This section is devoted to the proofs of Theorems 1.1 and 1.2.
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
$$ [b,T_{\alpha }](f) (x)=\bigl(b(x)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr)T_{\alpha }f(x)+T_{\alpha }\bigl((b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}}-b)f \bigr) (x). $$
And from the inequality \(|t^{\delta }-s^{\delta }|^{1/\delta }\leq |t-s|\) and Jensen’s inequality, we get
$$\begin{aligned} & \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert [b,T_{\alpha }](f)^{\delta }(y)-a^{\delta } \bigr\vert \,dy \biggr)^{1/\delta } \\ &\quad \leq \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert [b,T_{\alpha }](f) (y)-a \bigr\vert ^{\delta }\,dy \biggr)^{1/\delta } \\ &\quad \leq \frac{1}{ \vert B \vert } \int _{B} \bigl\vert \bigl(b(y)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr)T_{\alpha }f(y) \bigr\vert \,dy \\ &\qquad{} + \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha } \bigl((b_{\tilde{B}\cup \tilde{B}_{1} \cup \tilde{B}_{2}}-b)f_{1}\bigr) (y) \bigr\vert \,dy \\ &\qquad{} + \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha } \bigl((b_{\tilde{B}\cup \tilde{B}_{1} \cup \tilde{B}_{2}}-b) f_{2}\bigr) (y)-T_{\alpha } \bigl((b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}}-b)f_{2}\bigr) (c_{B}) \bigr\vert \,dy \\ &\quad =:I+\mathit{I I}+\mathit{I I I}. \end{aligned}$$
(4)
For I, by Lemma 2.1, we have
$$\begin{aligned} I&\leq C \Vert b \Vert _{\dot{\varLambda }_{\beta }} \vert B \vert ^{\beta /n} \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha }f(y) \bigr\vert \,dy \biggr) \\ &\leq C \Vert b \Vert _{\dot{\varLambda }_{\beta }}M_{\beta }(T_{\alpha }f) (x). \end{aligned}$$
(5)
For II, we know
$$\begin{aligned} \mathit{I I}&= \frac{1}{ \vert B \vert } \int _{B} \int _{\tilde{B}_{1}\cup \tilde{B}_{2}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \bigl\vert b(z)-b _{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f_{1}(z) \bigr\vert \,dz\,dy \\ &\leq \sum_{i=1}^{2} \frac{1}{ \vert B \vert } \int _{\tilde{B}_{i}} \bigl\vert b(z)-b_{\tilde{B} \cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f_{1}(z) \bigr\vert \int _{B} \bigl\vert K(y,z) \bigr\vert \,dy\,dz. \end{aligned}$$
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
$$ \int _{B} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy\leq \int _{\{{y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy + \int _{\{y\in B: \vert y-A_{2}z \vert \leq \vert y-A_{1}z \vert \}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy. $$
For \(j\in \mathbb{N}\), let us consider the set
$$ C_{j}^{1} := \bigl\{ y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert , \vert y-A_{1}z \vert \sim 2^{-j-1}R \bigr\} . $$
Notice that if \(y\in B\) and \(z\in \tilde{B}_{1}\), then \(|y-A_{1}z| \leq 3R<4R\). Thus,
