1 Introduction and main results
Let \(n, m \in \mathbb{N}\), \(0 \leq \alpha < n\). For any locally integrable bounded function f, define
where
\(\alpha =\alpha _{1}+\cdots +\alpha _{m}\), and for each \(1 \leq i \leq m\), \(k_{i}\) satisfies \((n-\alpha _{i})\)-order fractional size condition, \(A_{i}\) is a matrix such that Clearly, \(T_{\alpha ,1}=I_{\alpha }\), the Riesz potential, for \(m=1\), \(A_{1}\) is the n-order identity matrix, and \(k_{1}(x-A_{1}y)=1/|x-y|^{ \alpha }\). For general m and certain \(k_{i}\), \(T_{0,m}\) behaves like a singular integral operator and \(T_{\alpha ,m}\) has been studied in [1‐10]. In particular, Riveros and Urciuolo [5, 6, 11] considered each \(k_{i}\) as a rough fractional kernel, and each \(k_{i}\) satisfies an \(L^{\alpha _{i},\gamma _{i}}\)-Hörmander regular condition, or more general \(k_{i} \in H_{\alpha ,\gamma _{i}}\), that is, for all \(x \in \mathbb{R}^{n}\) and \(|x|< R\),
They showed that these operators are bounded from \(L^{p}\) into \(L^{q}\), for \(1< p\le q<\infty \), \(1/q =1/p-\alpha /n\). In [12], Ibañez-Firnkorn and Riveros analyzed operators of the form (1) with conditions of regularity more general than the \(L^{\alpha ,\gamma }\)-Hörmander condition and a fractional size condition. Before giving the definitions of the fractional size condition \({S_{n-{\alpha _{i},\varPsi _{i}}}}\) and the generalized fractional Hörmander condition \({H_{n-{\alpha _{i}, \varPsi _{i}, k}}}\), we first recall the definitions and properties for Young function.
$$ T_{\alpha ,m}f(x) := \int _{\mathbb{R}^{n}}K_{\alpha }(x,y)f(y)\,dy, $$
(1)
$$K_{\alpha }(x,y) = k_{1}(x - A_{1}y)k_{2}(x - A_{2}y) \cdots k_{m}(x - A_{m}y), $$
(H)
\(A_{i}\) is invertible and \(A_{i}-A_{j}\) is invertible for \(i \neq j\), \(1 \leq i,j \leq m\).
$$ \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{n-\alpha } \bigl\Vert \bigl(K_{\alpha }(\cdot -x)-K _{\alpha }(\cdot )\bigr)\chi _{B(x,2^{j+1}R)\setminus B(x,2^{j}R)} \bigr\Vert _{ \gamma _{i},B(x,2^{m}R)} < \infty . $$
A function \(\varPsi : [0,\infty ) \rightarrow [0,\infty )\) is said to be a Young function if Ψ is continuous, convex, nondecreasing and satisfies \(\varPsi (0) = 0 \) and \(\lim_{t\rightarrow \infty } \varPsi (t) = \infty \). For \(f \in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\) and each Young function Ψ, we can induce an average of the Luxemburg norm of a function f in the ball B defined by
and a fractional maximal operator \(M_{\alpha ,\varPsi }\) (\(0\leq \alpha < n\)) defined by
and we denote \(M_{0,\varPsi }\) by \(M_{\varPsi }\), the Orlicz maximal operator.
$$ \Vert f \Vert _{\varPsi ,B} := \inf \biggl\{ \lambda > 0 : \frac{1}{ \vert B \vert } \int _{B} \varPsi \biggl(\frac{ \vert f(x) \vert }{\lambda } \biggr)\,dx\leq 1 \biggr\} , $$
$$ M_{\alpha ,\varPsi }f(x):= \sup_{B \ni x} \vert B \vert ^{\alpha /n} \Vert f \Vert _{ \varPsi ,B}, $$
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In particular, for \(\varPsi (t) = t\), \(\|f\|_{\varPsi ,B} :=|B|^{-1}\int _{B}|f(x)|\,dx\) and \(M_{\alpha ,\varPsi } = M_{\alpha }\), the fractional maximal operator; for \(\varPsi (t) = t^{r}\) with \(1 < r < \infty \), \(\|f\|_{\varPsi ,B} =\|f\|_{r,B}:= (|B|^{-1}\int _{B}|f(x)|^{r}\,dx)^{1/r}\) and \(M_{\alpha ,\varPsi } = M_{\alpha ,r}\), and \(M_{0,r}f := \sup_{B \ni x}\|f\|_{r,B} := M(f^{r})^{1/r}\).
Next, we recall the definitions of the fractional size condition and the generalized fractional Hörmander condition. Normally, we use \(|x|\sim s\) to represent \(s < |x|\leq 2s\). For the Young function Ψ, we write
For \(0\leq \alpha < n\), the function \(K_{\alpha }\) is said to satisfy the fractional size condition if there exists a constant \(C > 0\) such that
And we denote \(K_{\alpha } \in S_{\alpha ,\varPsi }\) in this case. When \(\varPsi (t) = t\), we write \(S_{\alpha ,\varPsi } = S_{\alpha }\). Observe that if \({K_{\alpha }} \in S_{\alpha }\), then there exists a constant \(C > 0\) such that
$$ \Vert f \Vert _{\varPsi , \vert x \vert \sim s}= \Vert f\chi _{ \vert x \vert \sim s} \Vert _{\varPsi ,B(0,2s)}. $$
$$ \Vert K_{\alpha } \Vert _{\varPsi , \vert x \vert \sim s}\leq C s^{\alpha -n}. $$
$$ \int _{ \vert x \vert \sim s} \bigl\vert K_{\alpha }(x) \bigr\vert \,dx \leq C s^{\alpha }. $$
We say that the function \(K_{\alpha }\) satisfies the \(L^{\alpha , \varPsi ,k}\)-Hörmander condition denoted by \(K_{\alpha } \in H_{ \alpha ,\varPsi ,k}\) if there exist constants \(c_{\varPsi } > 1\) and \(C_{\varPsi } > 0\) such that, for all x and \(R > c_{\varPsi }|x|\),
When \(\varPsi (t) = t^{r}\), \(1 \leq r < \infty \), we simply write \(H_{\alpha ,r,k}\) instead of \(H_{\alpha ,\varPsi ,k}\). See [13] or [14] for more details.
$$ \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{n-\alpha }j^{k} \bigl\Vert K_{\alpha }(\cdot -x)-K_{\alpha }(\cdot ) \bigr\Vert _{\varPsi , \vert y \vert \sim 2^{j}R}\leq C_{\varPsi }. $$
In this paper, we consider the k-order commutators \(T_{\alpha ,m,b} ^{k}\) generated by Lipschitz functions and the operator \(T_{\alpha ,m}\), where \(k_{i} \in {S_{n-{\alpha _{i},\varPsi _{i}}}}\cap {H_{n-{\alpha _{i}, \varPsi _{i}, k}}}\), and for \(k \in \mathbb{N}\cup \{0\}\),
Clearly, \(T_{\alpha ,m,b}^{0}=T_{\alpha ,m}\).
$$ T_{\alpha ,m,b}^{k}(f) (x)= \int _{\mathbb{R}^{n}}\bigl(b(x)-b(y)\bigr)^{k}K_{\alpha}(x,y)f(y) \,dy. $$
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Also, we consider the following condition for the weights: there exists \(C > 0\) such that
for all \(1 \leq i \leq m\). Let \(\omega =\bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} \log \frac{1}{ \vert x \vert },& \vert x \vert \le e^{-1}, \\ 1,& \vert x \vert >e^{-1} \end{array}}\). It is easy to check that \(\omega \in A_{1}\) and satisfies (2) (see [15]).
$$ \omega (A_{i}x) \leq C\omega (x), \quad\mathrm{a.e.}\ x \in \mathbb{R}^{n} $$
(2)
In [14], Gallo, Ibañez-Firnkorn, and Riveros obtained the weighted estimates for this kind of operator and certain weights satisfying (2). Precisely as for the classical fractional integral operator \(I_{\alpha }\) with \(0 < \alpha < n\), or the singular integral operator with \(\alpha = 0\), they proved the \(L^{p}(\mathbb{R}^{n},\omega ^{p})\rightarrow L^{q}(\mathbb{R}^{n}, \omega ^{q})\) boundedness of \(T_{\alpha ,m}\) for weights \(\omega \in A(p, q)\), \(1 < p < n/\alpha \), \(1/q = 1/p - \alpha /n\), and \(0 \leq \alpha < n\). In [15], for \(b\in \mathrm{BMO}\), Ibañez-Firnkorn and Riveros obtained the weighted Coifman type estimates, weighted \(L^{p}(\omega ^{p})\to L^{q}(\omega ^{q})\) estimates, and weighted BMO estimates as well as two-weighted inequalities. Inspired by these results, we consider the weighted boundedness of \(T_{\alpha ,m,b}^{k}\) for \(b\in \dot{\varLambda }_{\beta }\) and a weighted \(\dot{\varLambda }_{\beta }\) estimate for weights in the class \(A(n/(\alpha +k\beta ) r,\infty )\). Our results can be formulated as follows.
