frac{1}{2},\frac{1}{2}-\frac{1}{\max\{2,\gamma'\}})\), \(P_{3}=(\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}},\frac{1}{2})\), \(P_{4}=(\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}}, \frac{1}{2}+\frac{1}{\max\{2,\gamma'\}})\), \(P_{5}=(\frac{1}{2},\frac{1}{2}+\frac{1}{\max\{2,\gamma'\}})\), \(P_{6}=(\frac{1}{2}-\frac{1}{\max\{2,\gamma'\}},\frac{1}{2})\), \(R_{2}=(1-\frac{1}{2\gamma}, \frac{1}{2\gamma})\), and \(R_{4}=(\frac{1}{2\gamma},1-\frac{1}{2\gamma})\).

    (Note that if \(1<\gamma\le2\), the octagon \(P_{1}P_{2}R_{2}P_{3}P_{4}P_{5}R_{4}P_{6}\) reduces to the hexagon \(P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}\).)

  2. (ii)

    If \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of \(Q_{1}Q_{2}Q_{3}Q_{4}\), there exists \(C>0\) such that

    $$\begin{aligned}& \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \sum _{k\in\mathbb{Z}}\vert \sigma_{\Omega,h,\phi,k}*g_{k,j} \vert ^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\& \quad \le C\|h\|_{\Delta_{\gamma}}\|\Omega\|_{L^{1}({{S}^{n-1}})} \biggl\Vert \biggl( \sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb {Z}}|g_{k,j}|^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})}, \end{aligned}$$
    (2.12)

    where \(Q_{1}=(0,0)\), \(Q_{2}=(\frac{1}{\gamma'},0)\), \(Q_{3}=(1,1)\), and \(Q_{4}=(\frac{1}{\gamma},1)\).

Proof

(a) Let \(1<\gamma\le\infty\). Since

$$ \sup_{k\in\mathbb{Z}}|\sigma_{\Omega,h,\phi,k}*g_{k,j}| \le\sup _{k\in\mathbb{Z}}|\sigma_{\Omega,h,\phi,k}|*\sup_{\ell \in\mathbb{Z}}|g_{\ell,j}| \le M_{\Omega,h,\phi} \Bigl(\sup_{\ell\in\mathbb{Z}}|g_{\ell ,j}| \Bigr), $$

we get using Lemma 2.3

$$\begin{aligned} \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \Bigl( \sup_{k\in\mathbb{Z}}|\sigma_{\Omega,h,\phi ,k}*g_{k,j}| \Bigr)^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})} &\le\biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \Bigl(M_{\Omega,h,\phi} \Bigl(\sup_{k\in\mathbb{Z}}|g_{k,j}| \Bigr) \Bigr)^{q} \biggl)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\ &\le C\biggl\Vert \biggl(\sum _{j\in\mathbb{Z}} \Bigl(\sup_{k\in\mathbb{Z}}|g_{k,j}| \Bigr)^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})}. \end{aligned}$$
(2.13)

On the other hand, there exists \(\{h_{j}\}\in L^{p'}(\ell^{q'})\) with \(\|\{h_{j}\}\|_{L^{p'}(\ell^{q'})}=1\) such that

$$\begin{aligned}& \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl( \sum_{k\in\mathbb{Z}}\vert \sigma_{\Omega,h,\phi ,k}*g_{k,j} \vert \biggr)^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\& \quad = \sum_{j\in\mathbb{Z}}\int _{\mathbb{R}^{n}} \sum_{k\in\mathbb{Z}}\bigl\vert \sigma_{\Omega,h,\phi,k}*g_{k,j}(x)\bigr\vert h_{j}(x)\,dx \\& \quad \le\sum_{j\in\mathbb{Z}}\int _{\mathbb{R}^{n}} \sum_{k\in\mathbb{Z}}\bigl\vert g_{k,j}(x)\bigr\vert \vert \tilde{\sigma}_{\Omega,h,\phi ,k}\vert *h_{j}(x)\,dx \\& \quad \le\sum_{j\in\mathbb{Z}}\int _{\mathbb{R}^{n}} \sum_{k\in\mathbb{Z}}\bigl\vert g_{k,j}(x)\bigr\vert M_{\tilde{\Omega},h,\phi}h_{j}(x)\,dx \\& \quad \le C \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl(\sum_{k\in\mathbb{Z}}|g_{k,j}| \biggr)^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \bigl(M_{\tilde{\Omega},h,\phi}h_{j}(x) \bigr)^{q'} \biggr)^{\frac{1}{q'}} \biggr\Vert _{L^{p'}({\mathbb{R}^{n}})}, \end{aligned}$$

where \(\tilde{\Omega}(y')=\Omega(-y')\). So by Lemma 2.3 we obtain for \(\gamma'< p',q'<\infty\), i.e., \(1< p,q<\gamma\),

$$\begin{aligned}& \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl(\sum_{k\in\mathbb{Z}}|\sigma_{\Omega,h,\phi ,k}*g_{k,j}| \biggr)^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\& \quad \le C\biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl( \sum_{k\in\mathbb{Z}}|g_{k,j}| \biggr)^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \biggl\Vert \biggl(\sum _{j\in\mathbb{Z}} \bigl(\bigl\vert h_{j}(x)\bigr\vert \bigr)^{q'} \biggr)^{\frac{1}{q'}} \biggr\Vert _{L^{p'}({\mathbb{R}^{n}})} \\& \quad \le C\biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl( \sum_{k\in\mathbb{Z}}|g_{k,j}| \biggr)^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})}. \end{aligned}$$
(2.14)

Now, let \(R_{1}=(\frac{1}{2\gamma}, \frac{1}{2\gamma})\), \(R_{2}=(1-\frac{1}{2\gamma}, \frac{1}{2\gamma})\), \(R_{3}=(1-\frac{1}{2\gamma}, 1-\frac{1}{2\gamma})\), and \(R_{4}=(\frac{1}{2\gamma}, 1-\frac{1}{2\gamma})\). Then, if \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the square \(R_{1}R_{2}R_{3}R_{4}\), there are two points \((\frac{1}{p_{1}},\frac{1}{q_{1}})\) and \((\frac{1}{p_{2}},\frac{1}{q_{2}})\) such that

$$\begin{aligned}& \frac{1}{p}=\frac{1}{2}\frac{1}{p_{1}}+\frac{1}{2} \frac{1}{p_{2}},\qquad \frac{1}{q}=\frac{1}{2}\frac{1}{q_{1}}+ \frac{1}{2}\frac{1}{q_{2}}, \\& 1< p_{1},q_{1}<\gamma\quad \text{and}\quad \gamma'<p_{2},q_{2}<\infty. \end{aligned}$$

Hence, interpolating (2.13) with (2.14), we obtain (2.11) if \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the square \(R_{1}R_{2}R_{3}R_{4}\).