$$\begin{aligned} & \int _{\{{y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \\ &\quad \leq \sum_{j=-2}^{\infty } \int _{C_{j}^{1}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \\ & \quad\leq \sum_{j=-2}^{\infty } \frac{ \vert B(A_{1}z,2^{-j}R) \vert }{ \vert B(A_{1}z,2^{-j}R) \vert } \int _{B(A_{1}z,2^{-j}R)} \bigl\vert K_{\alpha}(y,z) \bigr\vert \chi _{\{y: \vert y-A_{1}z \vert \sim 2^{-j-1}R\}}\,dy \\ & \quad\leq C \sum_{j=-2}^{\infty }{ \bigl\vert B\bigl(A_{1}z,2^{-j}R\bigr) \bigr\vert } \bigl\Vert k_{1}(\cdot -A_{1}z) \chi _{\{y: \vert y-A_{1}z \vert \sim 2^{-j-1}R\}} \bigr\Vert _{\varPsi _{1},B(A_{1}z,2^{-j}R)} \\ &\qquad{} \times \bigl\Vert k_{2}(\cdot -A_{2}z)\chi _{\{y: \vert y-A_{1}z \vert \sim 2^{-j-1}R \}} \bigr\Vert _{\varPsi _{2},B(A_{1}z,2^{-j}R)} \\ &\quad \leq C \sum_{j=-2}^{\infty }{ \bigl\vert B\bigl(A_{1}z,2^{-j}R\bigr) \bigr\vert } \bigl\Vert k_{1}(\cdot -A_{1}z) \bigr\Vert _{\varPsi _{1}, \vert y-A_{1}z \vert \sim 2^{-j-1}R} \\ & \qquad{}\times \bigl\Vert k_{2}(\cdot -A_{2}z) \bigr\Vert _{\varPsi _{2}, \vert y-A_{1}z \vert \sim 2^{-j-1}R}. \end{aligned}$$
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
$$\begin{aligned} \bigl\Vert k_{2}(\cdot -A_{2}z) \bigr\Vert _{\varPsi _{2}, \vert y-A_{1}z \vert \sim 2^{-j-1}R} &\leq \sum_{k\geq 0} \bigl\Vert k_{2}(\cdot -A_{2}z) \bigr\Vert _{\varPsi _{2}, \vert y-A_{2}z \vert \sim 2^{-j+k-1}R} \\ &\leq \sum_{k\geq 0} \bigl\Vert k_{2}( \cdot ) \bigr\Vert _{\varPsi _{2}, \vert y \vert \sim 2^{-j+k-1}R} \\ &\leq C \sum_{k\geq 0}\bigl(2^{-j+k-1}R \bigr)^{-\alpha _{2}}\leq C \bigl(2^{-j}R\bigr)^{-\alpha _{2}}, \end{aligned}$$
and
$$ \bigl\Vert k_{1}(\cdot -A_{1}z) \bigr\Vert _{\varPsi _{1}, \vert y-A_{1}z \vert \sim 2^{-j-1}R} \leq C\bigl(2^{-j-1}R\bigr)^{- \alpha _{1}} \leq C \bigl(2^{-j}R\bigr)^{-\alpha _{1}}. $$
Consequently,
$$ \int _{\{{y\in B: \vert y-A_{1}z \vert \leq \vert y-A_{2}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \leq C \sum _{j=-2}^{\infty }\bigl(2^{-j}R \bigr)^{n-\alpha _{1}-\alpha _{2}}\leq CR^{\alpha }. $$
Similarly,
$$ \int _{\{{y\in B: \vert y-A_{2}z \vert \leq \vert y-A_{1}z \vert }\}} \bigl\vert K_{\alpha}(y,z) \bigr\vert \,dy \leq CR^{ \alpha }. $$
Then
$$\begin{aligned} \mathit{I I}&\leq CR^{\alpha } \sum _{i=1}^{2}\frac{1}{ \vert B \vert } \int _{\tilde{B}_{i}} \bigl\vert b(z)-b_{\tilde{B} \cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f_{1}(z) \bigr\vert \,dz \\ &\leq C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i=1}^{2} R^{\alpha +\beta } \frac{1}{ \vert \tilde{B}_{i} \vert } \int _{\tilde{B}_{i}} \bigl\vert f(z) \bigr\vert \,dz \\ &\leq C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i=1}^{2} R^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B}_{i}} \\ &\leq C { \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i=1}^{2} M_{\alpha +\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr). \end{aligned}$$
(6)
For III, we have