Theorem 1.1
For
\(0<\beta <1\), \(0\leq \alpha < n\), \(k\in \mathbb{N}{\cup \{0\}}\), \(m\in \mathbb{N}\), and
\(1\leq i \leq m\), let
\(b\in \dot{\varLambda }_{\beta }\), \(\varPsi _{i}\)
be Young functions and
\(0\leq \alpha _{i}< n\)
such that
\(\alpha _{1}+\cdots +\alpha _{n}=n- \alpha \). Let
\(T_{\alpha ,m}\)
be the integral operator defined by (1) and
\(T^{k}_{\alpha ,m,b}\)
be the
k-order commutator of
\(T_{\alpha ,m}\). Suppose that the matrices
\(A_{i}\)
satisfy hypothesis (H) and
\(k_{i} \in {S_{n-{\alpha _{i},\varPsi _{i}}}}\cap {H_{n-{\alpha _{i},\varPsi _{i},k}}}\). Moreover, for
\(\alpha = 0\), suppose that
\(T_{0,m}\)
is strong type
\((q_{*},q_{*})\)
for some
\(1 < q_{*} < \infty \). Let
\(\varphi _{k}(t) = t\log (e+t)^{k}\), ϕ
be a Young function satisfying
\(\varPsi _{1}^{-1}(t)\cdots {\varPsi _{m}^{-1}}(t){{\phi } ^{-1}}(t){{{\overline{{{\varphi _{k}}}}}^{-1}}(t)}\lesssim t\)
for
\(t\geq t_{0}\), some
\(t_{0} > 0\). Then there exists
\(0< C=C(n,\alpha ,A _{1},\ldots ,A_{m})\)
such that, for
\(0< \delta \leq 1\)
and
\(f \in {{L_{c}^{\infty }} (\mathbb{R}^{n})}\),
$$ M_{\delta }^{\sharp } \bigl( T^{k}_{\alpha ,m,b}f \bigr) (x)\leq C\sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l}M_{(k-l)\beta } \bigl(T^{l}_{\alpha ,m,b}f\bigr) (x)+C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum_{i= 0}^{m} M _{\alpha +k\beta ,\phi }f\bigl(A_{i}^{-1}x\bigr). $$
Theorem 1.2
Under the assumptions of Theorem 1.1, for
\(1 \leq r< p<p_{l}\leq q < \infty \), \(k,l\in \mathbb{N}\), \(1/q =1/p_{l}-(k-l)\beta /n\), \(1/q =1/p-(\alpha +k\beta )/n\), there exists
\(0< C=C(n,\alpha ,A_{1},\ldots ,A_{m})\)
such that, for
\(f \in {{L_{c}^{\infty }}{({\mathbb{R}}^{n}}})\)
and
\(\omega ^{r} \in A(p/r,q/r )\),
Furthermore, if
\(\omega ^{r} \in A(p/r,q/r )\)
and satisfying (2), then
$$ \bigl\Vert T^{k}_{\alpha ,m,b}f \bigr\Vert _{L^{q}(\omega ^{q})}\leq C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m}\sum _{l= 0}^{k} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha +l\beta ,\phi }f(x) \bigr\vert ^{p_{l}}\omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1 / p_{l}}. $$
(3)
$$ \bigl\Vert T^{k}_{\alpha ,m,b}f \bigr\Vert _{L^{q}(\omega ^{q})}\leq C { \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{k} \sum_{l= 0}^{k} \Vert M_{\alpha +l\beta , \phi }f \Vert _{L^{p_{l}}(\omega ^{p_{l}})}. $$
Theorem 1.3
Let
\(0 \leq \alpha < n\), \(1 < p < n/( \alpha +k\beta )\), \(1/q = 1/p-(\alpha +k\beta )/n\)
and
ϕ
be a Young function such that
\({\eta }^{-1}(t)t^{ \frac{(\alpha +k\beta )}{n}}\lesssim \phi ^{-1}(t)\)
for every
\(t > 0\), where
\(\phi ^{1+\frac{sn}{n-(\alpha +k\beta )}}\in B_{\frac{sn}{n-( \alpha +k\beta )}}\)
for every
\(s > r(n-(\alpha +k\beta ))/(n-(\alpha +k\beta ) r) \). Then, under the hypotheses of Theorem 1.2, for
\(\omega ^{r} \in A(p/r,q/r )\),
$$ \bigl\Vert T^{k}_{\alpha ,m,b}f \bigr\Vert _{L^{q}(\omega ^{q})}\leq C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{k} \Vert f \Vert _{L^{p}(\omega ^{p})}. $$
Theorem 1.4
Under the hypotheses of Theorem 1.3, if
\(\omega ^{r} \in A(n/(\alpha +k\beta )r, \infty )\)
and satisfies (2), then there exists
\(C > 0\)
such that, for
\(f \in {{L_{c}^{\infty }}({R^{n}}})\),
where
$$ \bigl|\!\bigl|\!\bigl|T^{k}_{\alpha ,m,b}f\bigr|\!\bigr|\!\bigr|_{\omega } \leq C{ \Vert b \Vert }_{\dot{\varLambda } _{\beta }}^{k} \Vert f \omega \Vert _{L^{n/{\alpha +k\beta }}}, $$
$$ \bigl|\!\bigl|\!\bigl|T^{k}_{\alpha ,m,b}f\bigr|\!\bigr|\!\bigr|_{\omega }=\sup _{B} \Vert \omega \chi _{B} \Vert _{\infty } \biggl(\frac{1}{ \vert B \vert } \int _{B} \biggl\vert T^{k}_{\alpha ,m,b}f(y)- \frac{1}{ \vert B \vert } \int _{B}T^{k}_{\alpha ,m,b}f(z)\,dz \biggr\vert \,dy \biggr). $$
The rest of this paper is organized as follows. In Sect. 2 we recall some relevant definitions and previous results that are needed to state the other results, which appear in Sect. 1. The proofs of sharp maximal functions estimates and Coifman type inequalities are given in Sect. 3. Finally, the weighted \(L^{p}(w^{p}) \rightarrow L^{q}(w ^{q})\) estimates and the weighted endpoint estimates are presented in Sect. 4.
2 Preliminaries
In this section we present some relevant concepts and previous results, which will be used in our proofs.
2.1 The generalized Hölder inequality and the fractional \(B_{p}\) condition
Now, we present some extra properties for Young functions. For more details of these topics, see [16] or [17].
The function Ψ̄ is called the complementary of the function Ψ if the generalized Hölder inequality holds:
If \(\varPsi _{1},\ldots,\varPsi _{m},\phi \) are Young functions satisfying \(\varPsi _{1}^{-1}(t)\cdots {\varPsi _{m}^{-1}}(t){{\phi }^{-1}}(t)\lesssim t\) for \(t\geq t_{0}\), some \(t_{0} > 0\), then
where the function ϕ is called the complementary of the functions \(\varPsi _{1},\ldots,\varPsi _{m}\).
