1 Introduction
To represent certain linear differential operators by means of singular integral operators, Calderón [Reference Calderón3] introduced the Calderón commutator, defined as
$$ \begin{align*}[b,\,H\frac{d}{dx}]f(x)=\mathrm{\,p.\,v.}\int^{\infty}_{-\infty}\frac{b(x)-b(y)} {(x-y)^2}f(y)dy,\end{align*} $$
where b is a Lipschitz function in
$\mathbb {R}$
. This operator is closely related to Hilbert transform, but is a typical non-convolution singular integral operator. As it was demonstrated in [Reference Grafakos12, Chapter 8], the Calderón commutator plays an important role in the theory of singular integral operators and has many applications in related topics.
In this article, we will focus on the high-dimensional Calderón commutator. Let
$\Omega $
be homogeneous of degree zero, integrable on
$S^{d-1}$
, the unit sphere in
$\mathbb {R}^d$
with
$d\geq 2$
, and enjoy vanishing moment that
$$ \begin{align} \int_{S^{d-1}}\Omega(x)x^{\gamma}dx=0,\ \gamma\in \mathbb{Z}^{d}_{+},\ |\gamma|=1. \end{align} $$
Let a be a function on
$\mathbb {R}^{d}$
such that
$\nabla a\in L^{\infty }(\mathbb {R}^{d})$
. Define the d-dimensional Calderón commutator
$T_{\Omega ,a}$
by
$$ \begin{align} T_{\Omega,a}f(x)=\mathrm{p. v.}\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{|x-y|^{d+1}}(a(x)-a(y))f(y)dy. \end{align} $$
This operator was also introduced by Calderón [Reference Calderón2]. Calderón proved that if
$\Omega $
satisfies (1.1), then
$\Omega \in L\log L(S^{d-1})$
is a sufficient condition such that
$T_{\Omega , a}$
is bounded on
$L^p(\mathbb {R}^d)$
for all
$p\in (1,\,\infty )$
. Since then, considerable attention has been paid to the mapping properties of
$T_{\Omega ,a}$
on various function spaces. Pan, Wu and Yang [Reference Pan, Wu and Yang17] showed that
$\Omega \in H^1(S^{d-1})$
guarantees the
$L^p(\mathbb {R}^d)$
boundedness of
$T_{\Omega , a}$
for all
$p\in (1,\,\infty )$
. Hofmann [Reference Hofmann14] considered the weighted
$L^p$
boundedness with
$A_p$
weights for
$T_{\Omega ,a}$
, and proved that if
$\Omega \in L^{\infty }(S^{d-1})$
, then
$T_{\Omega ,a}$
is bounded on
$L^p(\mathbb {R}^d, w)$
for
$p\in (1,\,\infty )$
and
$w\in A_p(\mathbb {R}^d)$
, where and in the following,
$A_p(\mathbb {R}^d)$
denotes the weight function class of Muckenhoupt, see [Reference Grafakos12, Chapter 9] for the definition and properties of
$A_p(\mathbb {R}^d)$
. Ding and Lai [Reference Ding and Lai8] considered the weak type endpoint estimate for
$T_{\Omega , a}$
, and proved that
$\Omega \in L\log L(S^{d-1})$
is a sufficient condition such that
$T_{\Omega ,a}$
is bounded from
$L^1(\mathbb {R}^d)$
to
$L^{1,\,\infty }(\mathbb {R}^d)$
.
Fairly recently, Chen, Hu, and Tao [Reference Chen, Hu and Tao5] considered the
$L^p(\mathbb {R}^d)$
boundedness of
$T_{\Omega ,a}$
when
$\Omega $
satisfies another minimum size condition. Let
$\beta>1$
and
$\Omega $
be a integrable function on
$S^{d-1}$
, we say that
$\Omega \in GS_{\beta }(S^{d-1})$
if
$$ \begin{align} \sup\limits_{\zeta\in S^{d-1}}\int_{S^{d-1}}|\Omega(\theta)|\mathrm{log}^{\beta}{\bigg(}\frac{1}{|\zeta\cdot\theta|}{\bigg)}d\theta<\infty. \end{align} $$
The condition (1.3) was introduced by Grafakos and Stefanov [Reference Grafakos and Stefanov13], in order to study the
$L^p(\mathbb {R}^d)$
boundedness for the homogeneous singular integral operator. Obviously,
$L(\log L)^{\beta }(S^{d-1})\subset GS_{\beta }(S^{d-1})$
. On the other hand, as it was pointed out in [Reference Grafakos and Stefanov13], there exist integrable functions on
$S^{d-1}$
which are not in
$H^1(S^{d-1})$
but satisfy (1.3) for all
$\beta \in (1,\,\infty )$
. Chen, Hu, and Tao [Reference Chen, Hu and Tao5] proved that,
Theorem 1.1 Let
$\Omega $
be homogeneous of degree zero, satisfy the vanishing moment (1.1), a be a function on
$\mathbb {R}^{d}$
such that
$\nabla a\in L^{\infty }(\mathbb {R}^{d})$
. Suppose that
$\Omega \in GS_{\beta }(S^{d-1})$
with
$\beta \in (1,\,\infty )$
, then for
$\frac {2\beta }{2\beta -1}<p<2\beta $
,
$T_{\Omega ,a}$
is bounded on
$L^p(\mathbb {R}^d)$
.