(b) Let \(1<\gamma<2\). Using the Cauchy-Schwarz inequality, we get

$$\begin{aligned} \bigl\vert \sigma_{\Omega,h,\phi,k}*g_{k,j}(x)\bigr\vert \le& \biggl(\int_{2^{k-1}\le|y|\le2^{k}}\frac{|\Omega (y')||h(|y|)|^{\gamma}}{|y|^{n}} \, dy \biggr)^{\frac{1}{2}} \\ &{} \times \biggl(\int_{2^{k-1}\le|y|\le2^{k}}\bigl\vert g_{k,j} \bigl(x-\phi\bigl(\vert y\vert \bigr)y'\bigr)\bigr\vert ^{2} \frac{|\Omega(y')||h(|y|)|^{2-\gamma}}{|y|^{n}}\, dy \biggr)^{\frac{1}{2}} \\ \le& C\|h\|_{\Delta_{\gamma}}^{\frac{\gamma}{2}}\|\Omega\| _{L^{1}({{S}^{n-1}})}^{\frac{1}{2}} \bigl(\sigma_{|\Omega|,|h|^{2-\gamma},\phi,k}*|g_{k,j}|^{2} \bigr) (x)^{\frac{1}{2}}. \end{aligned}$$

So, we have

$$\begin{aligned}& \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl( \sum_{k\in\mathbb{Z}}|\sigma_{\Omega,h,\phi ,k}*g_{k,j}|^{2} \biggr)^{\frac{q}{2}} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\& \quad \le C\|h\|_{\Delta_{\gamma}}^{\frac{\gamma}{2}}\|\Omega\| _{L^{1}({{S}^{n-1}})}^{\frac{1}{2}} \biggl\Vert \biggl(\sum _{j\in\mathbb{Z}} \biggl(\sum_{k\in\mathbb{Z}} \sigma_{|\Omega|,|h|^{2-\gamma},\phi ,k}*|g_{k,j}|^{2} \biggr)^{\frac{q}{2}} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})}. \end{aligned}$$

Hence, noting \(|h|^{2-\gamma}\in\Delta_{\gamma/(2-\gamma)}\) and using (2.14) for \(\gamma/(2-\gamma)\), \(p/2\) and \(q/2\) in place of γ, p, q, respectively, we see that (2.11) holds provided \(1< p/2,q/2<\gamma/(2-\gamma)\), i.e., \(1/2-1/\gamma'<1/p,1/q<1/2\). By duality, it holds also provided \(1/2<1/p,1/q<1/2+1/\gamma'\). Interpolating these two cases, we see that (2.11) holds if \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the hexagon \(P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}\).

(c) Noting \(\Delta_{\gamma}\subset\Delta_{2}\) for \(\gamma>2\), and interpolating cases (a) and (b) above, we see that (2.11) holds if \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(P_{1}P_{2}R_{2}P_{3}P_{4}P_{5}R_{4}P_{6}\). This completes the proof of Lemma 2.4(i).

(d) We shall prove Lemma 2.4(ii). If \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the parallelogram \(Q_{1}Q_{2}Q_{3}Q_{4}\), there are two points \((\frac{1}{p_{1}},\frac{1}{q_{1}})\) and \((\frac{1}{p_{2}},\frac{1}{q_{2}})\) such that

$$\begin{aligned}& \frac{1}{p}= \biggl(1-\frac{1}{q} \biggr)\frac{1}{p_{1}}+ \frac{1}{q}\frac {1}{p_{2}},\qquad \frac{1}{q}= \biggl(1- \frac{1}{q} \biggr)\frac{1}{q_{1}}+\frac{1}{q}\frac{1}{q_{2}}, \\& 1< p_{1},q_{1}<\gamma\quad \text{and}\quad \gamma'<p_{2},q_{2}<\infty. \end{aligned}$$

Hence, interpolating (2.13) with (2.14), we obtain (2.12). Thus, we finished the proof of Lemma 2.4. □

About the Fourier transform estimates of \({\sigma}_{\Omega,h,\phi,k}\) with \(\Omega\in H^{1}({{S}^{n-1}})\), we have the following.

Lemma 2.5

Let \(1< q\le+\infty\) and Ω be a regular ∞-atom in \(H^{1}(S^{n-1})\) supported in \({{S}^{n-1}}\cap B(\mathbf{e}_{1},\tau)\), where \(\mathbf{e}_{1}=(1,0,\ldots,0)\). Let ϕ be a positive increasing function on \((0,\infty)\) satisfying \(\varphi(t)=\phi(t)/(t\phi'(t))\in L^{\infty}(0,\infty)\), and \(h\in\Delta_{\gamma}\) for some \(1<\gamma\le\infty\). Then there exist positive constants C’s such that

$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi)\bigr\vert \le C \|h\|_{\Delta_{1}}\|\Omega\|_{L^{1}({{S}^{n-1}})}, \end{aligned}$$
(2.15)
$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi)\bigr\vert \le C \|h\|_{\Delta_{1}} \phi\bigl(2^{k}\bigr)\bigl\vert A_{\tau}(\xi)\bigr\vert \end{aligned}$$
(2.16)

and

$$ \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi)\bigr\vert \le \frac{C\|h\|_{\Delta_{\gamma}}}{ (\phi(2^{k-1})|A_{\tau}(\xi)|)^{1/\max\{\gamma',2\}}}. $$
(2.17)

These are shown by using Lemmas 2.1 and 2.2 as in the proofs of Lemmas 3.3 and 3.4 in [20], pp.247-248. There these are stated for the case where a parameter ρ of positive number arises, but one sees easily that these hold in our case (\(\rho=0\)), too.

To show Theorem 1.1, we need a characterization of the Triebel-Lizorkin space in terms of lacunary sequences. Let \(\{a_{j}\}_{j\in\mathbb{Z}}\) be a lacunary sequence with lacunarity \(a>1\), i.e.,

$$ \frac{a_{j+1}}{a_{j}}\ge a \quad \text{for }j\in\mathbb{Z}. $$
(2.18)

Let η be a radial function in \(C^{\infty}({\mathbb{R}^{n}})\) satisfying \(\chi_{|\xi|\le1}(\xi)\le\eta(\xi)\le\chi_{|\xi|\le a}(\xi)\) and \(|\partial^{\alpha}\eta(\xi)|\le c_{\alpha}(a-1)^{-|\alpha|}\) for \(\xi\in{\mathbb{R}^{n}}\) and \(\alpha\in\mathbb{Z}_{+}^{n}\). We define functions \(\psi_{j}\) on \({\mathbb{R}^{n}}\) by

$$ \psi_{j}(\xi) =\eta \biggl(\frac{\xi}{a_{j+1}} \biggr) - \eta \biggl(\frac{\xi}{a_{j}} \biggr) \quad \bigl(\xi\in{\mathbb{R}^{n}} \bigr). $$
(2.19)

Then observe that

$$ \psi_{j}(\xi)= \begin{cases} 0, &0\le|\xi|\le a_{j}, |\xi|\ge aa_{j+1}, \\ 1, & a a_{j}\le|\xi|\le a_{j+1}, \end{cases} $$
(2.20)

and that

$$\begin{aligned}& \operatorname{supp}\psi_{j}\subset\bigl\{ a_{j}\le \vert \xi \vert \le aa_{j+1}\bigr\} , \end{aligned}$$
(2.21)
$$\begin{aligned}& \operatorname{supp}\psi_{j}\cap\operatorname{supp} \psi_{\ell}=\emptyset\quad \text{for }|j-\ell|\ge2, \end{aligned}$$
(2.22)
$$\begin{aligned}& \bigl\vert \xi^{\alpha}\partial^{\alpha}\psi_{j}(\xi) \bigr\vert \le C_{\alpha}\quad \text{for }\alpha \in\mathbb{Z}_{+}^{n}, \end{aligned}$$
(2.23)
$$\begin{aligned}& \sum_{j\in\mathbb{Z}}\psi_{j}(\xi)=1\quad \bigl(\xi \in{\mathbb {R}^{n}}\setminus\{0\}\bigr). \end{aligned}$$
(2.24)

Let \(\Psi_{j}\) be defined on \({\mathbb{R}^{n}}\) by \(\widehat{\Psi} _{j}(\xi)=\psi_{j}(\xi)\) for \(\xi\in{\mathbb{R}^{n}}\), i.e., \(\Psi_{j}(x)=a_{j+1}^{n}\check{\eta}(a_{j+1}x)-a_{j}^{n}\check{\eta}(a_{j} x)\).