$$ \Vert fg \Vert _{L_{1},B}\leq 2 \Vert f \Vert _{\varPsi ,B} \Vert g \Vert _{\bar{\varPsi },B}. $$
$$ \Vert f_{1}\cdots f_{m}g \Vert _{L_{1},B} \leq c \Vert f_{1} \Vert _{\varPsi _{1},B} \cdots \Vert f_{m} \Vert _{\varPsi _{m},B} \Vert g \Vert _{\phi ,B} , $$
In 2013, Cruz-Uribe and Moen [18] introduced the fractional \(B_{p}\) condition: for \(1< p< n/\alpha \) and \(1/q=1/p-\alpha /n\), a Young function \(\phi \in B_{p}^{\alpha }\) if
And they proved that if \(\phi \in B_{p}^{\alpha }\), then \(M_{\alpha ,\phi } : L^{p}(dx)\rightarrow L^{q}(dx)\) and
$$ \int _{1}^{\infty }\frac{\phi (t)^{q/p}}{t^{q}} \frac{dt}{t} < \infty . $$
$$ \Vert M_{\alpha ,\phi } \Vert _{ L^{p}\rightarrow L^{q}}\leq \biggl( \int _{1} ^{\infty }\frac{\phi (t)^{q/p}}{t^{q}} \frac{dt}{t} \biggr)^{1/q}. $$
2.2 The Lipschitz function spaces
For a locally integrable function f defined in \(\mathbb{R}^{n}\), we say f belongs to the space \(\dot{\varLambda }_{\beta }(\mathbb{R}^{n})\), \(0<\beta <1\), if there exists a constant \(C>0\) such that
The smallest bound C satisfying upper inequality is taken to be the norm of f denoted by \(\|f\|_{\dot{\varLambda }_{\beta }}\). Here B is a ball in \(\mathbb{R}^{n}\), and
$$ \sup_{B\ni x}\frac{1}{ \vert B \vert ^{1+\beta /n}} \int _{B} \bigl\vert f(x)-f_{B} \bigr\vert \,dx < \infty . $$
$$ f_{B} =\frac{1}{ \vert B \vert } \int _{B}f(x)\,dx. $$
Lemma 2.1
If
\(f \in \dot{\varLambda }_{\beta }\), then
(1)
for every
\(x,y\in \mathbb{R}^{n}\),
$$ \bigl\vert f(x)-f(y) \bigr\vert \leq \Vert f \Vert _{\dot{\varLambda }_{\beta }} \vert x-y \vert ^{\beta }; $$
(2)
for any ball
B,
$$ \sup_{x\in B} \bigl\vert f(x)-f_{B} \bigr\vert \leq C \Vert f \Vert _{\dot{\varLambda }_{\beta }} \vert B \vert ^{ \beta /n}; $$
(3)
for
\(B\subset B^{\ast }\),
$$ \vert f_{B}-f_{B^{\ast }} \vert \leq \Vert f \Vert _{\dot{\varLambda }_{\beta }} \bigl\vert B^{ \ast } \bigr\vert ^{\beta /n}. $$
In particular, if
\(A_{i}\)
are matrices satisfying (H), B̃
is a measurable set, and
\(\tilde{B_{i}} = A_{i}^{-1}\tilde{B}\), \(1\leq i \leq m\), then
and for
\(B = B(c_{B},R)\), the ball centered at
\(c_{B}\)
with radius
R, and
\(B^{j}: = B(c_{B},2^{j}R)\), \(j\in \mathbb{N}\),
$$ \vert f_{\tilde{B}}-f_{(\bigcup _{i=1}^{m}\tilde{B}_{i})\cup \tilde{B}} \vert \leq C \Vert f \Vert _{\dot{\varLambda }_{\beta }} \vert \tilde{B} \vert ^{\beta /n}, $$
$$ \vert f_{B}-f_{B^{j}} \vert \leq C j \Vert f \Vert _{\dot{\varLambda }_{\beta }} \bigl\vert B^{j} \bigr\vert ^{ \beta / n}. $$
2.3 Weights and maximal operators
A weight function ω is in the Muckenhoupt class \(A_{p}\) for \(1 < p<\infty \) if there exists \(C>1\) such that, for any ball B,
where \(1/p +1/p{'} =1\) and the infimum of C satisfying the above inequality is denoted by \([\omega ]_{A_{p}}\). We define \(A_{\infty } = \bigcup_{1\leq p<\infty }A_{p}\). When \(p=1\), \(\omega \in A_{1}\) if there exists \(C>1\) such that, for almost every x,
and the infimum of C satisfying the above inequality is denoted by \([\omega ]_{A_{1}}\).
$$ \biggl(\frac{1}{ \vert B \vert } \int _{B}\omega (x)\,dx \biggr) \biggl( \frac{1}{ \vert B \vert } \int _{B}\omega (x)^{1-{p{'}}}\,dx \biggr)^{p-1}\leq C, $$
$$ M\omega (x)\leq C\omega (x), $$
A weight function ω belongs to \(A(p,q)\) for \(1< p< q<\infty \) if there exists \(C>1\) such that
where \(1/p +1/p' =1\) and the infimum of C satisfying the above inequality is denoted by \([\omega ]_{A_{p,q}}\). When \(p =1\), ω is in \(A(1,q)\) with \(1 < q<\infty \) if there exists \(C>1\) such that
and the infimum of C satisfying the above inequality is denoted by \([\omega ]_{A_{1,q}}\).
$$ \biggl(\frac{1}{ \vert B \vert } \int _{B}\omega (x)^{q}\,dx \biggr)^{1/q} \biggl( \frac{1}{ \vert B \vert } \int _{B}\omega (x)^{-p'}\,dx \biggr)^{1/p'}\leq C, $$
$$ \biggl(\frac{1}{ \vert B \vert } \int _{B}\omega (x)^{q}\,dx \biggr) \biggl( \mathrm{ess} \sup_{x\in B} \frac{1}{\omega (x)} \biggr)\leq C, $$
Remark 2.1
For \(1\leq r< p< p_{0}\leq p\), \(\omega \in A(p,q)\), by Hölder’s inequality, we can know \(\omega \in A(p_{0},q)\) and \(\omega \in A(p,p_{0})\). For \(\omega ^{r} \in A(p/r,q/r)\), we also can know \(\omega \in A(p,q)\).
The sharp maximal function is defined by
A locally integrable function f has bounded mean oscillation \((f \in \mathrm{BMO})\) if \(M^{\sharp }f(x) \in L^{\infty }\) and the norm \(\|f\|_{\mathrm{BMO}} = \|M^{\sharp }f\|_{\infty }\). Observe that the BMO norm is equivalent to
There is also a weighted version of BMO, which is denoted by \(\mathrm{BMO}(\omega )\), which is described by the semi-norm
It is easy to check that
$$ M^{\sharp }f(x)=\sup_{B\ni x}\frac{1}{ \vert B \vert } \int _{B} \biggl\vert f(y)- \frac{1}{ \vert B \vert } \int _{B}f(z)\,dz \biggr\vert \,dy . $$
$$ \Vert f \Vert _{\mathrm{BMO}} = \bigl\Vert M^{\sharp }f \bigr\Vert _{\infty }\sim \sup_{B} \inf _{a\in {\mathbb{C}}}\frac{1}{ \vert B \vert } \int _{B} \bigl\vert f(y)-a \bigr\vert \,dy . $$
$$ \bigl|\!\bigl|\!\bigl|f\bigr|\!\bigr|\!\bigr|_{\omega }=\sup_{B} \Vert \omega \chi _{B} \Vert _{\infty } \biggl( \frac{1}{ \vert B \vert } \int _{B} \biggl\vert f(y)-\frac{1}{ \vert B \vert } \int _{B}f(z)\,dz \biggr\vert \,dy \biggr). $$
$$ \bigl|\!\bigl|\!\bigl|f\bigr|\!\bigr|\!\bigr|_{\omega }\simeq \bigl\Vert \omega M^{\sharp }f \bigr\Vert _{\infty }. $$
Proposition 2.1
([19])
Let
Ψ
be a Young function. Then, for all
\(x\in \mathbb{R}^{n}\)
and
\(r>1\), there exists a constant
\(C_{r}\)
such that
$$ M_{\varPsi }f(x)=M_{0,\varPsi }f(x)\leq C_{r} M_{0,r}f(x). $$
Proposition 2.2
([15])
Let
Ψ
be a Young function and
A
be an invertible matrix. Set
\(\omega _{A}(x) = \omega (Ax)\). Then
for almost every
\(x \in \mathbb{R}^{n}\).
$$ M_{\alpha ,\varPsi }(\omega _{A}) \bigl(A^{-1}x\bigr) \leq c_{A,n}M_{\alpha ,\varPsi }( \omega ) (x) $$
2.4 Previous results
In this subsection, we illustrate some known results for the operator \(T_{\alpha ,m}\), which will be used below, see [12] for more details.
Theorem 2.1
([12])
Let
\(0 \leq \alpha < n\), \(m \in \mathbb{N}\), and
\(T_{\alpha ,m}\)
be the integral operator defined by (1). For
\(1\leq i \leq m\), let
\(\varPsi _{i}\)
be Young functions, \(0 \leq \alpha _{i} < n\)
such that
\(\alpha _{1}+\cdots + \alpha _{m} = n-\alpha \). Also suppose
\(k_{i} \in {S_{n-{\alpha _{i}, \varPsi _{i}}}}\cap {H_{n-{\alpha _{i},\varPsi _{i}}}}\)
and that matrices
\(A_{i}\)
satisfy hypothesis (H).
If
\(\alpha = 0\), suppose
\(T_{0,m}\)
is of strong type
\((q_{*},q_{*})\)
for some
\(1 < q_{*} < \infty \).