Let
$\psi \in C^{\infty }_{0}(\mathbb {R}^{d})$
be a radial function, have integral zero and
$\mathrm {supp}\,\psi \subset B(0,\,1)$
. Let
$Q_{s}$
be the operator defined by
$Q_{s}f(x)= \psi _{s} \ast f(x)$
, where
$\psi _{s}(x)=s^{-d}\psi (s^{-1}x)$
. We assume that
$$ \begin{align*}\int_{0}^{\infty}[\widehat{\psi}(s)]^{4}\frac{ds}{s}=1. \end{align*} $$
Then the Calderón reproducing formula
$$ \begin{align} \int_{0}^{\infty}Q_{s}^{4}\frac{ds}{s}=I \end{align} $$
holds true. The argument in [Reference Chen, Hu and Tao5] relies essentially on a refined decomposition of
$Q_s^4T_{\Omega , a,\,j}Q_t^4$
, where
$T_{\Omega ,a,j}$
is the dyadic truncated of
$T_{\Omega ,a}$
.
Let
$1<p,\,q<\infty $
and
$\beta \in \mathbb {R}$
. Define the homogeneous Triebel–Lizorkin space
$\dot {F}_{p}^{\beta ,q}(\mathbb {R}^d)$
as the space of all tempered distributions
$f\in \mathcal {S}'(\mathbb {R}^d)$
satisfying
$$ \begin{align*}\|f\|_{\dot{F}_p^{\beta,q}(\mathbb{R}^d)}=\Big\|\Big(\int^{\infty}_0|t^{-\beta}Q_tf|^q \frac{dt}{t}\Big)^{1/q}\Big\|_{L^p(\mathbb{R}^d)}<\infty. \end{align*} $$
As is well known, the Triebel–Lizorkin space is a unified setting of many function spaces including Lebesgue spaces, the Hardy spaces, and the Sobolev spaces. We remark that there have been many works about the boundedness on Triebel–Lizorkin space for singular integral operators with rough kernels, see [Reference Al-Qassem, Cheng and Pan1, Reference Chen, Fan and Ying4, Reference Chen and Zhang7, Reference Fan and Zhao10] and the references therein.
Hu and Liu [Reference Hu and Liu15] considered the boundedness of
$T_{\Omega ,a}$
on Trieble–Lizorkin spaces when
$\Omega $
satisfies size condition, and proved the following conclusion.
Theorem 1.2 Let
$\Omega $
be homogeneous of degree zero, satisfy the vanishing moment (1.1), and let a be a function on
$\mathbb {R}^n$
such that
$\nabla a\in L^{\infty }(\mathbb {R}^d)$
. Suppose that
${\Omega \in L(\log L)^{2\tilde {q}}(S^{d-1})}$
, then for
$p,\,q\in (1,\,\infty )$
,
$T_{\Omega ,a}$
is bounded on
$\dot {F}_{p}^{0,q}(\mathbb {R}^d)$
, where and in what follows,
$\tilde {q}=\mathrm {max}\{1/q,1/q'\}$
.
In this article, we consider the boundedness on
$\dot {F}_p^{0,q}(\mathbb {R}^d)$
for
$T_{\Omega ,a}$
when
$\Omega \in GS_{\beta }(S^{d-1})$
for some
$\beta>1$
. Our main result can be stated as follows.
Theorem 1.3 Let
$\Omega $
be homogeneous of degree zero, satisfy the vanishing moment (1.1), and let a be a function on
$\mathbb {R}^n$
such that
$\nabla a\in L^{\infty }(\mathbb {R}^d)$
. Suppose that
$\Omega \in GS_{\beta }(S^{d-1})$
with
$\beta \in (1,\infty )$
, then
$T_{\Omega ,a}$
is bounded on
$\dot {F}_{p}^{0,q}(\mathbb {R}^d)$
for all
$1+\frac {1}{2\beta -1}<p,q<2\beta $
.
Remark 1.4 Theorem 1.3 recovers and extends the result established in [Reference Chen, Hu and Tao5]. To prove the
$L^p(\mathbb {R}^d)$
boundedness for
$T_{\Omega , a}$
when
$\Omega \in GS_{\beta }(S^{d-1})$
, Chen, Hu, and Tao invoked the Calderón reproducing formula, and approximate
$T_{\Omega ,\,a}$
by singular integral operators whose kernels enjoy the Hörmander condition. Our argument in establishing Theorem 1.3 only involves the Calderón reproducing formula and provided a new proof of the main result in [Reference Chen, Hu and Tao5].
Remark 1.5 The argument in this article involves a vector-valued estimate for
$Q_sQ_t$
, and some other vector-valued estimates, see Lemmas 2.3 and 2.2 below. It seems that these lemmas will be used in the study of related questions. In fact, invoking Lemma 2.3, and mimicking the estimates in [Reference Hu and Liu15], we can prove that
${\Omega \in L\log L(S^{d-1})}$
guarantees the boundedness of
$T_{\Omega ,a}$
on
$\dot {F}_{p}^{0,q}(\mathbb {R}^d)$
for all
$p,\,q\in (1,\,\infty )$
. This is of interest, since it is an extension of Calderón’s result of [Reference Calderón2]. Moreover, Theorem 1.3 provides a framework and fundamental method for further studying the boundedness on Triebel-Lizorkin spaces for the nonstandard singular integral operators and the maximal singular integrals with rough kernels
$\Omega \in L(\log L)^2(S^{d-1})$
or
$\Omega \in GS_{\beta }(S^{d-1})$
, and may potentially extend the works of [Reference Chen, Hu and Tao6, Reference Hu, Tao, Wang and Xue16, Reference Tao and Hu18, Reference Tao and Hu19] to the Triebel–Lizorkin spaces.