Lemma 2.6

Define the multiplier \(S_{j}\) by \(S_{j}f=\Psi_{j}*f\). Then, for \(1< p, q<\infty\), we have

$$ \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl(\sum _{k\in\mathbb {Z}}|S_{k} f_{j}|^{2} \biggr)^{q/2} \biggr)^{1/q}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \le C \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}}|f_{j}|^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})}, $$

where C is independent of \(\{f_{j}\}_{j\in\mathbb{Z}}\).

This is a consequence of Proposition 4.6.4 in Grafakos [21]. For the sake of completeness, we will give a proof in the Appendix. From this lemma we have the following lemma with minor change of the proof of Lemma 2.2 in [8].

Lemma 2.7

Let \(\psi_{j}\) be as in Lemma  2.6. Denote \(A_{\tau}(\xi)= (\tau^{2}\xi_{1},\tau\xi_{2},\ldots,\tau\xi_{n})\) for \(\tau>0\) and \(\xi \in{\mathbb{R}^{n}}\). Define the multiplier \(S_{j,\tau}\) by \(\widehat{S_{j,\tau}f}(\xi)=\psi(a_{k}A_{\tau}(\xi))\hat{f}(\xi)\). Then, for \(1< p, q<\infty\), we have

$$ \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}} \biggl(\sum _{k\in\mathbb {Z}}|S_{k,\tau} f_{j}|^{2} \biggr)^{q/2} \biggr)^{1/q}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \le C \biggl\Vert \biggl(\sum_{j\in\mathbb{Z}}|f_{j}|^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})}, $$

where C is independent of \(\{f_{j}\}_{j\in\mathbb{Z}}\).

We need one more lemma. If \(\{a_{k}\}_{k\in\mathbb{Z}}\) satisfies furthermore \(a_{k+1}/a_{k}\le d\) for some \(d\ge a\), we can characterize Triebel-Lizorkin spaces in terms of this lacunary sequence.

Denote by \({\mathcal{P}}\) the set of all polynomials in \({\mathbb{R}}^{n}\). Let \(1< p,q<\infty\), and \(\alpha\in\mathbb{R}\). For \(f\in\mathcal {S}'({\mathbb{R}^{n}})/{\mathcal{P}}\), we define the norm \(\|f\|_{\dot{F}_{pq}^{\alpha,\{\Psi_{k}\}_{k\in\mathbb {Z}}}({\mathbb{R}^{n}})}\) by

$$ \|f\|_{\dot{F}_{pq}^{\alpha,\{\Psi_{k}\}_{k\in\mathbb{Z}}}({\mathbb{R}^{n}})} =\biggl\Vert \biggl(\sum _{k\in\mathbb{Z}}a_{k}^{\alpha q}|\Psi _{k}*f|^{q} \biggr)^{1/q} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})}. $$
(2.25)

Lemma 2.8

Let \(\alpha\in\mathbb{R}\) and \(1< p,q<\infty\). Let \(\{a_{k}\}_{k\in\mathbb{Z}}\) be a lacunary sequence of positive numbers with \(d\ge a_{k+1}/a_{k}\ge a>1\) (\(k\in\mathbb{Z}\)). Then \(\|f\|_{\dot{F}_{pq}^{\alpha,\{\Psi_{k}\}_{k\in\mathbb{Z}}}({\mathbb{R}^{n}})}\) is equivalent to the usual homogeneous Triebel-Lizorkin space norm \(\|f\|_{\dot{F}_{pq}^{\alpha}({\mathbb{R}^{n}})}\).

This is stated in Proposition 1 in [22] for \(\alpha\ne 0\), but the proof of this part works also for \(\alpha=0\).

2.2 Proof of Theorem 1.1

We have only to show Theorem 1.1 in the case Ω is a regular atom with \(\operatorname{supp}\Omega\subset S^{n-1}\cap B(\xi,\tau)\), where \(B(\xi,\tau)=\{y\in\mathbb{R}^{n}; |y-\xi|<\tau\}\). Using the definition of \(\sigma_{\Omega,h,\phi,k}\), we see that

$$ T_{\Omega,h,\phi}f(x) ={\mathrm{p.v.}}\int_{{\mathbb{R}^{n}}} \frac{h(|y|)\Omega (y')}{|y|^{n}}f\bigl(x-\phi\bigl(\vert y\vert \bigr) y'\bigr) \,dy =\sum_{k\in\mathbb{Z}}\sigma_{\Omega,h,\phi,k}*f(x). $$
(2.26)

Let \(a_{k}=1/\phi(2^{-k})\), \(k\in\mathbb{Z}\). Then as is known, \(\{ a_{k}\}_{k\in\mathbb{Z}}\) is a lacunary sequence with lacunarity \(a=2^{1/\|\varphi\|_{L^{\infty}(\mathbb{R}_{+})}}\). This follows from (1.10) (see, for example, [22]). Also, we have \(a_{k+1}/a_{k}\le c_{1}\), which follows from (1.9).

Let \(\psi_{k}\in C_{c}^{\infty}({\mathbb{R}^{n}})\) be radial functions defined by (2.19). Set \({\psi}_{k,\tau}(\xi)=\psi_{k}(A_{\tau}(\xi))\) and \(\widehat{S_{k,\tau}f}(\xi)=\psi_{k,\tau}(\xi)\hat{f}(\xi)\), \(\xi\in{\mathbb{R}^{n}}\). Then, noting \(\sum_{j\in\mathbb{Z}}\psi_{j}(\xi)=1\) (\(\xi\ne0\)) and \(\sum_{\ell=-1}^{1}\psi_{j+\ell}(\xi)=1\) on \(\operatorname {supp}\psi_{j}\), we have

$$ T_{\Omega,h,\phi}f(x) =\sum_{k\in\mathbb{Z}}\sum _{j\in\mathbb{Z}}\sum_{\ell=-1}^{1} S_{j-k+\ell,\tau}(\sigma_{\Omega,h,\phi,k}*S_{j-k,\tau}f) (x) =\sum _{j\in\mathbb{Z}}Q_{j}f(x), $$
(2.27)

where

$$ Q_{j}f(x) =\sum_{k\in\mathbb{Z}}\sum _{\ell=-1}^{1} S_{j-k+\ell,\tau}( \sigma_{\Omega,h,\phi,k}*S_{j-k,\tau}f) (x). $$
(2.28)

We follow the proof of Theorem 1 in [8], using our Lemma 2.7 and Lemma 2.4 in place of Lemma 2.2 and Lemma 2.4 in [8], respectively, and we see that if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(P_{1}P_{2}R_{2}P_{3}P_{4}P_{5}R_{4}P_{6}\), then we have

$$ \|Q_{j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}\le C\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}. $$
(2.29)

About \(L^{2}\) estimate, we have

$$ \|Q_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le Ca^{-|j|/\max(\gamma',2)}\|f \|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. $$
(2.30)

In fact, by Lemma 2.5, we get

$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi)\bigr\vert \le C \|h\|_{\Delta_{1}}\|\Omega\|_{L^{1}({{S}^{n-1}})}, \end{aligned}$$
(2.31)
$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi,\eta)\bigr\vert \le C\|h\|_{\Delta_{1}} \phi\bigl(2^{k}\bigr)\bigl\vert A_{\tau}(\xi)\bigr\vert \end{aligned}$$
(2.32)

and

$$ \bigl\vert \hat{\sigma}_{\Omega,h,\phi,\phi,k}(\xi)\bigr\vert \le \frac{C\|h\|_{\Delta_{\gamma}}}{ (\phi(2^{k-1})|A_{\tau}(\xi)|)^{1/\max(\gamma',2)}}. $$
(2.33)