If
ϕ
is the complementary of the functions
\(\varPsi _{1},\ldots,\varPsi _{m}\), then there exists
\(C > 0\)
such that, for
\(0 < \delta \leq 1\)
and
\(f \in L_{c}^{\infty }(\mathbb{R}^{n})\),
$$ M_{\delta }^{\sharp } ( T_{\alpha ,m}f ) (x):=M^{\sharp }\bigl( \vert T_{\alpha ,m}f \vert ^{ \delta }\bigr) (x)^{1/\delta }\leq C\sum _{i=1}^{m} M_{\alpha ,\phi }f \bigl(A_{i} ^{-1}x\bigr) . $$
Theorem 2.2
([12])
Let
\(0 \leq \alpha < n\), \(m \in \mathbb{N}\), and
\(T_{\alpha ,m}\)
be the integral operator defined by (1). For
\(1\leq i \leq m\), let
\(\varPsi _{i}\)
be Young functions, \(0 \leq \alpha _{i} < n\)
such that
\(\alpha _{1}+\cdots + \alpha _{m} = n-\alpha \). Also suppose
\(k_{i} \in {S_{n-{\alpha _{i}, \varPsi _{i}}}}\cap {H_{n-{\alpha _{i},\varPsi _{i}}}}\)
and that matrices
\(A_{i}\)
satisfy hypothesis (H).
When
\(\alpha = 0\), suppose that
\(T_{0,m}\)
is of strong type
\((q_{*},q_{*})\)
for some
\(1 < q_{*} < \infty \).
Let
\(0< p<\infty \). If
ϕ
is the complementary of the functions
\(\varPsi _{1},\ldots,\varPsi _{m}\), then there exists
\(C > 0\)
such that, for
\(\omega \in A_{\infty }\)
and
\(f \in L_{c}^{\infty }(\mathbb{R}^{n})\),
whenever the left-hand side is finite.
$$ \int _{\mathbb{R}^{n}} \bigl\vert T_{\alpha ,m}f(x) \bigr\vert ^{p}\omega (x)\,dx\leq C\sum_{i= 1}^{m} \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha ,\phi }f(x) \bigr\vert ^{p}\omega (A _{i}x)\,dx, $$
3 Sharp maximal function estimates and Coifman type inequality
Proof of Theorem 1.1
We just consider the case \(m = 2 \) and \(k = 1\), i.e., \(T_{\alpha ,2,b}^{1} = [b,T_{\alpha ,2}]\), and we will just write \([b,T_{\alpha }]\) for simplicity. The general case is proved in an analogous way.
Let f be a bounded function with compact support, \(0< \delta \leq 1\). For \(x\in \mathbb{R}^{n}\), let \(B = B(c_{B},R)\) be a ball that contains x, centered at \(c_{B}\) with radius R. We write \(\tilde{B} = B(c _{B},2R)\), and for \(1 \leq i \leq 2\), set \(\tilde{B}_{i} = A_{i}^{-1} \tilde{B}\). Let \(f_{1} = f\chi _{\bigcup _{i=1}^{2}\tilde{B}_{i}}\) and \(f_{2} = f-f_{1}\). Suppose that \(a:=T_{\alpha }((b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}}-b)f)(c_{B})<\infty \). For \(0<\delta \leq 1\), we write
And from the inequality \(|t^{\delta }-s^{\delta }|^{1/\delta }\leq |t-s|\) and Jensen’s inequality, we get
For I, by Lemma 2.1, we have
For II, we know
We estimate only the first summand, that is, \(z\in \tilde{B}_{1}\), since the case \(z\in \tilde{B}_{2}\) is analogous. Observe that
For \(j\in \mathbb{N}\), let us consider the set
Notice that if \(y\in B\) and \(z\in \tilde{B}_{1}\), then \(|y-A_{1}z| \leq 3R<4R\). Thus,
And for \(y\in C_{j}^{1}\), we have \(|y-A_{2}z|\geq |y-A_{1}z| \geq 2^{-j-1}R\). By \(k_{2}\in S_{n-\alpha _{2},\varPsi _{2}} \) and \(k_{1}\in S_{n-\alpha _{1},\varPsi _{1}} \), we get
and
Consequently,
Similarly,
Then
$$ [b,T_{\alpha }](f) (x)=\bigl(b(x)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr)T_{\alpha }f(x)+T_{\alpha }\bigl((b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}}-b)f \bigr) (x). $$
$$\begin{aligned} & \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert [b,T_{\alpha }](f)^{\delta }(y)-a^{\delta } \bigr\vert \,dy \biggr)^{1/\delta } \\ &\quad \leq \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert [b,T_{\alpha }](f) (y)-a \bigr\vert ^{\delta }\,dy \biggr)^{1/\delta } \\ &\quad \leq \frac{1}{ \vert B \vert } \int _{B} \bigl\vert \bigl(b(y)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr)T_{\alpha }f(y) \bigr\vert \,dy \\ &\qquad{} + \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha } \bigl((b_{\tilde{B}\cup \tilde{B}_{1} \cup \tilde{B}_{2}}-b)f_{1}\bigr) (y) \bigr\vert \,dy \\ &\qquad{} + \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha } \bigl((b_{\tilde{B}\cup \tilde{B}_{1} \cup \tilde{B}_{2}}-b) f_{2}\bigr) (y)-T_{\alpha } \bigl((b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}}-b)f_{2}\bigr) (c_{B}) \bigr\vert \,dy \\ &\quad =:I+\mathit{I I}+\mathit{I I I}. \end{aligned}$$
(4)
$$\begin{aligned} I&\leq C \Vert b \Vert _{\dot{\varLambda }_{\beta }} \vert B \vert ^{\beta /n} \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha }f(y) \bigr\vert \,dy \biggr) \\ &\leq C \Vert b \Vert _{\dot{\varLambda }_{\beta }}M_{\beta }(T_{\alpha }f) (x). \end{aligned}$$
(5)
$$\begin{aligned} \mathit{I I}&= \frac{1}{ \vert B \vert } \int _{B} \int _{\tilde{B}_{1}\cup \tilde{B}_{2}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \bigl\vert b(z)-b _{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f_{1}(z) \bigr\vert \,dz\,dy \\ &\leq \sum_{i=1}^{2} \frac{1}{ \vert B \vert } \int _{\tilde{B}_{i}} \bigl\vert b(z)-b_{\tilde{B} \cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f_{1}(z) \bigr\vert \int _{B} \bigl\vert K(y,z) \bigr\vert \,dy\,dz. \end{aligned}$$
$$ \int _{B} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy\leq \int _{\{{y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy + \int _{\{y\in B: \vert y-A_{2}z \vert \leq \vert y-A_{1}z \vert \}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy. $$
$$ C_{j}^{1} := \bigl\{ y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert , \vert y-A_{1}z \vert \sim 2^{-j-1}R \bigr\} . $$
$$\begin{aligned} & \int _{\{{y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \\ &\quad \leq \sum_{j=-2}^{\infty } \int _{C_{j}^{1}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \\ & \quad\leq \sum_{j=-2}^{\infty } \frac{ \vert B(A_{1}z,2^{-j}R) \vert }{ \vert B(A_{1}z,2^{-j}R) \vert } \int _{B(A_{1}z,2^{-j}R)} \bigl\vert K_{\alpha}(y,z) \bigr\vert \chi _{\{y: \vert y-A_{1}z \vert \sim 2^{-j-1}R\}}\,dy \\ & \quad\leq C \sum_{j=-2}^{\infty }{ \bigl\vert B\bigl(A_{1}z,2^{-j}R\bigr) \bigr\vert } \bigl\Vert k_{1}(\cdot -A_{1}z) \chi _{\{y: \vert y-A_{1}z \vert \sim 2^{-j-1}R\}} \bigr\Vert _{\varPsi _{1},B(A_{1}z,2^{-j}R)} \\ &\qquad{} \times \bigl\Vert k_{2}(\cdot -A_{2}z)\chi _{\{y: \vert y-A_{1}z \vert \sim 2^{-j-1}R \}} \bigr\Vert _{\varPsi _{2},B(A_{1}z,2^{-j}R)} \\ &\quad \leq C \sum_{j=-2}^{\infty }{ \bigl\vert B\bigl(A_{1}z,2^{-j}R\bigr) \bigr\vert } \bigl\Vert k_{1}(\cdot -A_{1}z) \bigr\Vert _{\varPsi _{1}, \vert y-A_{1}z \vert \sim 2^{-j-1}R} \\ & \qquad{}\times \bigl\Vert k_{2}(\cdot -A_{2}z) \bigr\Vert _{\varPsi _{2}, \vert y-A_{1}z \vert \sim 2^{-j-1}R}. \end{aligned}$$
$$\begin{aligned} \bigl\Vert k_{2}(\cdot -A_{2}z) \bigr\Vert _{\varPsi _{2}, \vert y-A_{1}z \vert \sim 2^{-j-1}R} &\leq \sum_{k\geq 0} \bigl\Vert k_{2}(\cdot -A_{2}z) \bigr\Vert _{\varPsi _{2}, \vert y-A_{2}z \vert \sim 2^{-j+k-1}R} \\ &\leq \sum_{k\geq 0} \bigl\Vert k_{2}( \cdot ) \bigr\Vert _{\varPsi _{2}, \vert y \vert \sim 2^{-j+k-1}R} \\ &\leq C \sum_{k\geq 0}\bigl(2^{-j+k-1}R \bigr)^{-\alpha _{2}}\leq C \bigl(2^{-j}R\bigr)^{-\alpha _{2}}, \end{aligned}$$
$$ \bigl\Vert k_{1}(\cdot -A_{1}z) \bigr\Vert _{\varPsi _{1}, \vert y-A_{1}z \vert \sim 2^{-j-1}R} \leq C\bigl(2^{-j-1}R\bigr)^{- \alpha _{1}} \leq C \bigl(2^{-j}R\bigr)^{-\alpha _{1}}. $$
$$ \int _{\{{y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \leq C \sum _{j=-2}^{\infty }\bigl(2^{-j}R \bigr)^{n-\alpha _{1}-\alpha _{2}}\leq CR^{\alpha }. $$
$$ \int _{\{{y\in B: \vert y-A_{2}z \vert \leq \vert y-A_{1}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \leq CR^{ \alpha }. $$