$$\begin{aligned} \mathit{I I I} &= \frac{1}{ \vert B \vert } \int _{B} \int _{(\tilde{B}_{1}\cup \tilde{B}_{2})^{c}} \bigl\vert K_{\alpha}(y,z)-K_{\alpha}(c _{B},z) \bigr\vert \bigl\vert b(z)-b_{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz\,dy \\ &\leq \sum_{l=1}^{2} \frac{1}{ \vert B \vert } \int _{B} \int _{Z^{l}} \bigl\vert K_{\alpha}(y,z)-K_{\alpha}(c_{B},z) \bigr\vert \bigl\vert b(z)-b _{\tilde{B}\cup \tilde{B}_{1}\cup \tilde{B}_{2}} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz\,dy, \end{aligned}$$
where
$$ Z^{l}=(\tilde{B}_{1}\cup \tilde{B}_{2})^{c} \cap \bigl\{ z: \vert c_{B}-A_{l}z \vert \leq \vert c_{B}-A_{r}z \vert , r\neq l,1\leq r \leq 2 \bigr\} . $$
Let us estimate \(|K_{\alpha}(y,z)-K_{\alpha}(c_{B},z)|\) for \(y\in B\) and \(z\in Z^{l}\):
$$\begin{aligned} \bigl\vert K_{\alpha}(y,z)-K_{\alpha}(c_{B},z) \bigr\vert \leq{}& \bigl\vert k_{1}(y-A_{1}z)-k_{1}(c_{B}-A_{1}z) \bigr\vert \bigl\vert k_{2}(y-A _{2}z) \bigr\vert \\ & {}+ \bigl\vert k_{2}(y-A_{2}z)-k_{2}(c_{B}-A_{2}z) \bigr\vert \bigl\vert k_{1}(c_{B}-A_{1}z) \bigr\vert . \end{aligned}$$
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
$$ D_{j}^{l}:=\bigl\{ {z\in Z^{l}: \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R\bigr\} }. $$
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
$$\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}$$
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,
$$\begin{aligned} & \bigl\Vert k_{l}(y-A_{l}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{l}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad\leq \bigl\Vert k_{l}(y-A_{l}\cdot ) \bigr\Vert _{\varPsi _{l}, \vert y-A_{l}z \vert \sim 2^{j}R} \\ & \qquad{}+ \bigl\Vert k_{l}(y-A_{l}\cdot ) \bigr\Vert _{\varPsi _{l}, \vert y-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad\leq \bigl\Vert k_{l}(\cdot ) \bigr\Vert _{\varPsi _{l}, \vert x \vert \sim 2^{j}R}+ \bigl\Vert k_{l}(\cdot ) \bigr\Vert _{\varPsi _{l}, \vert x \vert \sim 2^{j+1}R} \\ &\quad\leq C\bigl(2^{j}R\bigr)^{-\alpha _{l}}, \end{aligned}$$
where the last inequality holds since \(k_{l} \in S_{n-\alpha _{l},\varPsi _{l}}\). Also, by the hypothesis,
$$ \bigl\Vert k_{l}(c_{B}-A_{l}\cdot ) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{l}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}\leq C\bigl(2^{j+1}R \bigr)^{-\alpha _{l}}. $$
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
$$ \bigl(D_{j}^{l}\bigr)_{k,r}=\bigl\{ {z\in D_{j}^{l}: \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R\bigr\} }. $$
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
$$\begin{aligned} \bigl\Vert k_{r}(y-A_{r}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} &\leq \sum _{k\geq j} \bigl\Vert k_{r}(y-A_{r} \cdot )\chi _{(D_{j}^{l})_{k,r}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\leq \sum_{k\geq j} \bigl\Vert k_{r}(y-A_{r}\cdot )\chi _{(D_{j}^{l})_{k,r}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R} \\ &\leq \sum_{k\geq j} \bigl\Vert k_{r}(y-A_{r}\cdot ) \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R} \\ &\leq \sum_{k\geq j} \bigl\Vert k_{r}( \cdot ) \bigr\Vert _{\varPsi _{r}, \vert x \vert \sim 2^{k}R}+ \bigl\Vert k_{r}( \cdot ) \bigr\Vert _{\varPsi _{r}, \vert x \vert \sim 2^{k+1}R} \\ &\leq C \sum_{k\geq j}\bigl(2^{k}R \bigr)^{-\alpha _{r}}\leq C\bigl(2^{j}R\bigr)^{-\alpha _{r}}. \end{aligned}$$