Also, we have

$$\begin{aligned} \|Q_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &= \Biggl(\int _{\mathbb{R}^{n}}\Biggl\vert \sum_{k\in\mathbb{Z}} \sum_{\ell=-1}^{1} S_{j-k+\ell,\tau}( \sigma_{\Omega,h,\phi,k}* S_{j-k,\tau}f) (x)\Biggr\vert ^{2}\, dx \Biggr)^{1/2} \\ &= \Biggl(\int_{\mathbb{R}^{n}}\Biggl\vert \sum _{k\in\mathbb{Z}}\sum_{\ell=-1}^{1} \psi_{j-k+\ell}\bigl(A_{\tau}(\xi)\bigr)\hat{\sigma}_{\Omega,h,\phi,k}( \xi) \psi_{j-k}\bigl(A_{\tau}(\xi)\bigr)\hat{f}(\xi)\Biggr\vert ^{2}\, d\xi \Biggr)^{1/2}. \end{aligned}$$

So, for \(j\ge0\), we have, using (2.33) and \(\phi(2^{\ell})=1/a_{-\ell}\) and \(a_{\ell+1}/a_{\ell}\ge a=2^{1/\|\varphi\|_{L^{\infty}(\mathbb{R}_{+})}}\),

$$\begin{aligned}& \|Q_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \\& \quad \le C \biggl(\sum_{k\in\mathbb{Z}}\int _{a_{j-k}\le|A_{\tau}(\xi )|\le a_{j-k+2}} \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi)\hat{f}( \xi) \bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\& \quad \le {C\|h\|_{\Delta_{\gamma}}} \biggl(\sum _{k\in\mathbb{Z}} \int_{a_{j-k}\le|A_{\tau}(\xi)|\le a_{j-k+2}} \biggl( \frac{|A_{\tau}(\xi)|}{a_{-k+1}} \biggr)^{-2/\max(\gamma',2)} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\& \quad \le{C\|h\|_{\Delta_{\gamma}}} {a^{-(j-1)/\max(\gamma',2)}} \biggl(\sum _{k\in\mathbb{Z}}\int_{a_{j-k}\le|A_{\tau}(\xi)|\le a_{j-k+2}} \biggl( \frac{|A_{\tau}(\xi)|}{a_{j-k}} \biggr)^{-2/\max(\gamma',2)} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\& \quad \le{C\|h\|_{\Delta_{\gamma}}} {a^{-j/\max(\gamma',2)}} \biggl(\int _{{\mathbb{R}^{n}}}\bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d \xi \biggr)^{1/2} \\& \quad \le Ca^{-j/\max(\gamma',2)}\|f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. \end{aligned}$$

In the fourth inequality we used \(a_{j+1}\le c_{1} a_{j}\).

For \(j\le-1\), using (2.32) we get as before

$$\begin{aligned} \| Q_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &\le C \biggl(\sum _{k\in\mathbb{Z}} \int_{a_{j-k}\le|A_{\tau}(\xi)|\le a_{j-k+2}} \bigl\vert \hat{ \sigma}_{\Omega,\psi,h,k}(\xi)\hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le C\|h\|_{\Delta_{\gamma}} \biggl(\sum_{k\in\mathbb{Z}} \int_{a_{j-k}\le|A_{\tau}(\xi)|\le a_{j-k+2}} \biggl(\frac{|A_{\tau}(\xi)|}{a_{-k}} \biggr)^{2} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j} \biggl(\sum_{k\in\mathbb{Z}} \int_{a_{j-k}\le|A_{\tau}(\xi)|\le a_{j-k+2}} \biggl(\frac{|A_{\tau}(\xi)|}{a_{j-k}} \biggr)^{2} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j} \biggl(\sum_{k\in\mathbb{Z}} \int_{a_{j-k}\le|A_{\tau}(\xi)|\le a_{j-k+2}} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j} \biggl(\int_{{\mathbb{R}^{n}}}\bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j}\|f\|_{L^{2}(\mathbb{R},\dot{F}_{2,2}^{0}({\mathbb{R}^{n}}))}. \end{aligned}$$

Thus we have

$$ \|Q_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le Ca^{-|j|/\max(\gamma',2)}\|f \|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}, $$

which shows the required estimate (2.30).

Interpolating these two cases (2.29) and (2.30), we see that if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(P_{1}P_{2}Q_{2}P_{3}P_{4}P_{5}Q_{4}P_{6}\), then \(T_{\Omega,h,\phi}\) is bounded on \(\dot{F}_{p,q}^{\alpha}({\mathbb {R}^{n}})\). This completes the proof of Theorem 1.1(i).

Next, we prove (ii). Let \(|\frac{1}{2}-\frac{1}{p}|<\min\{\frac{1}{2},\frac{1}{\gamma'}\}\), \(1< q<\infty\), and \(\alpha\in\mathbb{R}\). Then, by Theorem 1.1(ii), \(T_{\Omega,h,\phi}\) is bounded on \(\dot{F}_{p,p}^{\alpha-1}({\mathbb{R}^{n}})\) and \(\dot{F}_{p,p}^{\alpha +1}({\mathbb{R}^{n}})\). Since \((\dot{F}_{p,p}^{\alpha-1}({\mathbb{R}^{n}}), \dot{F}_{p,p}^{\alpha +1}({\mathbb{R}^{n}}) )_{\frac{1}{2},q} =\dot{B}_{p,q}^{\alpha}({\mathbb{R}^{n}})\), we see by interpolation that \(T_{\Omega,h,\phi}\) is bounded on \(\dot{B}_{p,q}^{\alpha}({\mathbb{R}^{n}})\). This shows (ii) and completes the proof of Theorem 1.1.

3 Proof of Theorem 1.3

Let \(\sigma_{\Omega,h,\phi,k}\), \(a_{k}\), \(\psi_{k}\), and \(S_{k}\) be the same as in the proof of Theorem 1.1. Then, noting \(\sum_{j\in\mathbb{Z}}\psi_{j}(\xi)=1\) (\(\xi\ne0\)) and \(\sum_{\ell=-1}^{1}\psi_{j+\ell}(\xi)=1\) on \(\operatorname {supp}\psi_{j}\), we have

$$ T_{\Omega,h,\phi}f(x) =\sum_{k\in\mathbb{Z}}\sum _{j\in\mathbb{Z}}\sum_{\ell=-1}^{1} S_{j-k+\ell}(\sigma_{\Omega,h,\phi,k}*S_{j-k}f) (x) =\sum _{j\in\mathbb{Z}}\tilde{Q}_{j}f(x), $$
(3.1)

where

$$ \tilde{Q}_{j}f(x) =\sum_{k\in\mathbb{Z}} \sum_{\ell=-1}^{1} S_{j-k+\ell}( \sigma_{\Omega,h,\phi,k}*S_{j-k}f) (x). $$
(3.2)

Using our Lemma 2.6 and Lemma 2.4(i) in place of Lemma 2.2 and Lemma 2.4 in [8], respectively, we see, as in the proof of Theorem 1.1, that if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(P_{1}P_{2}R_{2}P_{3}P_{4}P_{5}R_{4}P_{6}\), then we have

$$ \|\tilde{Q}_{j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}\le C\|f \|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}. $$
(3.3)