$$\begin{aligned} \mathit{I I}&\leq CR^{\alpha } \sum _{i=1}^{2}\frac{1}{ \vert B \vert } \int _{\tilde{B}_{i}} \bigl\vert b(z)-b_{\tilde{B} \cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f_{1}(z) \bigr\vert \,dz \\ &\leq C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i=1}^{2} R^{\alpha +\beta } \frac{1}{ \vert \tilde{B}_{i} \vert } \int _{\tilde{B}_{i}} \bigl\vert f(z) \bigr\vert \,dz \\ &\leq C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i=1}^{2} R^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B}_{i}} \\ &\leq C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i=1}^{2} M_{\alpha +\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr). \end{aligned}$$
(6)
For III, we have
where
Let us estimate \(|K_{\alpha}(y,z)-K_{\alpha}(c_{B},z)|\) for \(y\in B\) and \(z\in Z^{l}\):
For simplicity we estimate the first summand of \(|K_{\alpha}(y,z)-K_{\alpha}(c_{B},z)|\), the other one follows in an analogous way. For \(j \in \mathbb{N}\), let
Observe that \(D_{j}^{l}\subset {\{z:|c_{B}-A_{l}z|\sim 2^{j+1}R\}} \subset A_{l}^{-1}B(c_{B},2^{j+2}R)=:\tilde{B}_{l,j}\). Using the generalized Hölder inequality, we have
Note that \(|c_{B}-A_{l}z|/2\leq |y-A_{l}z|\leq 2|c_{B}-A_{l}z|\), and if \(|c_{B}-A_{l}z|\sim 2^{j+1}R\), then \(2^{j}R \leq |y-A_{l}z|\leq 2^{j+2}R\). Thus,
where the last inequality holds since \(k_{l} \in S_{n-\alpha _{l},\varPsi _{l}}\). Also, by the hypothesis,
For \(r \neq l\), observe that if \(z\in D_{j}^{l}\), then \(|c_{B}-A_{r}z| \geq |c_{B}-A_{l}z|\geq 2^{j+1}R\). We decompose \(D_{j}^{l}=\bigcup_{k \geq j}(D_{j}^{l})_{k,r}\), where
Since \((D_{j}^{l})_{k,r}\subset \{{z:|c_{B}-A_{r}z|\sim 2^{k+1}R\}}\) and \(k_{r}\in S_{n-\alpha _{r},\varPsi _{r}}\), we have
By the same arguments, we can get
As a result, no matter \(l=1\) or \(l=2\), we have
Hence,
So, when \(l=1\), from \(k_{1} \in H_{n-\alpha _{1},\varPsi _{1}, 1} \), we can get
For \(l= 2\), note that
we have
where the last inequality follows from that \(k_{1} \in H_{n-\alpha _{1},\varPsi _{1}, 1} \). Hence,
Then
Summing up (4)–(7), we know that
For the case \(\alpha = 0\), we repeat the same argument to inequality (4) and get the desired conclusion.
$$\begin{aligned} \mathit{I I I} &= \frac{1}{ \vert B \vert } \int _{B} \int _{(\tilde{B}_{1}\cup \tilde{B}_{2})^{c}} \bigl\vert K_{\alpha}(y,z)-K_{\alpha}(c _{B},z) \bigr\vert \bigl\vert b(z)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz\,dy \\ &\leq \sum_{l=1}^{2} \frac{1}{ \vert B \vert } \int _{B} \int _{Z^{l}} \bigl\vert K_{\alpha}(y,z)-K_{\alpha}(c_{B},z) \bigr\vert \bigl\vert b(z)-b _{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz\,dy, \end{aligned}$$
$$ Z^{l}=(\tilde{B}_{1}\cup \tilde{B}_{2})^{c} \cap \bigl\{ z: \vert c_{B}-A_{l}z \vert \leq \vert c_{B}-A_{r}z \vert , r\neq l,1\leq r \leq 2 \bigr\} . $$
$$\begin{aligned} \bigl\vert K_{\alpha}(y,z)-K_{\alpha}(c_{B},z) \bigr\vert \leq{}& \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A_{1}z) \bigr\vert \bigl\vert k_{2}(y-A _{2}z) \bigr\vert \\ & {}+ \bigl\vert k_{2}(y-A_{2}z)-k_{2}(c_{B}-A_{2}z) \bigr\vert \bigl\vert k_{1}(c_{B}-A_{1}z) \bigr\vert . \end{aligned}$$
$$ D_{j}^{l}:=\bigl\{ {z\in Z^{l}: \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R\bigr\} }. $$
$$\begin{aligned} & \int _{Z_{l}} \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A_{1}z) \bigr\vert \bigl\vert k_{2}(y-A_{2}z) \bigr\vert \bigl\vert b(z)-b _{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz \\ &\quad \leq \sum_{j=1}^{\infty } \int _{D_{j}^{l}} \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A _{1}z) \bigr\vert \bigl\vert k_{2}(y-A_{2}z) \bigr\vert \bigl\vert b(z)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz \\ & \quad\leq \sum_{j=1}^{\infty } \frac{ \vert \tilde{B}_{l,j} \vert }{ \vert \tilde{B}_{l,j} \vert } \int _{\tilde{B}_{l,j}} \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A_{1}z) \bigr\vert \bigl\vert k_{2}(y-A _{2}z) \bigr\vert \chi _{D_{j}^{l}}\chi _{\{z: \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R\}} \\ &\qquad{} \times \bigl( \bigl\vert b(z)-b_{\tilde{B}_{l,j}} \bigr\vert + \vert b_{\tilde{B}_{l,j}}-b_{\tilde{B} _{l}} \vert + \vert b_{\tilde{B}_{l}}- b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \vert \bigr) \bigl\vert f(z) \bigr\vert \,dz \\ &\quad \leq \sum_{j=1}^{\infty } \frac{ \vert \tilde{B}_{l,j} \vert }{ \vert \tilde{B}_{l,j} \vert } \int _{\tilde{B}_{l,j}} \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A_{1}z) \bigr\vert \bigl\vert k_{2}(y-A _{2}z) \bigr\vert \chi _{D_{j}^{l}}\chi _{\{z: \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R\}} \\ &\qquad{} \times \bigl(C \Vert b \Vert _{\dot{\varLambda }_{\beta }} \vert \tilde{B}_{l,j} \vert ^{\beta /n} +Cj \Vert b \Vert _{\dot{\varLambda }_{\beta }} \vert \tilde{B}_{l,j} \vert ^{\beta /n}+C \Vert b \Vert _{\dot{\varLambda }_{\beta }} \vert {B} \vert ^{\beta /n}\bigr) \bigl\vert f(z) \bigr\vert \,dz \\ & \quad\leq C \Vert b \Vert _{\dot{\varLambda }_{\beta }} \sum_{j=1}^{\infty } \vert \tilde{B}_{l,j} \vert ^{1+\beta /n}j \frac{1}{ \vert \tilde{B}_{l,j} \vert } \int _{\tilde{B}_{l,j}} \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A _{1}z) \bigr\vert \\ &\qquad{} \times \bigl\vert k_{2}(y-A_{2}z) \bigr\vert \chi _{D_{j}^{l}} \chi _{\{z: \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R\}} \bigl\vert f(z) \bigr\vert \,dz \\ &\quad \leq C \Vert b \Vert _{\dot{\varLambda }_{\beta }} \sum_{j=1}^{\infty } \vert \tilde{B}_{l,j} \vert ^{1+\beta /n}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \\ &\qquad{} \times \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \Vert f \Vert _{\phi ,\tilde{B}_{l,j}}. \end{aligned}$$
$$\begin{aligned} & \bigl\Vert k_{l}(y-A_{l}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{l}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad\leq \bigl\Vert k_{l}(y-A_{l}\cdot ) \bigr\Vert _{\varPsi _{l}, \vert y-A_{l}z \vert \sim 2^{j}R} \\ & \qquad{}+ \bigl\Vert k_{l}(y-A_{l}\cdot ) \bigr\Vert _{\varPsi _{l}, \vert y-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad\leq \bigl\Vert k_{l}(\cdot ) \bigr\Vert _{\varPsi _{l}, \vert x \vert \sim 2^{j}R}+ \bigl\Vert k_{l}(\cdot ) \bigr\Vert _{\varPsi _{l}, \vert x \vert \sim 2^{j+1}R} \\ &\quad\leq C\bigl(2^{j}R\bigr)^{-\alpha _{l}}, \end{aligned}$$
$$ \bigl\Vert k_{l}(c_{B}-A_{l}\cdot ) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{l}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}\leq C\bigl(2^{j+1}R \bigr)^{-\alpha _{l}}. $$
$$ \bigl(D_{j}^{l}\bigr)_{k,r}=\bigl\{ {z\in D_{j}^{l}: \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R\bigr\} }. $$
$$\begin{aligned} \bigl\Vert k_{r}(y-A_{r}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} &\leq \sum _{k\geq j} \bigl\Vert k_{r}(y-A_{r} \cdot )\chi _{(D_{j}^{l})_{k,r}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\leq \sum_{k\geq j} \bigl\Vert k_{r}(y-A_{r}\cdot )\chi _{(D_{j}^{l})_{k,r}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R} \\ &\leq \sum_{k\geq j} \bigl\Vert k_{r}(y-A_{r}\cdot ) \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R} \\ &\leq \sum_{k\geq j} \bigl\Vert k_{r}( \cdot ) \bigr\Vert _{\varPsi _{r}, \vert x \vert \sim 2^{k}R}+ \bigl\Vert k_{r}( \cdot ) \bigr\Vert _{\varPsi _{r}, \vert x \vert \sim 2^{k+1}R} \\ &\leq C \sum_{k\geq j}\bigl(2^{k}R \bigr)^{-\alpha _{r}}\leq C\bigl(2^{j}R\bigr)^{-\alpha _{r}}. \end{aligned}$$