In what follows, C always denotes a positive constant that is independent of the main parameters involved but whose value may differ from line to line. We use the symbol
$A\lesssim B$
to denote that there exists a positive constant C such that
$A\le CB$
. Constant with subscript such as
$C_1$
, does not change in different occurrences. For any set
$E\subset \mathbb {R}^d$
,
$\chi _E$
denotes its characteristic function. For a cube
$Q\subset \mathbb {R}^d$
and
$\lambda \in (0,\,\infty )$
, we use
$\lambda Q$
to denote the cube with the same center as Q and whose side length is
$\lambda $
times that of Q. For
$p\in [1,\,\infty ]$
,
$p'$
denotes the dual exponent of p, namely,
$p'=p/(p-1)$
.
2 Proof of Theorem 1.3
Let
$\varphi \in C^{\infty }_{0}(\mathbb {R}^{d})$
be a radial function, supp
$\varphi \subset \{1/4\leq |x|\leq 4\}$
, and
$$ \begin{align*}\sum\limits_{j\in\mathbb{Z}}\varphi(2^{-j}x)\equiv1,\ \text{ for }|x|\neq 0. \end{align*} $$
Let
$\varphi _{j}(x)=\varphi (2^{-j}x)$
. For a function
$\Omega \in L^{1}(S^{d-1})$
, define the operator
$U_{\Omega ,j}$
by
$$ \begin{align*}U_{\Omega,j}h(x)=\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{|x-y|^{d+1}}\varphi_{j}(x-y)h(y)dy. \end{align*} $$
Also, we define
$T_{\Omega ,a;j}$
by
$$ \begin{align*}T_{\Omega,a;j}f(x)=\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{|x-y|^{d+1}}\varphi_{j}(x-y)(a(x)-a(y))f(y)dy. \end{align*} $$
Lemma 2.1 Let
$\Omega $
be homogeneous of degree zero and
$\Omega \in L^{1}(S^{d-1})$
. Then the inequality
$$ \begin{align*} &\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{\infty}_0|2^jU_{\Omega,j}f_{j,s,t}|^q\frac{ds}{s}\frac{dt}{t} \Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim \|\Omega\|_{L^1(S^{d-1})}\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int_{0}^{\infty}|f_{j,s,t}|^q \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\nonumber \end{align*} $$
holds for all
$p,\,q\in (1,\,\infty )$
.
For the proof of Lemma 2.1, see [Reference Hu and Liu15].
The weighted Littlewood–Paley theory tells us that, when
$w\in A_{2}(\mathbb {R}^{d})$
,
$$ \begin{align*}\Big\|\Big(\int_{0}^{\infty}|Q_sf|^2\frac{ds}{s}\Big)^{1/2}\Big\|_{L^2(\mathbb{R}^d,\,w)}\lesssim \|f\|_{L^2(\mathbb{R}^d,\,w)},\end{align*} $$
then by the extrapolation theorem (see [Reference Duoandikoetxea9, Theorem 3.1]), we know that, for all
$p\in (1,\infty )$
,
$$ \begin{align} \Big\|\Big(\int_0^{\infty}\int_{0}^{\infty}\sum_{j\in\mathbb{Z}}|Q_sf_{j,t}|^2\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^d)}\lesssim \Big\|\Big(\int^{\infty}_0\sum_{j\in\mathbb{Z}}|f_{j,t}|^2\frac{dt}{t}\Big)^{1/2}\Big\|_{L^p(\mathbb{R}^d)}. \end{align} $$
Let M be the Hardy–Littlewood maximal operator. A well-known result of Fefferman and Stein [Reference Fefferman and Stein11] tells us that for all
$p,q\in (1,\infty )$
,
$$ \begin{align} &\Big\|\Big(\sum_{j}\int^{\infty}_0\int^{\infty}_0|Mf_{j,s,t}|^q \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\Big(\sum_{j}\int^{\infty}_0\int^{\infty}_0|f_{j,s,t}|^q \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}.\notag \end{align} $$
This, together with the fact that
$|Q_sf(x)|\lesssim Mf(x)$
, implies that
$$ \begin{align} \Big\|\Big(\int_{0}^{\infty}|M(Q_sf)|^q\frac{ds}{s}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\lesssim \|f\|_{\dot{F}^{0,q}_p(\mathbb{R}^d)}. \end{align} $$
Lemma 2.2 Let
$\Omega $
be homogeneous of degree zero and
$\Omega \in GS_{\beta }(S^{d-1})$
for some
$\beta>1$
. Then for any
$0<\theta <1$
,
$\frac {\theta }{2}<\frac {1}{p}<1-\frac {\theta }{2}$
,
$\frac {\theta }{2}<\frac {1}{q}<1-\frac {\theta }{2}$
, it holds that
$$ \begin{align} &\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|2^j Q_sU_{\Omega,j}f_{j,s,t}|^q\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|f_{j,s,t}|^q\log^{-q\beta\theta}(\mathrm{e}+2^j/s)\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}.\nonumber \end{align} $$
Proof As it was proved in [Reference Chen, Hu and Tao5, Lemma 2.2], we have that, for
$j\in \mathbb {Z}$
and
$0<s\leq 2^j$
,
$$ \begin{align*}\|Q_sU_{\Omega, j}f\|_{L^2(\mathbb{R}^d)}\lesssim 2^{-j}\log^{-\beta}(\mathrm{e}+2^j/s)\|f\|_{L^2(\mathbb{R}^d)}.\end{align*} $$
Therefore,
$$ \begin{align} &\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|2^j Q_sU_{\Omega,j}f_{j,s,t}|^2\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{2}}\Big\|_{L^2(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|f_{j,s,t}|^2\log^{-2\beta}(\mathrm{e}+2^j/s)\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{2}} \Big\|_{L^2(\mathbb{R}^d)}.\nonumber \end{align} $$
On the other hand, we obtain from Lemma 2.1 that for
$1<p,q<\infty $
,
$$ \begin{align} &\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|2^j Q_sU_{\Omega,j}f_{j,s,t}|^q\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|f_{j,s,t}|^q\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}. \nonumber \end{align} $$
Inequalities (2.5) and (2.6), via interpolation, shows that for
$1<p,q<\infty $
, we get
$$ \begin{align*} &\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|2^j Q_sU_{\Omega,j}f_{j,s,t}|^q\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{\infty}_0\int^{2^j}_0|f_{j,s,t}|^q\log^{-q\beta\theta}(\mathrm{e}+2^j/s)\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)},\nonumber \end{align*} $$
for all
$0<\theta <1$
,
$\frac {\theta }{2}<\frac {1}{p}<1-\frac {\theta }{2}$
,
$\frac {\theta }{2}<\frac {1}{q}<1-\frac {\theta }{2}$
. This completes the proof of Lemma 2.2.