By the same arguments, we can get
$$\begin{aligned} \bigl\Vert k_{r}(c_{B}-A_{r} \cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} &\leq \sum _{k\geq j} \bigl\Vert k_{r}(c_{B}-A_{r} \cdot )\chi _{(D_{j}^{l})_{k,r}} \bigr\Vert _{\varPsi _{r}, \vert c_{B}-A_{r}z \vert \sim 2^{k+1}R} \\ &\leq C \sum_{k\geq j}\bigl(2^{k}R \bigr)^{-\alpha _{r}}\leq C\bigl(2^{j}R\bigr)^{-\alpha _{r}}. \end{aligned}$$
As a result, no matter \(l=1\) or \(l=2\), we have
$$\begin{aligned} & \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}\leq C\bigl(2^{j}R \bigr)^{-\alpha _{2}}, \\ &\bigl\Vert k_{1}(c_{B}-A_{1}\cdot ) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}\leq C\bigl(2^{j}R \bigr)^{-\alpha _{1}}. \end{aligned}$$
Hence,
$$\begin{aligned} & \sum_{j=1}^{\infty } \vert \tilde{B}_{l,j} \vert ^{1+\beta /n}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \\ &\qquad{} \times \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \Vert f_{2} \Vert _{\phi ,\tilde{B}_{l,j}} \\ &\quad \leq C \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{n+\beta -\alpha _{2}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \Vert f \Vert _{\phi ,\tilde{B}_{l,j}} \\ &\quad = C \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R\bigr)^{n-\alpha -\alpha _{2}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c _{B}-A_{1}\cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad = C \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R\bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A _{1}\cdot )\bigr)\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R}. \end{aligned}$$
So, when \(l=1\), from \(k_{1} \in H_{n-\alpha _{1},\varPsi _{1}, 1} \), we can get
$$\begin{aligned} & \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R \bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c_{B}-A _{1}\cdot )\bigr) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\qquad{} \times \bigl\Vert k_{2}(y-A_{2}\cdot )\chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{2}, \vert c_{B}-A _{l}z \vert \sim 2^{j+1}R} \\ & \quad\leq C M_{\alpha +\beta ,\phi } f\bigl(A_{1}^{-1}x\bigr) \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k _{1}(c_{B}-A_{1}\cdot )\bigr) \chi _{D_{j}^{l}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ &\quad \leq C M_{\alpha +\beta ,\phi }f\bigl(A_{1}^{-1}x\bigr). \end{aligned}$$
For \(l= 2\), note that
$$\begin{aligned} & \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A_{1}\cdot )\bigr)\chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{2}z \vert \sim 2^{j+1}R} \\ & \quad\leq \sum_{k\geq j} \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A_{1} \cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R}, \end{aligned}$$