$$\begin{aligned} \bigl\Vert k_{r}(c_{B}-A_{r} \cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} &\leq \sum _{k\geq j} \bigl\Vert k_{r}(c_{B}-A_{r} \cdot )\chi _{(D_{j}^{l})_{k,r}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R} \\ &\leq C \sum_{k\geq j}\bigl(2^{k}R \bigr)^{-\alpha _{r}}\leq C\bigl(2^{j}R\bigr)^{-\alpha _{r}}. \end{aligned}$$
$$\begin{aligned} & \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}\leq C\bigl(2^{j}R \bigr)^{-\alpha _{2}}, \\ &\bigl\Vert k_{1}(c_{B}-A_{1}\cdot ) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}\leq C\bigl(2^{j}R \bigr)^{-\alpha _{1}}. \end{aligned}$$
$$\begin{aligned} & \sum_{j=1}^{\infty } \vert \tilde{B}_{l,j} \vert ^{1+\beta /n}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \\ &\qquad{} \times \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \Vert f_{2} \Vert _{\phi ,\tilde{B}_{l,j}} \\ &\quad \leq C \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{n+\beta -\alpha _{2}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \Vert f \Vert _{\phi ,\tilde{B}_{l,j}} \\ &\quad = C \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R\bigr)^{n-\alpha -\alpha _{2}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c _{B}-A_{1}\cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad = C \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R\bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A _{1}\cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}. \end{aligned}$$
$$\begin{aligned} & \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R \bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A _{1}\cdot )\bigr) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\qquad{} \times \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \\ & \quad\leq C M_{\alpha +\beta ,\phi } f\bigl(A_{1}^{-1}x\bigr) \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k _{1}(c_{B}-A_{1}\cdot )\bigr) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad \leq C M_{\alpha +\beta ,\phi }f\bigl(A_{1}^{-1}x\bigr). \end{aligned}$$
$$\begin{aligned} & \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A_{1}\cdot )\bigr)\chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{2}z \vert \sim 2^{j+1}R} \\ & \quad\leq \sum_{k\geq j} \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R}, \end{aligned}$$
$$\begin{aligned} & \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot ) -k _{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{2}z \vert \sim 2^{j+1}R} \\ &\quad \leq \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha _{1}}j\sum_{k\geq j} \bigl\Vert \bigl(k_{1}(y-A _{1}\cdot )-k_{1}(c_{B}-A_{1}\cdot )\bigr)\chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha _{1}}\sum_{k\geq j} \frac{(2^{k}R)^{ \alpha _{1}}}{(2^{k}R)^{\alpha _{1}}}k \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1} (c _{B}-A_{1}\cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{j=1}^{\infty }\sum _{k\geq j}\frac{(2^{j}R)^{\alpha _{1}}}{(2^{k}R)^{ \alpha _{1}}}\bigl(2^{k}R \bigr)^{\alpha _{1}}k \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c _{B}-A_{1}\cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{k=1}^{\infty }\Biggl(\sum _{j=1}^{k}\bigl(2^{-\alpha _{1}} \bigr)^{k-j}\Biggr) \bigl(2^{k}R\bigr)^{ \alpha _{1}} k \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A_{1}\cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{k=1}^{\infty } \bigl(2^{k}R\bigr)^{\alpha _{1}}k \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot ) -k _{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{1}z \vert \sim 2^{k+1}R}< \infty , \end{aligned}$$
$$\begin{aligned} & \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R\bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A _{1}\cdot )\bigr)\chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ & \quad\leq CM_{\alpha +\beta ,\phi } f\bigl(A_{2}^{-1}x\bigr) \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k _{1}(c_{B}-A_{1}\cdot )\bigr) \chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ & \quad\leq CM_{\alpha +\beta ,\phi }f\bigl(A_{2}^{-1}x\bigr). \end{aligned}$$
$$ \mathit{I I I}\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i= 1}^{2} M_{\alpha +\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr). $$
(7)
$$ M_{\delta }^{\sharp } \bigl( T^{1}_{\alpha ,m,b}f \bigr) (x)\leq C { \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}M_{\beta }(T_{\alpha }f) (x)+C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}\sum_{i= 0}^{2} M_{\alpha +\beta ,\phi }f\bigl(A _{i}^{-1}x\bigr). $$
For the general case, from the definition of \(T^{k}_{\alpha ,m,b}\), we know that, for any λ,
Let \(\lambda :=b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}\cup\cdots \cup \tilde{B}_{m}}\), \(a:=T_{\alpha }((b-b_{\tilde{B}\cup \tilde{B} _{1}\cup \tilde{B}_{2}\cup\cdots \cup \tilde{B}_{m}})f_{2})(c_{B})\). We write
To estimate IV, by Hölder’s inequality and Lemma 2.1, we obtain
The terms V and VI are analogous to the ones in the case \(m = 2\) and \(k = 1\), we can get
Then we conclude
Theorem 1.1 is proved. □
$$\begin{aligned} &T^{k}_{\alpha ,m,b}(f) (y)\\ &\quad= \int _{\mathbb{R}^{n}}\bigl(b(y)-b(z)\bigr)^{k}K_{\alpha}(y,z)f(z) \,dz \\ &\quad= \int _{\mathbb{R}^{n}}\bigl(b(y)-\lambda +\lambda -b(z) \bigr)^{k}K_{\alpha}(y,z)f(z)\,dz \\ &\quad= \sum_{i= 0}^{k} \int _{\mathbb{R}^{n}}c_{ki}\bigl(b(y)-\lambda \bigr)^{i}\bigl(\lambda -b(z)\bigr)^{k-i}K_{\alpha}(y,z)f(z)\,dz \\ &\quad= \sum_{i= 0}^{k}\bigl(b(y)-\lambda \bigr)^{i} \int _{\mathbb{R}^{n}}c_{ki}\bigl(\lambda -b(z) \bigr)^{k-i}K_{\alpha}(y,z)f(z)\,dz \\ &\quad= \int _{\mathbb{R}^{n}}\bigl(\lambda -b(z)\bigr)^{k}K_{\alpha}(y,z)f(z) \,dz \\ &\qquad{} + \sum_{i= 1}^{k}c_{ki} \bigl(b(y)-\lambda \bigr)^{i} \int _{\mathbb{R}^{n}}\bigl(\lambda -b(y)+b(y)-b(z)\bigr)^{k-i}K_{\alpha}(y,z)f(z) \,dz \\ &\quad= \sum_{i= 1}^{k}c_{ki} \bigl(b(y)-\lambda \bigr)^{i}\sum_{j= 0}^{k-i}c_{kj} \bigl( \lambda -b(y)\bigr)^{j} \int _{\mathbb{R}^{n}}\bigl(b(y)-b(z)\bigr)^{k-i-j}K_{\alpha}(y,z)f(z) \,dz \\ &\qquad{} + T_{\alpha ,m}\bigl(\bigl(\lambda -b(\cdot )\bigr)^{k}f \bigr) (y) \\ &\quad = \sum_{i= 1}^{k}\sum _{j= 0}^{k-i}c_{kij}\bigl(b(y)-\lambda \bigr)^{i+j} \int _{\mathbb{R}^{n}}\bigl(b(y)-b(z)\bigr)^{k-i-j}K_{\alpha}(y,z)f(z) \,dz \\ &\qquad{} + T_{\alpha ,m}\bigl(\bigl(\lambda -b(\cdot )\bigr)^{k}f \bigr) (y) \\ &\quad=T_{\alpha ,m}\bigl(\bigl(\lambda -b(\cdot )\bigr)^{k}f\bigr) (y)+ \sum_{l= 0}^{k-1}c_{kl} \bigl(b(y)-\lambda \bigr)^{k-l}T^{l}_{\alpha ,m,b}f(y). \end{aligned}$$
$$\begin{aligned} & \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T^{k}_{\alpha ,m,b}(f) (y)-a \bigr\vert ^{\delta }\biggr)\,dy )^{1/ \delta } \\ &\quad \leq \Biggl( \frac{1}{ \vert B \vert } \int _{B} \Biggl\vert \sum_{i=0}^{k-1} \bigl(b(y)-\lambda \bigr)^{k-i}T_{\alpha ,m,b}^{i}f(y) \Biggr\vert ^{\delta }\,dy \Biggr)^{{1/\delta }} \\ & \qquad{}+ \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha }\bigl(\lambda -b( \cdot )\bigr)^{k}f_{1}) (y) \bigr\vert ^{ \delta }\,dy \biggr)^{{1/\delta }} \\ &\qquad{} + \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha }\bigl(\lambda -b( \cdot )\bigr)^{k}f_{2}) (y)-T _{\alpha } \bigl(\lambda -b(\cdot )\bigr)^{k}f_{2}) (c_{B}) \bigr\vert ^{\delta }\,dy\biggr)^{ {1/\delta }} \\ & \quad=:\mathit{I V}+V+\mathit{V I} . \end{aligned}$$