The following lemma also plays an important role in the proof of Theorem 1.3.
Lemma 2.3 Let
$p,\,q\in (1,\,\infty )$
. Then there exists a constant
$\varrho =\varrho (p,\,q)\in (0,1)$
such that
$$ \begin{align} &\Big\|\Big(\sum_j\int_{0}^{\infty}\int_{0}^{\infty}|Q_sQ_tg_{j,s,t}|^q \frac{ds}{s}\frac{dt}{t}\Big)^{1/q} \Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim\Big\|\Big(\sum_{j}\int_{0}^{\infty}\int_{0}^{\infty}|g_{j,s,t}|^q\min\{(\frac{s}{t})^{\varrho},\,(\frac{t}{s})^{\varrho}\} \frac{ds}{s}\frac{dt}{t}\Big)^{1/q}\Big\|_{L^p(\mathbb{R}^d)}.\nonumber \end{align} $$
Proof Recalling that
$\psi $
has integral zero, it then follows that, for
$s\leq t$
,
$$ \begin{align*} \Big|\int_{\mathbb{R}^d}\psi_s(x-y)\psi_t(y)dy\Big|&=\Big|\int_{\mathbb{R}^d}\psi_s(x-y)\big (\psi_{t}(y)-\psi_t(x)\big)dy\Big|\\ &\leq \frac{s}{t^{d+1}}\chi_{\{|x|\leq 2t\}}(x), \end{align*} $$
and for
$0<t<s$
,
$$ \begin{align*} \Big|\int_{\mathbb{R}^d}\psi_s(x-y)\psi_t(y)dy\Big|&=\Big|\int_{\mathbb{R}^d}\psi_t(x-y)\big (\psi_{s}(y)-\psi_s(x)\big)dy\Big|\\ &\leq \frac{t}{s^{d+1}}\chi_{\{|x|\leq 2s\}}(x). \end{align*} $$
This, in turn, implies that for
$p\in (1,\,\infty )$
,
$$ \begin{align*}\|Q_sQ_tf\|_{L^p(\mathbb{R}^d)}\lesssim \min\{\frac{s}{t},\,\frac{t}{s}\}\|Mf\|_{L^p(\mathbb{R}^d)}\lesssim\min\{\frac{s}{t},\,\frac{t}{s}\} \|f\|_{L^p(\mathbb{R}^d)}. \end{align*} $$
Therefore, we deduce that
$$ \begin{align*} &\sum_{j}\int_{0}^{\infty}\int_{0}^{\infty}\|Q_sQ_tf_{j,s,t}\|^p_{L^p(\mathbb{R}^d)} \frac{ds}{s}\frac{dt}{t}\\ &\quad\lesssim\sum_{j}\int_{0}^{\infty}\int_{0}^{\infty}\big(\min\{\frac{s}{t},\,\frac{t}{s}\}\big)^{p} \|f_{j,s,t}\|_{L^p(\mathbb{R}^d)}^p\frac{ds}{s}\frac{dt}{t}. \end{align*} $$
On the other hand, it is obvious that
$$ \begin{align*}\|\sup_{j\in\mathbb{Z}}\sup_{s, t>0}|Q_sQ_tf_{j,s,t}|\big\|_{L^p(\mathbb{R}^d)}\lesssim \|\sup_{j\in\mathbb{Z}}\sup_{s,t>0}|f_{j,s,t}|\|_{L^p(\mathbb{R}^d)}. \end{align*} $$
A trivial computation involving interpolation then gives (2.7) for the case of
$1<p<q<\infty $
. The case of
$1<q<p<\infty $
for (2.7) follows from a standard duality argument.