Next, we approach the above estimate (3.3) by another method. We calculate the \(\dot{F}_{p,q}^{\alpha}\) norm of \(\tilde{Q}_{j}\) more directly. Considering the support property of \(\psi_{k}\), we have

$$\begin{aligned} \|\tilde{Q}_{j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} =&\Biggl\Vert \Biggl( \sum_{m\in\mathbb{Z}}a_{m}^{\alpha q} \Biggl\vert S_{m}\sum_{k\in\mathbb{Z}} \sum _{\ell=-1}^{1} S_{j-k+\ell}(\sigma_{\Omega,h,\phi,k}*S_{j-k}f) \Biggr\vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\ \le& \Biggl\Vert \Biggl(\sum_{m\in\mathbb{Z}}a_{m}^{\alpha q} \Biggl\vert S_{m}\sum_{\ell=-1}^{1} S_{m+\ell}(\sigma_{\Omega,h,\phi,j-m}*S_{m}f)\Biggr\vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\ &{}\times\Biggl\Vert \Biggl(\sum_{m\in\mathbb{Z}}a_{m}^{\alpha q} \Biggl\vert S_{m}\sum_{\ell=0}^{1} S_{m+\ell}(\sigma_{\Omega,h,\phi,j-m-1}*S_{m+1}f)\Biggr\vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\ &{}\times\Biggl\Vert \Biggl(\sum_{m\in\mathbb{Z}}a_{m}^{\alpha q} \Biggl\vert S_{m}\sum_{\ell=-1}^{0} S_{m+\ell}(\sigma_{\Omega,h,\phi,j-m-1}*S_{m-1}f)\Biggr\vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p}({\mathbb{R}^{n}})}. \end{aligned}$$

By Fefferman-Stein’s vector-valued inequality for maximal functions, Lemma 2.4(ii), and \(a_{m+1}/c_{1}\le a_{m}\le a_{m+1}/a\), we get

$$\begin{aligned} \|\tilde{Q}_{j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} &\le C\sum _{\ell=-1}^{1}\biggl\Vert \biggl(\sum _{m\in\mathbb {Z}}a_{m}^{\alpha q}\vert \sigma_{\Omega,h,\phi,j-m}*S_{m+\ell}f \vert ^{q} \biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \\ &\le C \sum_{\ell=-1}^{1}\biggl\Vert \biggl(\sum_{m\in\mathbb {Z}}a_{m}^{\alpha q} \vert S_{m+\ell}f \vert ^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb {R}^{n}})} \\ &\le C \biggl\Vert \biggl(\sum_{m\in\mathbb{Z}}a_{m}^{\alpha q} \vert S_{m}f\vert ^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}({\mathbb{R}^{n}})} \le C\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \end{aligned}$$
(3.4)

if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the parallelogram \(Q_{1}Q_{2}Q_{3}Q_{4}\).

Interpolating (3.3) and (3.4), we obtain

$$ \|\tilde{Q}_{j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}\le C\|f \|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$
(3.5)

if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)).

About \(L^{2}\) estimate, we have

$$ \|\tilde{Q}_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le C \biggl( \frac{1}{1+|j|} \biggr)^{\beta/\max(\gamma',2)}. $$
(3.6)

In fact, let \(\sigma_{k}=\sigma_{\Omega,h,\phi,k}\). Then we have

$$ \hat{\sigma}_{k}(\xi) =\int_{2^{k-1}}^{2^{k}} \int_{{{S}^{n-1}}}{\Omega\bigl(y'\bigr)h(r)}e^{-i\phi (r)y'\cdot\xi} \,d\sigma\bigl(y'\bigr)\frac{dr}{r}. $$

First we have

$$ \bigl\vert \hat{\sigma}_{k}(\xi)\bigr\vert \le2\|h \|_{\Delta_{\gamma}}\|\Omega\| _{L^{1}({{S}^{n-1}})}. $$
(3.7)

Next, using Hölder’s inequality and assuming \(\|\Omega\| _{L^{1}({{S}^{n-1}})}\le1\) without loss of generality, we have

$$\begin{aligned} \bigl\vert \hat{\sigma}_{k}(\xi)\bigr\vert \le& \biggl(\int _{2^{k-1}}^{2^{k}}\bigl\vert h(r)\bigr\vert ^{\gamma}\frac{dr}{r} \biggr)^{1/\gamma} \biggl(\int _{2^{k-1}}^{2^{k}}\biggl\vert \int_{{{S}^{n-1}}} \Omega\bigl(y'\bigr)e^{-i\phi(r)y'\cdot\xi}\,d\sigma \bigl(y' \bigr)\biggr\vert ^{\gamma'} \frac{dr}{r} \biggr)^{1/\gamma'} \\ \le&2\|h\|_{\Delta_{\gamma}} \biggl(\int_{2^{k-1}}^{2^{k}} \biggl\vert \int_{{{S}^{n-1}}}\Omega\bigl(y' \bigr)e^{-i\phi(r)y'\cdot\xi}\,d\sigma \bigl(y'\bigr)\biggr\vert ^{2} \frac{dr}{r} \biggr)^{\frac{1}{\max(2,\gamma')}} \\ =& 2\|h\|_{\Delta_{\gamma}} \biggl(\int_{\phi(2^{k-1})}^{\phi(2^{k})} \biggl\vert \int_{{{S}^{n-1}}}\Omega\bigl(y' \bigr)e^{-iry'\cdot\xi}\,d\sigma\bigl(y'\bigr)\biggr\vert ^{2} \frac{\phi(\phi^{-1}(r))}{\phi^{-1}(r)\phi'(\phi^{-1}(r))}\frac{dr}{r} \biggr)^{\frac{1}{\max(2,\gamma')}} \\ \le&2\|h\|_{\Delta_{\gamma}}\|\varphi\|_{L^{\infty}(\mathbb{R}_{+})} \biggl(\int _{\phi(2^{k-1})}^{\phi(2^{k})}\biggl\vert \int_{{{S}^{n-1}}} \Omega\bigl(y'\bigr)e^{-iry'\cdot\xi}\,d\sigma\bigl(y' \bigr)\biggr\vert ^{2} \frac{dr}{r} \biggr)^{\frac{1}{\max(2,\gamma')}} \\ =&2\|h\|_{\Delta_{\gamma}}\|\varphi\|_{L^{\infty}(\mathbb{R}_{+})} \\ &{}\times\biggl(\int _{{{S}^{n-1}}}\int_{{{S}^{n-1}}}\Omega \bigl(y'\bigr)\overline {\Omega\bigl(z'\bigr)} \int _{\phi(2^{k-1})}^{\phi(2^{k})}e^{-ir(y'-z')\cdot\xi} \frac{dr}{r} \,d \sigma\bigl(y'\bigr)\, d\sigma\bigl(z'\bigr) \biggr)^{\frac{1}{\max(2,\gamma')}}. \end{aligned}$$

We see that

$$ \biggl\vert \int_{\phi(2^{k-1})}^{\phi(2^{k})}e^{-ir(y'-z')\cdot\xi} \frac{dr}{r}\biggr\vert \le\log\frac{\phi(2^{k})}{\phi(2^{k-1})}\le\log c_{1}. $$

We see also

$$ \biggl\vert \int_{\phi(2^{k-1})}^{\phi(2^{k})}e^{-ir(y'-z')\cdot\xi} \frac{dr}{r}\biggr\vert \le\frac{2}{\phi(2^{k-1})|\xi||\xi'\cdot(x'-y')|}. $$

So, as in [10], p.458 (using Lemma 3.1 in [12]), we have for \(\beta>1\),

$$\begin{aligned}& \biggl\vert \int_{\phi(2^{k-1})}^{\phi(2^{k})}e^{-ir(y'-z')\cdot\xi} \frac{dr}{r} \biggr\vert \\& \quad \le\frac{C}{\log^{\beta}(\log c_{1})e\phi(2^{k-1})|\xi|}\log ^{\beta} \frac{2e}{|(y'-z')\cdot\xi'|}\quad \text{for }\phi\bigl(2^{k}\bigr)|\xi|\ge \frac{c_{1}}{\log c_{1}}. \end{aligned}$$

Hence we have

$$ \bigl\vert \hat{\sigma}_{k}(\xi)\bigr\vert \le \frac{2\|h\|_{\Delta_{\gamma}} (W\mathcal{F}_{\beta}({\Omega}) )^{2/{\max\{\gamma',2\}}} \|\Omega\|_{L^{1}({{S}^{n-1}})}^{1-2/{\max\{\gamma',2\}}}}{ (\log(e(\log c_{1})\phi(2^{k})|\xi|/c_{1}))^{\beta/{\max\{\gamma',2\}}}} $$
(3.8)

for \(\phi(2^{k})|\xi|\ge\frac{c_{1}}{\log c_{1}}\ge{e}\).