$$\begin{aligned} \mathit{I V}&\leq \sum_{l=0}^{k-1} \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert \bigl(b(y)-\lambda \bigr)^{k-l}T _{\alpha ,m,b}^{l}f(y) \bigr\vert ^{\delta }\,dy \biggr)^{1/\delta } \\ &\leq \sum_{l=0}^{k-1}C \Vert b \Vert _{\dot{\varLambda }_{\beta }}^{k-l} \vert B \vert ^{(k-l) \beta /n} \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha ,m,b}^{l}f(y) \bigr\vert ^{\delta }\,dy \biggr)^{1/\delta } \\ &\leq \sum_{l=0}^{k-1}C \Vert b \Vert _{\dot{\varLambda }_{\beta }}^{k-l} \vert B \vert ^{(k-l) \beta /n} \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha ,m,b}^{l}f(y) \bigr\vert \,dy \\ &\leq C \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l}M_{(k-l) \beta } \bigl(T^{l}_{\alpha ,m,b}f\bigr) (x). \end{aligned}$$
$$\begin{aligned} &V\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum _{i= 1}^{m}M_{\alpha +k\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr), \\ &\mathit{V I}\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum _{i= 1}^{m} M_{ \alpha +k\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr). \end{aligned}$$
$$ M_{\delta }^{\sharp } \bigl\vert T^{k}_{\alpha ,m,b}f \bigr\vert (x)\leq C\sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l}M_{(k-l)\beta } \bigl(T^{l}_{\alpha ,m,b}f\bigr) (x)+C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum_{i= 1}^{m} M _{\alpha +k\beta ,\phi }f\bigl(A_{i}^{-1}x\bigr). $$
Next, we prove Theorem 1.2.
Proof of Theorem 1.2
By the extrapolation result Theorem 1.1 in [20], we need only to show that (3) is true for some \(0 < q_{*} <\infty \) and all \(\omega ^{r} \in A(p/r,q_{*}/r )\) with \((n-\alpha ) /n < q_{*} < \infty \). Without loss of generality, we may assume \({\|b\|}_{ \dot{\varLambda }_{\beta }} = 1\). We will prove the desired conclusion by induction.
When \(k = 0\), \(T_{\alpha ,m,b}^{0} = T_{\alpha ,m}\). As \(k_{i} \in H _{n-\alpha _{i}, \varPsi _{i},0} = H_{n-\alpha _{i},\varPsi _{i}}\), Theorem 3.3 in [15] tells us that
$$ \int _{\mathbb{R}^{n}} \bigl\vert T_{\alpha ,m}f(x) \bigr\vert ^{q_{*}}\omega ^{q_{*}}(x)\,dx \le C \sum _{i= 1}^{m} \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha ,\phi }f(x) \bigr\vert ^{q_{*}}\omega ^{q_{*}}(A _{i}x)\,dx. $$
Now, for any \(k\in \mathbb{N}\), we assume that the results hold for all \(0 \leq j \leq k-1\), and let us see how to derive the case k. For \(\omega ^{r} \in A(p/r,q_{*}/r )\), by Remark 2.1, we know that \(\omega \in A({p,q_{*}})\). Then \(\omega ^{q_{*}} \in A_{q_{*}}\). By Lemma 5.1 in [12], we have \(\|T_{\alpha ,m}f\|_{L^{q_{*}}( \omega ^{q_{*}})} < \infty \). Therefore \(\omega ^{r} \in A(p/r,q_{*}/r )\) and \(b \in L^{\infty }\), we have
Besides, for \(p< p_{l}\leq q_{*}\), \(\omega \in A({p,q_{*}})\) implies \(\omega \in A({p_{l},q_{*}})\), and \(1/q_{*}=1/p_{l}-(k-l)\beta /n\) implies \(q_{*}=p_{k}\) when \(l=k\). Then there exists \(C>0\) such that
By the induction hypothesis, for \(0 \leq l \leq k-1\) and \(1/p_{l}=1/q _{j}-(l-j)\beta /n\), we get
Since \(1/q_{*}=1/p_{l}-(k-l)\beta /n\) and \(1/p_{l}=1/q_{j}-(l-j) \beta /n\), which implies \(p_{l}=q_{j}\) when \(l=j\), we have
Namely,
$$ \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})}= \Biggl\Vert \sum_{j=0} ^{k}c_{k,j}b^{k-j}T_{\alpha ,m} \bigl(b^{j}f\bigr) \Biggr\Vert _{L^{q_{*}}(\omega ^{q _{*}})}< \infty . $$
$$ \bigl\Vert M_{(k-l)\beta }\bigl(T^{l}_{\alpha ,m,b}f \bigr) \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})} \leq C \bigl\Vert T^{l}_{\alpha ,m,b}f \bigr\Vert _{{L^{p_{l}}}(\omega ^{p_{l}})} . $$
$$ \bigl\Vert T^{l}_{\alpha ,m,b}f \bigr\Vert _{{L^{p_{l}}}(\omega ^{p_{l}})}\leq C{ \Vert b \Vert } _{\dot{\varLambda }_{\beta }}^{l} \sum_{i= 1}^{m}\sum _{j= 0}^{l} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha +j\beta ,\phi }f(x) \bigr\vert ^{q_{j}}\omega ^{q_{j}}(A_{i}x)\,dx \biggr)^{1/q_{j}}. $$
$$\begin{aligned} & \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})} \\ &\quad\leq \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M\bigl(T^{k}_{\alpha ,m,b}f \bigr)^{\delta }(x) \bigr\vert ^{q _{*}/\delta }\omega ^{q_{*}}(x)\,dx \biggr)^{1/q_{*}} \\ &\quad\leq \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\delta }^{\sharp } \bigl(T^{k}_{\alpha ,m,b}f\bigr) (x) \bigr\vert ^{q _{*}}\omega ^{q_{*}}(x)\,dx \biggr)^{1/q_{*}} \\ &\quad\leq C \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l} \bigl\Vert M_{(k-l) \beta } \bigl(T^{l}_{\alpha ,m,b}f\bigr) (x) \bigr\Vert _{{L^{q_{*}}}{(\omega ^{q_{*}})}} \\ &\qquad{} +C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum _{i= 1}^{m} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +k\beta ,\phi }f\bigl(A_{i}^{-1}x \bigr)\bigr)^{q_{*}} \omega ^{q_{*}}(x)\,dx \biggr)^{1/q_{*}} \\ &\quad\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m}\sum_{l= 0}^{k-1} \sum_{j= 0}^{l} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha +j\beta ,\phi }f(x) \bigr\vert ^{q_{j}}\omega ^{q_{j}}(A_{i}x)\,dx \biggr)^{1/q_{j}} \\ &\qquad{}+C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +k\beta ,\phi }f\bigl(A _{i}^{-1}x\bigr)\bigr)^{q_{*}}\omega ^{q_{*}}(x)\,dx \biggr)^{1/q_{*}} \\ &\quad\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m} \sum_{j= 0}^{k-1} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha +j\beta ,\phi }f(x) \bigr\vert ^{q_{j}}\omega ^{q_{j}}(A_{i}x)\,dx \biggr)^{1/q_{j}} \\ &\qquad{} +C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +k\beta ,\phi }f(x)\bigr)^{q_{*}} \omega ^{q_{*}}(A_{i}x)\,dx \biggr)^{1/q_{*}} \\ &\quad\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m}\sum_{l= 0}^{k-1} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha +l\beta ,\phi }f(x) \bigr\vert ^{p_{l}}\omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1/p_{l}} \\ &\qquad{} +C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +k\beta ,\phi }f(x)\bigr)^{p_{k}} \omega ^{p_{k}}(A_{i}x)\,dx \biggr)^{1/p_{k}} \\ &\quad\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m}\sum_{l= 0}^{k} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +l\beta ,\phi }f(x)\bigr)^{p_{l}} \omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1/p_{l}}. \end{aligned}$$
$$ \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})} \leq C{ \Vert b \Vert } _{\dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m}\sum _{l= 0}^{k} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +l\beta ,\phi }f(x)\bigr)^{p_{l}} \omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1/p_{l}}. $$
(8)
For the general case, if \(b\in \dot{\varLambda }_{\beta }\), for any \(N\in \mathbb{N}\), we define \(b_{N}=b\chi _{\{x:-N< b(x)< N\}}+N{\chi _{\{x:b(x) \geq N\}}}-N{\chi _{\{x:b(x)\leq -N\}}}\), then \(\|b_{N}\|_{ \dot{\varLambda }_{\beta }} \leq c{\|b\|}_{\dot{\varLambda }_{\beta }}\). Using convergence theorems, for details see [21], we conclude that (8) holds for any \(b\in \dot{\varLambda }_{ \beta }\) and \(\omega ^{r} \in A(p/r,q_{*}/r )\). Thus, as mentioned, using the extrapolation results obtained in [20], (3) holds for all \(0 < q < \infty \), \(b\in \dot{\varLambda }_{\beta }\), and \(\omega ^{r} \in A(p/r,q/r )\).