Proof of Theorem 1.3
By homogeneity, we assume that
$\left \|\nabla a\right \|_{L^{\infty }(\mathbb {R}^{d})}=1$
. Recall that
$T_{\Omega ,\,a}$
is a self-adjoint operator. By (1.4), it suffices to prove that for
$f, g\in C^{\infty }_{0}(\mathbb {R}^{d})$
,
$$ \begin{align} {\bigg|}\int_{0}^{\infty}\int_{0}^{t}\int_{\mathbb{R}^{d}}Q_{s}^{4}T_{\Omega,a}Q_{t}^{4}f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}{\bigg|}\lesssim \left\|f\right\|_{\dot{F}^{0,q}_{p}(\mathbb{R}^{d})}\left\|g\right\|_{\dot{F}^{0,q^{'}}_{p^{'}}(\mathbb{R}^{d})}. \end{align} $$
Let
$\alpha \in (\frac {d+1}{d+2},\,1)$
be a constant. As it was proved in [Reference Hu and Liu15], for
$p,q\in (1,\,\infty )$
,
$$ \begin{align} &\Big|\sum_{j\in\mathbb{Z}}\int^{\infty}_{2^j}\int_{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}}^t \int_{\mathbb{R}^d}Q_s{T}_{\Omega,\,a;j}Q_t^4f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}\Big|\\ &\quad\lesssim \|\Omega\|_{L^1(S^{d-1})}\|f\|_{\dot{F}_{p}^{0,q}(\mathbb{R}^d)}\|g\|_{\dot{F}_{p'}^{0,q'}(\mathbb{R}^d)}.\nonumber \end{align} $$
The proof of (2.8) is reduced to proving that for
$1+\frac {1}{2\beta -1}<p,q<2\beta $
,
$$ \begin{align} &\Big|\sum_{j\in\mathbb{Z}}\int^{\infty}_{2^j}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}} \int_{\mathbb{R}^d}Q_s{T}_{\Omega,\,a;j}Q_t^4f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}\Big|\\ &\quad\lesssim \|f\|_{\dot{F}_{p}^{0,q}(\mathbb{R}^d)}\|g\|_{\dot{F}_{p'}^{0,q'}(\mathbb{R}^d)},\nonumber \end{align} $$
and
$$ \begin{align} &\Big|\sum_{j\in\mathbb{Z}}\int^{2^j}_{0}\int_{0}^{t} \int_{\mathbb{R}^d}Q_s{T}_{\Omega,\,a;j}Q_t^4f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}\Big|\\ &\quad\lesssim \|f\|_{\dot{F}_{p}^{0,q}(\mathbb{R}^d)}\|g\|_{\dot{F}_{p'}^{0,q'}(\mathbb{R}^d)}.\nonumber \end{align} $$
We first prove (2.10). We will employ a refined decomposition for
$Q_sT_{\Omega ,a;j}Q_t$
in [Reference Chen, Hu and Tao5]. For each fixed
$j\in \mathbb {Z}$
, let
$\{I_{j,l}\}_{l}$
be a sequence of cubes having disjoint interiors and side lengths
$2^j$
, such that
$ \mathbb {R}^d=\cup _{l}I_{j,l}.$
For each fixed
$j,l$
, let
$\omega _{j,l}\in C^{\infty }_0(\mathbb {R}^d)$
such that
$\mathrm {supp}\,\omega _{j,l}\subset 48dI_{j,l}$
,
$0\leq \omega _{j,l}\leq 1$
and
$\omega _{j,l}(x)\equiv 1$
when
$x\in 32dI_{j,l}$
. Let
$I_{j,l}^*=64dI_{j,l}$
and
$x_{j,l}$
be the center of
$I_{j,l}$
. For each l, set
$a_{j,l}(y)=(a(y)-a(x_{j,l}))\omega _{j,l}(y)$
, and
$h_{s,j,l}(y)=Q_s^2g(y)\chi _{I_{j,l}}(y)$
. It is obvious that for all l,
$$ \begin{align*}\|a_{j,l}\|_{L^{\infty}(\mathbb{R}^d)}\lesssim 2^j,\,\,\|\nabla a_{j,l}\|_{L^{\infty}(\mathbb{R}^d)}\lesssim 1. \end{align*} $$
As in [Reference Chen, Hu and Tao5] (see also [Reference Hu and Liu15]), for each
$l\in \mathbb {Z}_+$
,
$j\in \mathbb {Z}$
and
$s\in (0,\,2^j]$
, we have that
$$ \begin{align} \int_{\mathbb{R}^{d}}Q^{4}_{s}T_{\Omega,a;j}Q^{4}_{t}f(x)g(x)dx =\sum_{n=1}^{5}\sum\limits_{l}D_{j,l,n}(s,t), \end{align} $$
where
$$ \begin{align*} D_{j,l,1}(s,t) & =-\int_{\mathbb{R}^{d}}[a_{j,l},Q_{s}]U_{\Omega,j}Q^{4}_{t}f(x)Q_{s}h_{s,j,l}(x)dx, \\ D_{j,l,2}(s,t) & =\int_{\mathbb{R}^{d}}Q_{s}U_{\Omega,j}Q^{4}_{t}f(x)[a_{j,l},Q_{s}]h_{s,j,l}(x)dx, \\ D_{j,l,3}(s,t) & =\int_{\mathbb{R}^{d}}Q_{s}U_{\Omega,j}Q^{4}_{t}f(x)Q_{s}(a_{j,l}h_{s,j,l})(x)dx, \\ D_{j,l,4}(s,t) & =\int_{\mathbb{R}^{d}}Q_{s}U_{\Omega,j}[a_{j,l},Q_{s}]Q^{4}_{t}f(x)h_{s,j,l}(x)dx, \end{align*} $$
and
$$ \begin{align*}D_{j,l,5}(s,t)=-\int_{\mathbb{R}^{d}}Q_{s}U_{\Omega,j}(a_{j,l}Q_{s}Q^{4}_{t}f)(x)h_{s,j,l}(x)dx, \end{align*} $$
where and in the following, for a locally integrable function b and an operator U,
$[b, U]$
denotes the commutator of U with symbol b, namely,
$$ \begin{align*}[b, U]h(x) = b(x)Uh(x)- U(bh)(x). \end{align*} $$
We now estimate
$\sum _j\sum _l \int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm {D}_{j,l,1}(s,t)\frac {ds}{s}\frac {dt}{t}$
. It follows from Hölder’s inequality that
$$ \begin{align*} &\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}\mathrm{D}_{j,l,1}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\&\quad\le \Big\|\Big(\sum_j\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}|\chi_{I_{j,l}^*}Q_t^4f|^q (2^{-j}s)^{\frac{q}{q^{'}}} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\&\qquad\times\Big\|\Big(\sum_j\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}} |U_{\Omega,j}[a_{j,l},Q_s]Q_sh_{s,j,l} |^{q'}\frac{1}{2^{-j}s} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}}\Big\|_{L^{p'}(\mathbb{R}^d)}. \end{align*} $$