we have
$$\begin{aligned} & \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot ) -k _{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{2}z \vert \sim 2^{j+1}R} \\ &\quad \leq \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha _{1}}j\sum_{k\geq j} \bigl\Vert \bigl(k_{1}(y-A _{1}\cdot )-k_{1}(c_{B}-A_{1}\cdot )\bigr)\chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha _{1}}\sum_{k\geq j} \frac{(2^{k}R)^{ \alpha _{1}}}{(2^{k}R)^{\alpha _{1}}}k \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1} (c _{B}-A_{1}\cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{j=1}^{\infty }\sum _{k\geq j}\frac{(2^{j}R)^{\alpha _{1}}}{(2^{k}R)^{ \alpha _{1}}}\bigl(2^{k}R \bigr)^{\alpha _{1}}k \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k_{1}(c _{B}-A_{1}\cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{k=1}^{\infty }\Biggl(\sum _{j=1}^{k}\bigl(2^{-\alpha _{1}} \bigr)^{k-j}\Biggr) \bigl(2^{k}R\bigr)^{ \alpha _{1}} k \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A_{1}\cdot )\bigr) \chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{1}z \vert \sim 2^{k+1}R} \\ &\quad \leq \sum_{k=1}^{\infty } \bigl(2^{k}R\bigr)^{\alpha _{1}}k \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot ) -k _{1}(c_{B}-A_{1} \cdot )\bigr)\chi _{(D_{j}^{2})_{k,1}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A _{1}z \vert \sim 2^{k+1}R}< \infty , \end{aligned}$$
where the last inequality follows from that \(k_{1} \in H_{n-\alpha _{1},\varPsi _{1}, 1} \). Hence,
$$\begin{aligned} & \sum_{j=1}^{\infty } \bigl(2^{j}R\bigr)^{\alpha +\beta } \Vert f \Vert _{\phi ,\tilde{B} _{l,j}}\bigl(2^{j}R\bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1}\cdot )-k_{1}(c_{B}-A _{1}\cdot )\bigr)\chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ & \quad\leq CM_{\alpha +\beta ,\phi } f\bigl(A_{2}^{-1}x\bigr) \sum_{j=1}^{\infty }\bigl(2^{j}R \bigr)^{\alpha _{1}}j \bigl\Vert \bigl(k_{1}(y-A_{1} \cdot )-k _{1}(c_{B}-A_{1}\cdot )\bigr) \chi _{D_{j}^{2}} \bigr\Vert _{\varPsi _{1}, \vert c_{B}-A_{l}z \vert \sim 2^{j+1}R} \\ & \quad\leq CM_{\alpha +\beta ,\phi }f\bigl(A_{2}^{-1}x\bigr). \end{aligned}$$
Then
$$ \mathit{I I I}\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }} \sum _{i= 1}^{2} M_{\alpha +\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr). $$
(7)
Summing up (4)–(7), we know that
$$ M_{\delta }^{\sharp } \bigl( T^{1}_{\alpha ,m,b}f \bigr) (x)\leq C { \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}M_{\beta }(T_{\alpha }f) (x)+C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}\sum_{i= 0}^{2} M_{\alpha +\beta ,\phi }f\bigl(A _{i}^{-1}x\bigr). $$
For the case \(\alpha = 0\), we repeat the same argument to inequality (4) and get the desired conclusion.
