Suppose that $\mathbb{S}^{n-1}$ is the unit sphere in $\mathbb{R}^{n}(n\geq2)$ equipped with the normalized Lebesgue measure $d\sigma$. Let $\Omega\in L^s(\mathbb{S}^{n-1})(1<s\leq\infty)$ be homogeneous of degree zero and satisfy the cancellation condition

$ \int_{S^{n-1}}\Omega(x^{'})d\sigma(x^{'})=0, $

(1.1)

where $x^{'}=\frac{x}{|x|}$ for any $x\neq 0$. The Lusin-area integral $\mu_{\Omega, S}$ is defined by

$ \mu_{\Omega, S}(f)(x)={\biggl(\int\int_{\Gamma(x)}\biggl|\frac{1}{t}\int_{|y-z|<t}\frac{\Omega(y-z)} {|y-z|^{n-1}}f(z)dz\biggr|^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}, $

(1.2)

where

$ \Gamma(x)=\{(y, t)\in\mathbb{R}^{n+1}_+:|x-y|<t\}\;\; \mbox{and} \quad\mathbb{R}^{n+1}_+=\mathbb{R}\times(0, \infty). $

Moreover, let $\vec{b}=(b_1, b_2, ..., b_m), $ where $b_i\in L_{loc}{(\mathbb{R}^n)}$ for $1\leq i\leq m.$ Then the multilinear commutator generated by $\vec{b}$ and $\mu_{\Omega, S}$ can be defined as follows:

$ \mu_{\Omega, S}^{\vec{b}}f(x)= {\biggl(\int\int_{\Gamma(x)}\biggl|\frac{1}{t}\int_{|y-z|<t}\prod\limits_{i=1}^{m} \frac{\Omega(y-z)}{|y-z|^{n-1}}(b_i(x)-b_i(z))f(z)dz\biggr|^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}. $

(1.3)

It is well known that the Lusin-area integral plays an important role in harmonic analysis and PDE (for example, see [1-8]). Therefore, it is a very interesting problem to discuss the boundedness of the Lusin area integral. In [2], Ding, Fan and Pan studied the weighted $L^p$ boundedness of the area integral $\mu_{\Omega, S}.$ In [3], the authors investigated the boundeness of $\mu_{\Omega, S}$ on the weighted Morrey spaces. The commutators generated by $\mu_{\Omega, S}$ attract much attention, also. In [5] and [6], the authors discuss the weighted $L^p$ boundedness and endpoint estimates for the higher order commutators generated by $\mu_{\Omega, S}$ and BMO function, respectively. In [8], the authors showed that the commutator generated by $\mu_{\Omega, S}$ and $VMO$ is a compact operator in the Morrey space.

Moreover, the classical Morrey space $M_{p, \lambda}$ were first introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. And, in [10], the authors introduced the local generalized Morrey space $LM_{p, \varphi}^{\{x_{0}\}}$, and they also studied the boundedness of the homogeneous singular integrals with rough kernel on these spaces.

Motivated by the works of [2, 3, 5, 8, 10, 13], we are going to consider the boundedness of $\mu_{\Omega, S}$ on the local generalized Morrey space $LM_{p, \varphi}^{\{x_0\}}, $ as well as the boundedness of the commutators generated by $\mu_{\Omega, S}$ and local Campanato functions.

Definition 2.1 [10] Let $\varphi(x, r)$ be a positive measurable function on $\mathbb{R}^n\times(0, \infty)$ and $1\leq p\leq\infty.$ For any fixed ${x}_{0}\in\mathbb{R}^n, $ a function $f\in L_{\rm{loc}}^{q}$ is said to belong to the local Morrey space, if

$ \|f\|_{LM^{\{x_0\}}_{p, \varphi}}=\sup\limits_{r>0}\varphi(x_0, r)^{-1}|B(x_0, r)|^{-\frac{1}{p}}\|f\|_{{L}^p(B(x_0, r))}<\infty. $

And, we denote

$ {LM}^{\{x_0\}}_{p, \varphi}\equiv{LM}^{\{x_0\}}_{p, \varphi}(\mathbb{R}^n)=\{f\in L_{loc}^{q}(\mathbb{R}^n):\|f\|_{{LM}^{\{x_0\}}_{p, \varphi}}<\infty\}. $

According to this definition, we recover the local Morrey space$ {LM}^{\{x_0\}}_{p, \lambda}$ under the choice $\varphi(x_0, r)=r^{\frac{\lambda-n}{p}}.$

Definition 2.2 [10] Let $1\leq q<\infty$ and $0\leq\lambda<\frac{1}{n}.$ A function $f\in L_{\rm{loc}}^{q}(\mathbb{R}^n)$ is said to belong to the space $LC_{q, \lambda}^{\{x_0\}}$ (local Campanato space), if

$ \|f\|_{LC_{q, \lambda}^{\{x_0\}}}=\sup\limits_{r>0}\biggl(\dfrac{1}{|B(x_0, r)|^{1+\lambda q}}\int_{B(x_, r)}|f(y)-f_{B(x_0, r)}|^{q}dy\biggr)^{\frac{1}{q}}<\infty, $

where

$ f_{B(x_0, r)}=\dfrac{1}{|B(x_0, r)|}\int_{B(x_0, r)}f(y)dy. $

Define

$ LC_{q, \lambda}^{\{x_0\}}(\mathbb{R}^n)=\{f\in L_{loc}^{q}(\mathbb{R}^n):\|f\|_{LC_{q, \lambda}^{\{x_0\}}}<\infty\}. $