On the other hand, using the cancelation property of Ω, we get easily

$$ \bigl\vert \hat{\sigma}_{k}(\xi)\bigr\vert \le2\|h \|_{\Delta_{1}}\|\Omega\| _{L^{1}({{S}^{n-1}})} \phi\bigl(2^{k}\bigr)| \xi|. $$
(3.9)

Now we can estimate the \(L^{2}\) norm of \(\tilde{Q}_{j}f\):

$$\begin{aligned} \begin{aligned} \|\tilde{Q}_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &= \Biggl(\int _{\mathbb{R}^{n}}\Biggl\vert \sum_{k\in\mathbb{Z}} \sum_{\ell=-1}^{1} S_{j-k+\ell}( \sigma_{\Omega,h,\phi,k}* S_{j-k}f) (x)\Biggr\vert ^{2}\, dx \Biggr)^{1/2} \\ &= \Biggl(\int_{\mathbb{R}^{n}}\Biggl\vert \sum _{k\in\mathbb{Z}}\sum_{\ell=-1}^{1} \psi_{j-k+\ell}(\xi)\hat{\sigma}_{\Omega,h,\phi,k}(\xi) \psi_{j-k}( \xi)\hat{f}(\xi)\Biggr\vert ^{2}\, d\xi \Biggr)^{1/2}. \end{aligned} \end{aligned}$$

Note that \(\phi(2^{k})|\xi|=|\xi|/a_{-k}\ge a^{j}|\xi|/a_{j-k}\ge a^{j}\) for \(a_{j-k}\le|\xi|\le a_{j-k+2}\) and \(j\ge0\), where \(a_{\ell+1}/a_{\ell}\ge a=2^{1/\|\varphi\|_{L^{\infty}(\mathbb {R}_{+})}}\). So, for \(j\ge2\log_{a}(c_{1}/\log c_{1})\) and \(a_{j-k}\le|\xi|\le a_{j-k+2}\), we have

$$ \frac{\log c_{1}}{c_{1}}\phi\bigl(2^{k}\bigr)|\xi|\ge a^{\frac{j}{2}} \ge1, $$

and hence we have

$$\begin{aligned} \|\tilde{Q}_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &\le C \biggl(\sum_{k\in\mathbb{Z}}\int_{a_{j-k}\le|\xi|\le a_{j-k+2}} \bigl\vert \hat{\sigma}_{\Omega,h,\phi,k}(\xi)\hat{f}(\xi) \bigr\vert ^{2} \, d\xi \biggr)^{1/2} \\ &\le {C\|h\|_{\Delta_{\gamma}}} \biggl(\sum_{k\in\mathbb{Z}} \int_{a_{j-k}\le|\xi|\le a_{j-k+2}} \biggl(\frac{1}{1+j} \biggr)^{2\beta/\max(\gamma',2)} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le{C\|h\|_{\Delta_{\gamma}}} \biggl(\frac{1}{1+j} \biggr)^{\beta /\max(\gamma',2)} \biggl(\sum_{k\in\mathbb{Z}}\int _{a_{j-k}\le|\xi|\le a_{j-k+2}} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le{C\|h\|_{\Delta_{\gamma}}} \biggl(\frac{1}{1+j} \biggr)^{\beta /\max(\gamma',2)} \biggl(\int_{{\mathbb{R}^{n}}}\bigl\vert \hat{f}( \xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le C \biggl(\frac{1}{1+j} \biggr)^{\beta/\max(\gamma',2)} \|f \|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. \end{aligned}$$

For \(j\le-2\), we have \(\phi(2^{k})|\xi|=|\xi|/a_{-k}\le a^{2+j}\) for \(a_{j-k}\le|\xi|\le a_{j-k+2}\). So, using (3.9) we get as before

$$\begin{aligned} \|\tilde{Q}_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &\le C \biggl(\sum _{k\in\mathbb{Z}} \int_{a_{j-k}\le|\xi|\le a_{j-k+2}} \bigl\vert \hat{\sigma}_{\Omega,\psi,h,k}(\xi)\hat{f}(\xi) \bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le C\|h\|_{\Delta_{\gamma}} \biggl(\sum_{k\in\mathbb{Z}} \int_{a_{j-k}\le|\xi|\le a_{j-k+2}} a^{4+2j}\bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j} \biggl(\sum_{k\in\mathbb{Z}} \int_{a_{j-k}\le|\xi|\le a_{j-k+2}} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j} \biggl(\int_{{\mathbb{R}^{n}}}\bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{1/2} \\ &\le Ca^{j}\|f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. \end{aligned}$$

For \(-2\le j< 2\log_{a}(c_{1}/\log c_{1})\), using (3.7), we get

$$ \|\tilde{Q}_{j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le C \biggl(\sum _{k\in\mathbb{Z}}\int_{a_{j-k}\le|\xi|\le a_{j-k+2}} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\, d\xi \biggr)^{1/2} \le C \|f \|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. $$

Thus we have (3.6) for \(j\in\mathbb{Z}\). Now, let \(\mathcal{Q}_{1}= (\frac{\max(\gamma',2)}{2\beta}, \frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{Q}_{2}= (\frac{1}{\gamma'}+ \frac{\max(\gamma',2)}{\beta}(\frac{1}{2}-\frac{1}{\gamma'}), \frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{P}_{3}= (\frac{1}{2}+\frac{1}{\max(\gamma',2)}-\frac {1}{\beta}, \frac{1}{2} )\), \(\mathcal{Q}_{3}= (1-\frac{\max(\gamma',2)}{2\beta}, 1-\frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{Q}_{4} = (\frac{1}{\gamma}-\frac{\max(\gamma',2)}{\beta}(\frac {1}{\gamma} -\frac{1}{2}), 1-\frac{\max(\gamma',2)}{2\beta} )\), \(\mathcal{P}_{6}= (\frac{1}{2}-\frac{1}{\max(\gamma',2)}+\frac {1}{\beta}, \frac{1}{2} )\), \(\mathcal{R}_{2}= (1-\frac{1}{2\gamma}-\frac{\max(\gamma ',2)}{2\beta\gamma'}, \frac{1}{2\gamma}+\frac{\max(\gamma',2)}{2\beta\gamma'} )\), and \(\mathcal{R}_{4}= (\frac{1}{2\gamma}+\frac{\max(\gamma ',2)}{2\beta\gamma'}, 1-\frac{1}{2\gamma}-\frac{\max(\gamma',2)}{2\beta\gamma'} )\). Then, for \((\frac{1}{p},\frac{1}{q})\) belonging to the interior of the octagon \(\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{R}_{2}\mathcal{P}_{3}\mathcal{Q}_{3}\mathcal{Q}_{4} \mathcal{R}_{4}\mathcal{P}_{6}\) (hexagon \(\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{P}_{3}\mathcal{Q}_{3}\mathcal{Q}_{4} \mathcal{P}_{6}\) in the case \(1<\gamma\le2\)), we can find \((\frac{1}{p_{1}},\frac{1}{q_{1}})\) in the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)) such that \(\frac{1}{p}=\frac{\theta}{2}+\frac{1-\theta}{p_{1}}\), \(\frac{1}{q}=\frac{\theta}{2}+\frac{1-\theta}{q_{1}}\), and \(1>\theta>\frac{\max(\gamma',2)}{\beta}\). Hence, for \(\alpha\in\mathbb{R}\), taking \(\alpha_{1}\) with \(\alpha=(1-\theta)\alpha_{1}\) and interpolating between (3.6) and (3.5), we obtain the desired estimate

$$ \|\tilde{Q}_{j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\bigl(1+\vert j\vert \bigr)^{-\theta\beta/{\max\{\gamma',2\}}} \|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}. $$
(3.10)

Summing up this with respect to j, we finish the proof of Theorem 1.3(i). The proof of (ii) is the same as in Theorem 1.1(i).