For the general case, from the definition of \(T^{k}_{\alpha ,m,b}\), we know that, for any λ,
$$\begin{aligned} &T^{k}_{\alpha ,m,b}(f) (y)\\ &\quad= \int _{\mathbb{R}^{n}}\bigl(b(y)-b(z)\bigr)^{k}K_{\alpha}(y,z)f(z) \,dz \\ &\quad= \int _{\mathbb{R}^{n}}\bigl(b(y)-\lambda +\lambda -b(z) \bigr)^{k}K_{\alpha}(y,z)f(z)\,dz \\ &\quad= \sum_{i= 0}^{k} \int _{\mathbb{R}^{n}}c_{ki}\bigl(b(y)-\lambda \bigr)^{i}\bigl(\lambda -b(z)\bigr)^{k-i}K_{\alpha}(y,z)f(z)\,dz \\ &\quad= \sum_{i= 0}^{k}\bigl(b(y)-\lambda \bigr)^{i} \int _{\mathbb{R}^{n}}c_{ki}\bigl(\lambda -b(z) \bigr)^{k-i}K_{\alpha}(y,z)f(z)\,dz \\ &\quad= \int _{\mathbb{R}^{n}}\bigl(\lambda -b(z)\bigr)^{k}K_{\alpha}(y,z)f(z) \,dz \\ &\qquad{} + \sum_{i= 1}^{k}c_{ki} \bigl(b(y)-\lambda \bigr)^{i} \int _{\mathbb{R}^{n}}\bigl(\lambda -b(y)+b(y)-b(z)\bigr)^{k-i}K_{\alpha}(y,z)f(z) \,dz \\ &\quad= \sum_{i= 1}^{k}c_{ki} \bigl(b(y)-\lambda \bigr)^{i}\sum_{j= 0}^{k-i}c_{kj} \bigl( \lambda -b(y)\bigr)^{j} \int _{\mathbb{R}^{n}}\bigl(b(y)-b(z)\bigr)^{k-i-j}K_{\alpha}(y,z)f(z) \,dz \\ &\qquad{} + T_{\alpha ,m}\bigl(\bigl(\lambda -b(\cdot )\bigr)^{k}f \bigr) (y) \\ &\quad = \sum_{i= 1}^{k}\sum _{j= 0}^{k-i}c_{kij}\bigl(b(y)-\lambda \bigr)^{i+j} \int _{\mathbb{R}^{n}}\bigl(b(y)-b(z)\bigr)^{k-i-j}K_{\alpha}(y,z)f(z) \,dz \\ &\qquad{} + T_{\alpha ,m}\bigl(\bigl(\lambda -b(\cdot )\bigr)^{k}f \bigr) (y) \\ &\quad=T_{\alpha ,m}\bigl(\bigl(\lambda -b(\cdot )\bigr)^{k}f\bigr) (y)+ \sum_{l= 0}^{k-1}c_{kl} \bigl(b(y)-\lambda \bigr)^{k-l}T^{l}_{\alpha ,m,b}f(y). \end{aligned}$$
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
$$\begin{aligned} & \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T^{k}_{\alpha ,m,b}(f) (y)-a \bigr\vert ^{\delta }\biggr)\,dy )^{1/ \delta } \\ &\quad \leq \Biggl( \frac{1}{ \vert B \vert } \int _{B} \Biggl\vert \sum_{i=0}^{k-1} \bigl(b(y)-\lambda \bigr)^{k-i}T_{\alpha ,m,b}^{i}f(y) \Biggr\vert ^{\delta }\,dy \Biggr)^{{1/\delta }} \\ & \qquad{}+ \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha }\bigl(\lambda -b( \cdot )\bigr)^{k}f_{1}) (y) \bigr\vert ^{ \delta }\,dy \biggr)^{{1/\delta }} \\ &\qquad{} + \biggl( \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha }\bigl(\lambda -b( \cdot )\bigr)^{k}f_{2}) (y)-T _{\alpha } \bigl(\lambda -b(\cdot )\bigr)^{k}f_{2}) (c_{B}) \bigr\vert ^{\delta }\,dy\biggr)^{ {1/\delta }} \\ & \quad=:\mathit{I V}+V+\mathit{V I} . \end{aligned}$$
To estimate IV, by Hölder’s inequality and Lemma 2.1, we obtain
$$\begin{aligned} \mathit{I V}&\leq \sum_{l=0}^{k-1} \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert \bigl(b(y)-\lambda \bigr)^{k-l}T _{\alpha ,m,b}^{l}f(y) \bigr\vert ^{\delta }\,dy \biggr)^{1/\delta } \\ &\leq \sum_{l=0}^{k-1}C \Vert b \Vert _{\dot{\varLambda }_{\beta }}^{k-l} \vert B \vert ^{(k-l) \beta /n} \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha ,m,b}^{l}f(y) \bigr\vert ^{\delta }\,dy \biggr)^{1/\delta } \\ &\leq \sum_{l=0}^{k-1}C \Vert b \Vert _{\dot{\varLambda }_{\beta }}^{k-l} \vert B \vert ^{(k-l) \beta /n} \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\alpha ,m,b}^{l}f(y) \bigr\vert \,dy \\ &\leq C \sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l}M_{(k-l) \beta } \bigl(T^{l}_{\alpha ,m,b}f\bigr) (x). \end{aligned}$$