If ω satisfies (2), we have
which completes the proof of Theorem 1.2. □
$$\begin{aligned} \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q}(\omega ^{q})} &\leq C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m}\sum _{l= 0}^{k} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +l\beta ,\phi }f(x)\bigr)^{p_{l}} \omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1/p_{l}} \\ &\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{l= 0}^{k} \Vert M_{\alpha +l\beta ,\phi }f \Vert _{L^{p_{l}}(\omega ^{p _{l}})}, \end{aligned}$$
4 The weighted inequalities of commutators
This section is concerned with the proofs of Theorems 1.3 and 1.4. For the proof of Theorem 1.3, we need the Coifman inequality (3) and the boundedness of the maximal operator, given in [22] (see Theorem 2.6). Let us begin with the following previous result.
Theorem 4.1
([22])
Let
\(0 \leq \alpha < n\), ω
be a weight, \(1 \leq r < p < n/(\alpha +k\beta )\), and
\(1/q = 1/p-(\alpha +k\beta )/n\). Let
ϕ
be a Young function such that
\(\phi ^{1+ \frac{\rho (\alpha +k\beta )}{n-(\alpha +k\beta )}} \in B_{\frac{\rho n}{n-(\alpha +k\beta )}}\)
for every
\(\rho > r(n-( \alpha +k\beta ))/(n-(\alpha +k\beta )r)\), and let
η
be a Young function such that
\(\eta ^{-1}(t)t^{(\alpha +k\beta )/n}\lesssim \phi ^{-1}(t)\)
for every
\(t > 0 \). If
\(\omega ^{r} \in A(p/r,q/r )\), then
\(M_{\alpha +k\beta ,\phi }\)
is bounded from
\(L^{p}(\omega ^{p})\)
to
\(L^{q}(\omega ^{q})\).
Now we are in a position to prove Theorem 1.3.
Proof of Theorem 1.3
For \(1/q =1/p_{l}-(k-l)\beta /n\) and \(1/q = 1/p-(\alpha +k\beta )/n\), we can know \(p< p_{l}< q\). Then, for \(\omega ^{r} \in A(p/r,q/r )\), by Remark 2.1, we can get \(\omega ^{r} \in A(p/r,p_{l}/r )\) and \(\omega \in A(p,q)\); moreover, \(\omega \in A({p_{l},q})\). Therefore, from Theorem 4.1, we know that \(M_{\alpha +l\beta ,\phi }\) is bounded from \(L^{p}(\omega ^{p})\) into \(L^{p_{l}}(\omega ^{p_{l}})\). Then, by Theorem 1.2 and ω satisfies (2), we have
This proves the conclusion of Theorem 1.3. □
$$ \bigl\Vert T^{k}_{\alpha ,m,b}f \bigr\Vert _{L^{q}(\omega ^{q})}\leq C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{k} \sum_{l= 0}^{k} \Vert M_{\alpha +l\beta ,\phi }f \Vert _{L^{p_{l}}(\omega ^{p _{l}})}\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \Vert f \Vert _{L^{p}(\omega ^{p})}. $$
Finally, we give the proof of Theorem 1.4.
Proof of Theorem 1.4
Following the ideas in [21], for \(\omega ^{r} \in {A(\frac{n}{(\alpha +k\beta )r}, \infty )}\), we know
Note that \(\omega ^{r} \in {A(\frac{n}{(\alpha +k\beta )r}, \infty )}\) implies \(\omega ^{r} \in {A(\frac{n}{(k-l)\beta r}, \infty )}\), we have
Now, by (9), (10), (2), Proposition 2.1, Theorems 1.1 and 1.3, we get
This completes the proof of Theorem 1.4. □
$$ \Vert \omega M_{\alpha +k\beta , r}f \Vert _{\infty } \leq \Vert f\omega \Vert _{n/( \alpha +k\beta )}. $$
(9)
$$ \bigl\Vert \omega M_{(k-l)\beta , r}T^{l}_{\alpha ,m,b}f \bigr\Vert _{\infty }\leq \bigl\Vert T ^{l}_{\alpha ,m,b}f \omega \bigr\Vert _{n/(k-l)\beta }. $$
(10)
$$\begin{aligned} \bigl|\!\bigl|\!\bigl|T^{k}_{\alpha ,m,b}f\bigr|\!\bigr|\!\bigr|_{\omega }\simeq{}& \bigl\Vert \omega M^{\sharp }T ^{k}_{\alpha ,m,b}f \bigr\Vert _{\infty } \\ \leq{}& C \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l} \bigl\Vert \omega M _{(k-l)\beta }\bigl(T^{l}_{\alpha ,m,b}f\bigr) \bigr\Vert _{\infty } \\ & {}+C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{i= 1}^{m} \bigl\Vert \omega M_{\alpha +k\beta ,\phi }f\bigl(A_{i}^{-1}\cdot \bigr) \bigr\Vert _{\infty } \\ \leq{}& C\kappa _{r_{1}} \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l} \bigl\Vert \omega M _{(k-l)\beta ,r}\bigl(T^{l}_{\alpha ,m,b}f\bigr) \bigr\Vert _{\infty } \\ &{}+C\kappa _{r_{2}} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m} \bigl\Vert \omega M_{\alpha +k\beta ,r}f\bigl(A_{i}^{-1}\cdot \bigr) \bigr\Vert _{\infty } \\ \leq{}& C\kappa _{r_{1}} \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l} \bigl\Vert \omega T ^{l}_{\alpha ,m,b}f \bigr\Vert _{n/(k-l)\beta } \\ &{} +C\kappa _{r_{2}}{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m} \bigl\Vert \omega f \bigl(A_{i}^{-1}\cdot \bigr) \bigr\Vert _{n/(\alpha +k\beta )} \\ \leq{}& C\kappa _{r_{1}} \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l}{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{l} \Vert \omega f \Vert _{n/(\alpha +k\beta )} \\ &{}+C\kappa _{r_{2}}{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m} \bigl\Vert \omega f \bigl(A_{i}^{-1}\cdot \bigr) \bigr\Vert _{n/(\alpha +k\beta )} \\ \leq {}&C\kappa _{r_{1}} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \Vert \omega f \Vert _{n/(\alpha +k\beta )} \\ &{}+C\kappa _{r_{2}}{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m} \bigl\Vert \omega (A_{i}\cdot ) f \bigr\Vert _{n/(\alpha +k\beta )} \\ \leq{}& C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \Vert f \omega \Vert _{L^{n/{\alpha +k \beta }}}. \end{aligned}$$
Acknowledgements
The authors thank the referees cordially for their valuable suggestions and comments.
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