Invoking the fact that
$\sum _{l}\chi _{I_{j,l}^*}\lesssim 1$
, and the inequality (2.3), we deduce that
$$ \begin{align*} & \Big\|\Big(\sum_j\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}|\chi_{I_{j,l}^*}Q_t^4f|^q (2^{-j}s)^{\frac{q}{q^{'}}} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\Big( \int^{\infty}_{0}|Q_t^4f|^q\int_{0}^{t}\sum_{j:\, 2^j\geq s^{\alpha}t^{1-\alpha}}(2^{-j}s)^{\frac{q}{q^{'}}} \frac{ds}{s} \frac{dt}{t}\Big)^{1/q}\Big\|_{L^p(\mathbb{R}^d)}\lesssim\|f\|_{\dot{F}^{0,q}_p(\mathbb{R}^d)}. \end{align*} $$
Observing that when
$s\in (0,\,2^j]$
,
$$ \begin{align*}\big|[a_{j,l},\,Q_s]h(x)\big|\le \int_{\mathbb{R}^d}|\psi_s(x-y)||a_{j,l}(x)-a_{j,l}(y)||h(y)|dy\lesssim sMh(x), \end{align*} $$
and that
$|U_{\Omega , j}h(x)|\lesssim 2^{-j}M_\Omega h(x)$
, we then obtain from Lemma 2.1 and (2.2) that
$$ \begin{align*} &\Big\|\Big(\sum_j\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}|U_{\Omega,j} [a_{j,l},Q_s]Q_sh_{s,j,l} |^{q'}\frac{1}{2^{-j}s} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}}\Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\sum_j\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}} \big(M_\Omega MMh_{s,j,l}\big)^{q'}(2^{-j}s)^{q'-1} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}}\Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\quad\lesssim \Big\|\sum_j\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}} \big|h_{s,j,l}\big|^{q'}(2^{-j}s)^{q'-1} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}}\Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\quad \lesssim\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}, \end{align*} $$
where the last inequality follows from the fact that
$$ \begin{align*} \int_{s}^{\infty}\sum_{j:2^j\geq s^{\alpha}t^{1-\alpha}}(2^{-j}s)^{q'-1}\frac{dt}{t}\lesssim 1. \end{align*} $$
Therefore,
$$ \begin{align} \Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}}\mathrm{D}_{j,l,1}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\lesssim \|f\|_{\dot{F}_p^{0,q}(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}. \end{align} $$
Similar to the estimate (2.13), we have that for
$1<p,\,q<\infty $
,
$$ \begin{align} &\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}}\mathrm{D}_{j,l,2}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\ &\quad+\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}}\mathrm{D}_{j,l,4}(s,t)\frac{ds}{s}\frac{dt}{t}\Big|\lesssim \|f\|_{\dot{F}_p^{0,q}(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}.\nonumber\end{align} $$
We now estimate the term
$\sum _j\sum _l \int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm {D}_{j,l,3}(s,t)\frac {ds}{s}\frac {dt}{t}$
. By Hölder’s inequality, we can write
$$ \begin{align*} &\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}}\mathrm{D}_{j,l,3}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\&\quad\le \Big\|\Big(\sum_{j}\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}} |Q_s^2(2^jU_{\Omega,j})Q_t^3f|^q\big(\frac{t}{s}\big)^{\sigma\frac{q}{q'}} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\&\qquad\times \Big\|\Big(\sum_{j}\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}} \big|2^{-j}Q_t\big(\sum_la_{j,l}h_{s,j,l}\big)\big|^{q'}\big(\frac{s}{t}\big)^{\sigma} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\&\quad:=\mathrm{I}_1\mathrm{I}_2, \end{align*} $$
where
$\sigma>0$
is a constant such that
$\sigma q/q'<\varrho $
. Invoking the fact that
$$ \begin{align*}\int^{\infty}_s\big(\frac{s}{t}\big)^{{\sigma}/2}\frac{dt}{t}\lesssim 1, \end{align*} $$
we obtain that
$$ \begin{align*} \quad\quad\mathrm{I}_2&\lesssim\Big\|\Big(\sum_{j}\int_0^{2^j} \big|2^{-j}M\big(\sum_la_{j,l}h_{s,j,l}\big)\big|^{q'}\big(\frac{s}{2^j}\big)^{{\sigma}/2} \frac{ds}{s}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\lesssim\Big\|\Big(\int_0^{\infty} \big|M\big(Q^2_sg\big)\big|^{q'}\sum_{j:2^j\geq s}\big(\frac{s}{2^j}\big)^{{\sigma}/2} \frac{ds}{s}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\lesssim \|g\|_{\dot{F}_{p'}^{0,q'}(\mathbb{R}^d)}.\nonumber \end{align*} $$