The terms V and VI are analogous to the ones in the case \(m = 2\) and \(k = 1\), we can get
$$\begin{aligned} &V\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum _{i= 1}^{m}M_{\alpha +k\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr), \\ &\mathit{V I}\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum _{i= 1}^{m} M_{ \alpha +k\beta ,\phi }f \bigl(A_{i}^{-1}x\bigr). \end{aligned}$$
Then we conclude
$$ M_{\delta }^{\sharp } \bigl\vert T^{k}_{\alpha ,m,b}f \bigr\vert (x)\leq C\sum_{l= 0}^{k-1} { \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k-l}M_{(k-l)\beta } \bigl(T^{l}_{\alpha ,m,b}f\bigr) (x)+C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k}\sum_{i= 1}^{m} M _{\alpha +k\beta ,\phi }f\bigl(A_{i}^{-1}x\bigr). $$
Theorem 1.1 is proved. □
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
$$ \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})}= \Biggl\Vert \sum_{j=0} ^{k}c_{k,j}b^{k-j}T_{\alpha ,m} \bigl(b^{j}f\bigr) \Biggr\Vert _{L^{q_{*}}(\omega ^{q _{*}})}< \infty . $$
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
$$ \bigl\Vert M_{(k-l)\beta }\bigl(T^{l}_{\alpha ,m,b}f \bigr) \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})} \leq C \bigl\Vert T^{l}_{\alpha ,m,b}f \bigr\Vert _{{L^{p_{l}}}(\omega ^{p_{l}})} . $$
By the induction hypothesis, for \(0 \leq l \leq k-1\) and \(1/p_{l}=1/q _{j}-(l-j)\beta /n\), we get
$$ \bigl\Vert T^{l}_{\alpha ,m,b}f \bigr\Vert _{{L^{p_{l}}}(\omega ^{p_{l}})}\leq C{ \Vert b \Vert } _{\dot{\varLambda }_{\beta }}^{l} \sum_{i= 1}^{m}\sum _{j= 0}^{l} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert M_{\alpha +j\beta ,\phi }f(x) \bigr\vert ^{q_{j}}\omega ^{q_{j}}(A_{i}x)\,dx \biggr)^{1/q_{j}}. $$
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
$$\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}$$
Namely,
$$ \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q_{*}}(\omega ^{q_{*}})} \leq C{ \Vert b \Vert } _{\dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m}\sum _{l= 0}^{k} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +l\beta ,\phi }f(x)\bigr)^{p_{l}} \omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1/p_{l}}. $$
(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
$$\begin{aligned} \bigl\Vert T_{\alpha ,m,b}^{k}f \bigr\Vert _{L^{q}(\omega ^{q})} &\leq C{ \Vert b \Vert }_{ \dot{\varLambda }_{\beta }}^{k} \sum_{i= 1}^{m}\sum _{l= 0}^{k} \biggl( \int _{\mathbb{R}^{n}}\bigl(M_{\alpha +l\beta ,\phi }f(x)\bigr)^{p_{l}} \omega ^{p_{l}}(A_{i}x)\,dx \biggr)^{1/p_{l}} \\ &\leq C{ \Vert b \Vert }_{\dot{\varLambda }_{\beta }}^{k} \sum _{l= 0}^{k} \Vert M_{\alpha +l\beta ,\phi }f \Vert _{L^{p_{l}}(\omega ^{p _{l}})}, \end{aligned}$$
which completes the proof of Theorem 1.2. □