4 Proof of Theorem 1.2

In this section we shall give the proof of Theorem 1.2.

(A) \(L\log L\) case. Let \(\Omega\in L\log L({{S}^{n-1}})\) satisfying the cancelation property. Then letting \(A_{m}=\|\Omega\chi_{2^{m-1}\le|\Omega(y')|<2^{m}}\| _{L^{1}({{S}^{n-1}})}\) and \(\Lambda=\{m\in\mathbb{N}:A_{m}>2^{-m}\}\), we can construct \(\Omega_{m}\in L^{2}({{S}^{n-1}})\) (\(m\in\Lambda\)) and \(\Omega_{0}\in \bigcap_{1< r<2}L^{r}({{S}^{n-1}})\) such that

$$\begin{aligned}& \|\Omega_{m}\|_{L^{2}({{S}^{n-1}})}\le C2^{m},\qquad \| \Omega_{m}\| _{L^{1}({{S}^{n-1}})}\le C, \end{aligned}$$
(4.1)
$$\begin{aligned}& \sum_{m\in\Lambda}m A_{m}\le C\|\Omega \|_{L\log L({{S}^{n-1}})}, \end{aligned}$$
(4.2)
$$\begin{aligned}& \int_{{{S}^{n-1}}}\Omega_{m}\bigl(y'\bigr) \,d\sigma\bigl(y'\bigr)=0\quad (m=0,m\in\Lambda ), \qquad \Omega= \Omega_{0}+\sum_{m\in\Lambda}A_{m} \Omega_{m}. \end{aligned}$$
(4.3)

From the above, we see that

$$ T_{\Omega,\psi,h} f =T_{\Omega_{0},h,\phi} f+\sum _{m\in\Lambda}A_{m}T_{\Omega_{m},h,\phi} f. $$
(4.4)

So, we consider \(T_{\Omega_{m},h,\phi}\). We use the notations in Section 3 with minor change such as \(\tilde{Q}_{m,j}\) for \(\Omega_{m}\) instead of \(\tilde{Q}_{j}\) for Ω. Since \(\|\Omega_{m}\|_{L^{1}({{S}^{n-1}})}\le C\), we have as in Section 3 that

$$ \|\tilde{Q}_{m,j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\|f \|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$
(4.5)

if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)).

About \(L^{2}\) estimate, we have

$$ \|\tilde{Q}_{m,j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le C a^{-\frac{\beta}{m}|j|}\|f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} $$
(4.6)

for some β with \(0<\beta<1/2\). In fact, let \(\sigma_{m,k}=\sigma_{\Omega_{m},h,\phi,k}\). Since \(\|\Omega_{m}\|_{L^{1}({{S}^{n-1}})}\le C\) and \(\|\Omega_{m}\| _{L^{2}({{S}^{n-1}})}\le C2^{m}\), we get by Lemma 3.1 in [23], p.1567,

$$\begin{aligned}& \bigl\vert \hat{\sigma}_{m,k}(\xi)\bigr\vert \le C\|h \|_{\Delta_{1}}, \end{aligned}$$
(4.7)
$$\begin{aligned}& \bigl\vert \hat{\sigma}_{m,k}(\xi)\bigr\vert \le C \frac{\|h\|_{\Delta_{\gamma}}(1+\|\varphi\|_{\infty})}{ |\phi(2^{k-1})\xi|^{\frac{\beta}{m}}}, \end{aligned}$$
(4.8)
$$\begin{aligned}& \bigl\vert \hat{\sigma}_{m,k}(\xi)\bigr\vert \le C\|h \|_{\Delta_{1}} \bigl\vert \phi\bigl(2^{k}\bigr)\xi\bigr\vert ^{\frac{\beta}{m}}, \end{aligned}$$
(4.9)

where β is a fixed constant with \(0<\beta<1/2\). Using Plancherel’s theorem, (4.8), the support property of \(\psi_{j}\), and \(a_{k+1}/c_{1}\le a_{k}\le a_{k+1}/a\), we get for \(j\ge0\),

$$\begin{aligned} \|\tilde{Q}_{m,j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &\le C \biggl(\sum _{k\in\mathbb{Z}}\int_{a_{j-k}\le|\xi|< a_{j-k+2}} \bigl\vert \hat{ \sigma}_{m,k}(\xi)\hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{\frac{1}{2}} \\ &\le C \biggl(\sum_{k\in\mathbb{Z}}\int_{a_{j-k}\le|\xi|<a_{j-k+2}} \frac{1}{|\phi(2^{k-1})\xi|^{\frac{\beta}{m}}} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{\frac{1}{2}} \\ &\le Ca^{-\frac{\beta}{m}j} \biggl(\sum_{k\in\mathbb{Z}}\int _{a_{j-k}\le|\xi|<a_{j-k+2}} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{\frac{1}{2}} \\ &\le Ca^{-\frac{\beta}{m}j}\|f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. \end{aligned}$$

For \(j<0\), using (4.9) in place of (4.8), we get

$$ \|\tilde{Q}_{m,j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le Ca^{\frac{\beta}{m}j}\|f \|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. $$

This shows (4.6). Interpolating (4.6) and (4.5), we obtain for some \(0<\theta<1\),

$$ \|\tilde{Q}_{m,j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le Ca^{-\frac{\beta\theta}{m}|j|}\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$
(4.10)

provided \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)).

From (4.10) and the definition of \(\tilde{Q}_{m,j}\) it follows

$$ \|T_{\Omega_{m},h,\phi}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\sum_{j\in\mathbb{Z}}a^{-\frac{\beta\theta}{m}|j|} \|f\| _{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le\frac{C}{1-a^{-\frac{\beta\theta}{m}}}\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C \frac{m}{\beta\theta}\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}. $$

We can see that the same estimate holds for \(\Omega_{0}\). Thus, by (4.4) we have

$$ \|T_{\Omega,h,\phi}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\biggl(1+\sum _{m\in\Lambda}A_{m} m\biggr)\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\|f \|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$

provided \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)). This completes the proof of Theorem 1.2 in the case \(\Omega\in L\log L({{S}^{n-1}})\).