Note that
$Q_s^2(2^jU_{\Omega ,j})=Q_s(2^jU_{\Omega ,j})Q_s$
. It follows from Lemma 2.1, inequality (2.2) and Lemma 2.3 that
$$ \begin{align*} \mathrm{I}_1 &\lesssim\Big\|\Big(\sum_{j}\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{\frac{1}{\alpha}}} |Q_sQ_t^3f|^q\big(\frac{t}{s}\big)^{\sigma\frac{q}{q'}} \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}}\Big\|_{L^p(\mathbb{R}^d)}\nonumber\\ &\lesssim\Big\|\Big(\int_{0}^{\infty}|Q_t^2f|^q\int_0^t \sum_{j:2^j\geq s}\big(\frac{s}{2^j}\big)^{\big(\varrho-\sigma q/q'\big)/2}\big(\frac{s}{t}\big)^{\big(\varrho-\sigma q/q'\big)/2}\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\lesssim \|f\|_{\dot{F}^{0,q}_p(\mathbb{R}^d)}.\nonumber \end{align*} $$
The estimates for
$\mathrm {I}_1$
and
$\mathrm {I}_2$
give us that
$$ \begin{align} &\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}\mathrm{D}_{j,l,3}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\ &\quad\lesssim \|f\|_{\dot{F}^{0,q}_p(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)},\,\,p,q\in (1,\,\infty).\nonumber \end{align} $$
The estimate for term
$\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm {D}_{j,l,5}(s,t)\frac {ds}{s}\frac {dt}{t}$
is similar to the estimate for
$\mathrm {D}_{j,l,3}(s,t)$
. In fact, we have by Hölder’s inequality that
$$ \begin{align*} &\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}\mathrm{D}_{j,l,5}(s,t)\frac{ds}{s}\frac{dt}{t}\Big |\\&\quad\leq \Big\|\Big(\sum_j\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}|Q_sQ_t^3f|^q \big(\frac{t}{s}\big)^{\sigma\frac{q}{q'}}\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\&\qquad\times \Big\|\Big(\sum_j\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}\Big|Q_t\Big(\sum_la_{j,l} U_{\Omega,j}Q_sh_{s,j,l}\Big)\Big|^{q'}\big(\frac{s}{t}\big)^{\sigma}\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\&\quad:=\mathrm{I}_3\mathrm{I_4}. \end{align*} $$
It follows from Lemma 2.3 that
$$ \begin{align*}\mathrm{I}_3\lesssim \Big\|\Big(\sum_j\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}} |Q^2_tf|^q\big(\frac{s}{t}\big)^{\varrho-\sigma q/q'}\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\lesssim\|f\|_{\dot{F}_{p}^{0,q}(\mathbb{R}^d)}. \end{align*} $$
On the other hand, we have that
$$ \begin{align*} \mathrm{I}_4\lesssim\Big\|\Big(\int_{0}^{\infty}\Big [M\Big(\sum_la_{j,l} U_{\Omega,\,j}Q_sh_{s,j,l}\Big)\Big]^{q'}\frac{ds}{s}\Big)^{\frac{1}{q'}}\|_{L^{p'}(\mathbb{R}^d)}\lesssim \|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}, \end{align*} $$
since
$\|a_{j,l}\|_{L^{\infty }(\mathbb {R}^d)}\lesssim 2^j$
, and the supports of functions
$\{a_{j,l} U_{\Omega ,\,j}Q_sh_{s,j,l}\}$
have bounded overlaps. The estimates for
$\mathrm {I}_3$
,
$\mathrm {I}_4$
tell us that
$$ \begin{align} &\Big|\sum_{j}\sum_l\int_{2^j}^{\infty}\int_{0}^{(2^jt^{\alpha-1})^{1/\alpha}}\mathrm{D}_{j,l,5}(s,t)\frac{ds}{s}\frac{dt}{t}\Big |\\ &\quad\lesssim \|f\|_{\dot{F}^{0,q}_{p}(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)},\,\,p,\,q\in (1,\,\infty).\nonumber \end{align} $$
Combining inequalities (2.13), (2.14), (2.15), and (2.16) leads to (2.10).
It remains to prove the inequality (2.11). As in the estimate for (2.10), we can verify that, for
$p,q\in (1,\infty )$
,
$$ \begin{align*} &\Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,1}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|+\Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,2}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\ &\quad+\Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,4}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\lesssim \|f\|_{\dot{F}_p^{0,q}(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}. \end{align*} $$
Recall the assumption of Theorem 1.3 that
$1+\frac {1}{2\beta -1}<p,q<2\beta $
with
$\beta>0$
, we could take
$\theta>\frac {1}{\beta }$
and close to
$\frac {1}{\beta }$
enough, such that
$\frac {\theta }{2}<\frac {1}{p}<1-\frac {\theta }{2}$
and
$\frac {\theta }{2}<\frac {1}{q}<1-\frac {\theta }{2}$
, which implies that inequality (2.4) holds. Furthermore, we choose
$\gamma>1$
and
$ \gamma \frac {q}{q'}<q\beta \theta -1$
for
$q\in (2,\,2\beta )$
;
$ \gamma \frac {q'}{q}<q'\beta \theta -1$
for
$q\in (1+\frac {1}{2\beta -1},\,2]$
.