(B) Block space case. Let \(r>1\). Then if \(\Omega\in B_{r}^{(0,0)}(S^{n-1})\) and satisfies the cancelation condition, it can be written as \(\Omega=\sum_{\ell=1}^{\infty}\lambda_{\ell}\breve{\Omega}_{\ell}\), where \(\lambda_{\ell}\in\mathbb{C}\) and \(\breve{\Omega}_{\ell}\) is an r-block supported on a cap \(B_{\ell}=B(x_{\ell},\tau_{\ell})\cap S^{n-1}\) on \(S^{n-1}\) and

$$ \sum_{\ell=1}^{\infty}| \lambda_{\ell}| \bigl\{ 1+{\log \bigl(|B_{\ell}|^{-1} \bigr)} \bigr\} < 2\|\Omega\| _{B_{r}^{(0,0)}(S^{n-1})}<\infty. $$
(4.11)

To each block \(\breve{\Omega}_{\ell}\), we define

$$ \Omega_{\ell}\bigl(y'\bigr) =\breve{\Omega}_{\ell}\bigl(y'\bigr) -\frac{1}{|{{S}^{n-1}}|}\int_{S^{n-1}} \breve{\Omega}_{\ell}\bigl(x'\bigr)\, d\sigma \bigl(x'\bigr). $$

Let \(\Lambda=\{\ell\in\mathbb{N}; |B_{\ell}|\le1/2\}\) and set

$$ \Omega_{0}=\Omega-\sum_{\ell\in\Lambda} \lambda_{\ell}\Omega_{\ell}. $$
(4.12)

Then there exists a positive constant C such that the following hold for all \(\ell\in\Lambda\):

$$\begin{aligned}& \int_{S^{n-1}}\Omega_{\ell}\bigl(x'\bigr) \,d\sigma\bigl(x'\bigr)=0, \end{aligned}$$
(4.13)
$$\begin{aligned}& \|\Omega_{\ell}\|_{L^{r}(S^{n-1})}\le C|B_{\ell}|^{-1/r'}, \end{aligned}$$
(4.14)
$$\begin{aligned}& \|\Omega_{\ell}\|_{L^{1}(S^{n-1})}\le2, \end{aligned}$$
(4.15)
$$\begin{aligned}& \Omega=\Omega_{0}+\sum_{\ell\in\Lambda} \lambda_{\ell}\Omega_{\ell}. \end{aligned}$$
(4.16)

Moreover, from (4.11) and the definition of \(\Omega_{\ell}\) it follows that

$$\begin{aligned}& \|\Omega_{0}\|_{L^{r}(S^{n-1})}\le C\sum_{\ell\in\mathbb{N}\setminus\Lambda}2^{-1/r'}| \lambda_{\ell}| \le C\|\Omega\|_{B_{r}^{(0,0)}(S^{n-1})} , \end{aligned}$$
(4.17)
$$\begin{aligned}& \int_{S^{n-1}}\Omega_{0}\bigl(x'\bigr) \,d\sigma\bigl(x'\bigr)=0. \end{aligned}$$
(4.18)

By (4.16), we have

$$ {{T}}_{\Omega,h,\phi}f(x) = \sum_{\ell\in\Lambda\cup{0}} \lambda_{\ell} {{T}}_{\Omega_{\ell},h,\phi}f(x). $$
(4.19)

So, we have only to show the boundedness of \({{T}}_{\Omega_{\ell},h,\phi} f\). We use the notations in Section 3 with minor change such as \(\tilde{Q}_{\ell,j}\) for \(\Omega_{\ell}\) instead of \(\tilde{Q}_{j}\) for Ω. Since \(\|\Omega_{\ell}\|_{L^{1}({{S}^{n-1}})}\le C\), we have as in Section 3 that

$$ \|\tilde{Q}_{\ell,j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\|f \|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$
(4.20)

if \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)).

About \(L^{2}\) estimate, we have

$$ \|\tilde{Q}_{\ell,j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le C a^{-\frac{\beta}{m_{\ell}}|j|}\|f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} $$
(4.21)

for some β with \(0<\beta<r\). In fact, let \(\sigma_{\ell,k}=\sigma_{\Omega_{\ell},h,\phi,k}\). For \(\ell\in\Lambda\cup\{0\}\), we set \(m_{\ell}=[\log_{2}|B_{\ell}|^{-1/r'}]+1\), where \([\, \cdot\, ]\) denotes the greatest integer function.

Since \(\|\Omega_{\ell}\|_{L^{1}({{S}^{n-1}})}\le2\) and \(\|\Omega_{\ell}\|_{L^{2}({{S}^{n-1}})}\le C2^{m_{\ell}}\), we get by Lemma 3.1 in [23], p.1567

$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\ell,k}(\xi)\bigr\vert \le C\|h \|_{\Delta_{1}}, \end{aligned}$$
(4.22)
$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\ell,k}(\xi)\bigr\vert \le C \frac{\|h\|_{\Delta_{\gamma}}(1+\|\varphi\|_{\infty})}{ |\phi(2^{k-1})\xi|^{\frac{\beta}{m_{\ell}}}}, \end{aligned}$$
(4.23)
$$\begin{aligned}& \bigl\vert \hat{\sigma}_{\ell,k}(\xi)\bigr\vert \le C\|h \|_{\Delta_{1}} \bigl\vert \phi\bigl(2^{k}\bigr)\xi\bigr\vert ^{\frac{\beta}{m_{\ell}}}, \end{aligned}$$
(4.24)

where β is a fixed constant with \(0<\beta<r\). Using Plancherel’s theorem, (4.23), the support property of \(\psi_{j}\), and \(a_{k+1}/c_{1}\le a_{k}\le a_{k+1}/a\), we get for \(j\ge0\),

$$\begin{aligned} \|\tilde{Q}_{\ell,j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} &\le C \biggl(\sum _{k\in\mathbb{Z}}\int_{a_{j-k}\le|\xi|< a_{j-k+2}} \bigl\vert \hat{ \sigma}_{\ell,k}(\xi)\hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{\frac{1}{2}} \\ &\le C \biggl(\sum_{k\in\mathbb{Z}}\int_{a_{j-k}\le|\xi|<a_{j-k+2}} \frac{1}{|\phi(2^{k-1})\xi|^{\frac{\beta}{m_{\ell}}}} \bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d\xi \biggr)^{\frac{1}{2}} \\ &\le Ca^{-\frac{\beta}{m_{\ell}}j} \biggl(\sum_{k\in\mathbb{Z}} \int _{a_{j-k}\le|\xi|<a_{j-k+2}}\bigl\vert \hat{f}(\xi)\bigr\vert ^{2}\, d \xi \biggr)^{\frac{1}{2}} \\ &\le Ca^{-\frac{\beta}{m_{\ell}}j}\|f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. \end{aligned}$$

For \(j<0\), using (4.9) in place of (4.8), we get

$$ \|\tilde{Q}_{\ell,j}f\|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})} \le Ca^{\frac{\beta}{m_{\ell}}j}\|f \|_{\dot{F}_{2,2}^{0}({\mathbb{R}^{n}})}. $$

This shows (4.21). Interpolating (4.21) and (4.20), we obtain for some \(0<\theta<1\),

$$ \|\tilde{Q}_{\ell,j}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le Ca^{-\frac{\beta\theta}{m_{\ell}}|j|}\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$
(4.25)

provided \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)).

From (4.25) and the definition of \(\tilde{Q}_{\ell ,j}\) it follows

$$ \|T_{\Omega_{\ell},h,\phi}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\sum_{j\in\mathbb{Z}}a^{-\frac{\beta\theta}{m_{\ell}}|j|} \|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le\frac{C}{1-a^{-\frac{\beta\theta}{m_{\ell}}}}\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C \frac{m_{\ell}}{\beta\theta}\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb {R}^{n}})}. $$

We can see that the same estimate holds for \(\Omega_{0}\). Thus, by (4.14) we have

$$ \|T_{\Omega,h,\phi}f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} \le C\biggl(1+\sum _{\ell\in\Lambda}\lambda_{\ell}m_{\ell}\biggr) \|f \|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})}\le C\|f\|_{\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})} $$

provided \(\alpha\in\mathbb{R}\) and \((\frac{1}{p},\frac{1}{q})\) belongs to the interior of the octagon \(Q_{1}Q_{2}R_{2}P_{3}Q_{3}Q_{4}R_{4}P_{6}\) (hexagon \(Q_{1}Q_{2}P_{3}Q_{3}Q_{4}P_{6}\) in the case \(1<\gamma\le2\)).

This completes the proof of Theorem 1.2.