For the case of
$q\in (2,\,2\beta )$
, write
$$ \begin{align*} &\Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,3}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\&\quad\le \Big\|\Big(\sum_{j}\int_0^{2^j} \int_{0}^{t} |Q_s^2(2^jU_{\Omega,j})Q_t^3f|^q\big(\frac{t}{s}\big)^{ {\sigma}\frac{q}{q'}}\log^{\gamma\frac{q}{q'}}(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\&\qquad\times \Big\|\Big(\sum_{j}\int_0^{2^j} \int_{0}^{t} \big|2^{-j}Q_t\big(\sum_la_{j,l}h_{s,j,l}\big)\big|^{q'}\big(\frac{s}{t}\big)^{\sigma}\log^{-\gamma}(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\&\quad:=\mathrm{I}_5\mathrm{I}_6. \end{align*} $$
It follows from Lemmas 2.2 and 2.3 that
$$ \begin{align*} \mathrm{I}_5&\lesssim \Big\|\Big(\sum_{j}\int_0^{2^j} \int_{0}^{t} |Q_s(2^jU_{\Omega,j})Q^2_tf|^q\big(\frac{s}{t}\big)^{\varrho- \sigma \frac{q}{q'}}\log^{\gamma\frac{q}{q'}}(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\ &\lesssim\Big\|\Big(\int_0^{\infty}|Q^2_tf|^q\int^t_0 \big(\frac{s}{t}\big)^{\varrho- \sigma \frac{q}{q'}}\sum_{j:2^j\geq s}\log^{- q\beta\theta+\gamma\frac{q}{q'} }(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\ &\lesssim\Big\|\Big(\int_0^{\infty} |Q^2_tf|^q\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}, \end{align*} $$
provided that
$1+\frac {1}{2\beta -1}<p,q<2\beta $
. Also, we have that
$$ \begin{align*} \mathrm{I}_6&\lesssim\Big\|\Big(\sum_{j}\int_0^{2^j} \int_{0}^{t} \big|M\big(Q_s^2g\big)\big|^{q'}\big(\frac{s}{t}\big)^{ \sigma }\log^{-\gamma}(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\lesssim\Big\|\Big(\int_0^{\infty} \big|M\big(Q_s^2g\big)\big|^{q'}\frac{ds}{s}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}. \end{align*} $$
For the case of
$q\in (1+\frac {1}{2\beta -1},\,2]$
, we write
$$ \begin{align*} &\Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,3}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\\ &\quad\le \Big\|\Big(\sum_{j}\int_0^{2^j} \int_{0}^{t} |Q_sQ_t^3f|^q\big(\frac{t}{s}\big)^{\sigma\frac{q}{q'}}\log^{-\gamma}(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\ &\qquad\times \Big\|\Big(\sum_{j}\int_0^{2^j} \int_{0}^{t} \big|Q_sU_{\Omega,j}Q_t\big(\sum_la_{j,l}h_{s,j,l}\big)\big|^{q'} \big(\frac{s}{t}\big)^{\sigma}\log^{\gamma\frac{q'}{q}}(\mathrm{e}+\frac{2^j}{s}) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\quad:=\mathrm{I}_5'\mathrm{I}_6'. \end{align*} $$
It follows from Lemma 2.3 that
$$ \begin{align*} \mathrm{I}_5'&\lesssim\Big\|\Big(\sum_{j}\int_0^{2^j}|Q^2_tf|^q \int_{0}^{t} \big(\frac{s}{t}\big)^{\varrho- \sigma\frac{q}{q'}}\log^{-\gamma}(\mathrm{e}+2^j/s) \frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^p(\mathbb{R}^d)}\\ &\lesssim\Big\|\Big(\int_0^{\infty}|Q^2_tf|^q \frac{dt}{t}\Big)^{\frac{1}{q}} \Big\|_{L^{p}(\mathbb{R}^d)}. \end{align*} $$
We obtain from Lemma 2.2 that
$$ \begin{align*} \mathrm{I}_6'&\lesssim\Big\|\Big(\sum_{j\in\mathbb{Z}}\int^{2^j}_0\int^t_0|M(Q_s^2g)\big|^{q'} \big(\frac{s}{t}\big)^{\sigma}\log^{-q'\beta\theta+\gamma\frac{q'}{q}}(\mathrm{e}+2^j/s)\frac{ds}{s}\frac{dt}{t}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}\\ &\lesssim\Big\|\Big(\int_0^{\infty} \big|M\big(Q_s^2g\big)\big|^{q'}\frac{ds}{s}\Big)^{\frac{1}{q'}} \Big\|_{L^{p'}(\mathbb{R}^d)}. \end{align*} $$
The estimates for
$\mathrm {I}_5$
,
$\mathrm {I}_6$
,
$\mathrm {I}_5'$
and
$\mathrm {I}_6'$
, via inequality (2.3), state that
$$ \begin{align*} \Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,3}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\lesssim\|f\|_{\dot{F}^{0,q}_{p}(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}. \end{align*} $$
Similarly, we can show that
$$ \begin{align*} \Big|\sum_{j}\sum_l\int_0^{2^j}\int_{0}^{t}\mathrm{D}_{j,l,5}(s,t)\frac{ds}{s}\frac{dt}{t} \Big|\lesssim\|f\|_{\dot{F}^{0,q}_{p}(\mathbb{R}^d)}\|g\|_{\dot{F}^{0,q'}_{p'}(\mathbb{R}^d)}. \end{align*} $$
This verifies the inequality (2.11) and then completes the proof of Theorem 1.3.
Acknowledgements
The authors want to express their sincere thanks to the referee for his/her valuable remarks and suggestions, which made this article more readable.