Remark 2.1 [10] Note that, the central $BMO$ space $CBMO_{q}(\mathbb{R}^n)=LC_{q, 0}^{\{0\}}(\mathbb{R}^n)$ and $CBMO_{q}^{\{x_0\}}(\mathbb{R}^n)=LC_{q, 0}^{\{x_0\}}(\mathbb{R}^n).$ Moreover, imagining that the behavior of $CBMO_{q}^{\{x_0\}}(\mathbb{R}^n)$ may be quite different from that of $BMO(\mathbb{R}^n), $ since there is no analogy of the John-Nirenberg inequality of $BMO$ for the space $CBMO_{q}^{\{x_0\}}(\mathbb{R}^n).$

Lemma 2.1 [10] Let $1<q<\infty, $ $0<r_2<r_1$ and $b\in LC_{q, \lambda}^{\{x_0\}}, 0\leq\lambda<\frac{1}{n}, $ then

$ \begin{align*} \bigg(\dfrac{1}{|B(x_0, r_1)|^{1+\lambda q}}\int_{B(x_0, r_1)}|b(x)-b_{B(x_0, r_2)}|^{q}dx\biggr)^{\frac{1}{q}}\leq C\biggl(1+\ln\dfrac{r_1}{r_2}\biggr)\|b\|_{LC_{q, \lambda}^{\{x_0\}}}. \end{align*} $

And, from this inequality, we have

$ \begin{align*} |b_{B(x_0, r_1)}-b_{B(x_0, r_2)}|\leq C\biggl(1+\ln\dfrac{r_1}{r_2}\biggr)|B(x_0, r_1)|^{\lambda}\|b\|_{LC_{q, \lambda}^{\{x_0\}}}. \end{align*} $

In this section, we are going to use the following statement on the boundedness of the weighted Hardy operator:

$ {H}_{w}g(t):=\int^\infty_{t}g(s)w(s)ds, ~0<t<\infty, $

where $w$ is a fixed function non-negative and measurable on $(0, \infty).$

Lemma 2.2 [11, 12] Let $v_1, v_2$ and $w$ be positive almost everywhere and measurable functions on $(0, \infty).$ The inequality

$ \mbox{ess}\sup\limits_{t>0}v_2(t)H_wg(t)\leq C \mbox{ess}\sup\limits_{t>0}v_1(t)g(t) $

(2.1)

holds for some $C>0$ and all non-negative and non-decreasing $g$ on$~(0, \infty)~$if and only if

$ B:=\;\mbox{ess}\sup\limits_{t>0}v_2(t)\int^\infty_{t}\frac{w(s)}{\mbox{ess}\sup\limits_{s<\tau<\infty} v_1(\tau)}ds<\infty. $

Moreover, if $\tilde{C}$ is the minimum value of $C$ in (2.1), then $\tilde{C}=B$.

Lemma 2.3 [2] Suppose that $1<q, s\leq\infty$ and $\Omega\in L^s(\mathbb{S}^{n-1})$ satisfying (1.1). If $q, s$ and weighted function $w$ satisfy one of the following conditions:

(ⅰ) max$\{s^{'}, 2\}=\eta<q<\infty, $ and $w\in A_{q/{\eta}}, $

(ⅱ) $2<q<s, $ and $w^{1-(q/2)^{'}}\in A_{q^{'}/s^{'}}, $

(ⅲ) $2\leq q<\infty, $ and $w^{s^{'}}\in A_{q/2}, $

then the operator $\mu_{\Omega, S}$ is bounded on $L^q(w, \mathbb{R}^{n})$ space, where $s'=\frac{s}{s-1}$ is the conjugate exponent of $s.$

Remark 2.2 From Lemma 2.3, it's obvious that when $\Omega\in L^s(\mathbb{S}^{n-1}) (1<s\leq\infty)$ satisfies the condition (1.1), the operator $\mu_{\Omega, S}$ is bounded on $L^q(R^n)$ space for $2\leq q<\infty.$

Theorem 3.1 Let $\Omega\in L^s(\mathbb{S}^{n-1}) (1<s\leq\infty)$ satisfy the condition (1.1) and $\max\{2, s'\}<q<\infty, $ where $s'=\frac{s}{s-1}$ is the conjugate exponent of $s.$ Then, the inequality

$ \|\mu_{\Omega, S}(f)\|_{L^q(B(x_0, r))}\lesssim r^{\frac{n}{q}}\int^\infty_{2r}\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}} $

holds for any ball $B(x_0, r).$

Proof Let $B=B(x_0, r).$ We write $f=f_1+f_2, $ where $f_1=f\chi_{2B}$ and $f_2=f\chi_{(2B)^c}.$ Thus, we have

$ \|\mu_{\Omega, S}(f)\|_{L^q(B)}\leq\|\mu_{\Omega, S}(f_1)\|_{L^q(B)}+\|\mu_{\Omega, S}(f_2)\|_{L^q(B)}. $

Since $\mu_{\Omega, S}$ is bounded on $L^q(\mathbb{R}^n)$ space (see Lemma 2.3), then it follows that

$ \|\mu_{\Omega, S} f_1\|_{L^q(B)}\lesssim\|f\|_{L^q(B)} \lesssim r^{\frac{n}{q}}\int_{2r}^\infty\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}}. $

(3.1)

Our attention will be focused now on $|\mu_{\Omega, S}f_2(x)|$ for $x\in B.$

$ \begin{array}{cl} |\mu_{\Omega, S}f_2(x)| &\leq{\biggl(\int\int_{\Gamma(x)}\biggl|\frac{1}{t}\int_{(2B)^c\bigcap{\{z:|y-z|<t}\}}\frac{\Omega(y-z)} {|y-z|^{n-1}}f(z)dz\biggr|^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}\\ &\leq{\biggl(\int\int_{\Gamma(x)}\biggl|\frac{1}{t}\sum\limits_{j=1}^{\infty}\int_{{(2^{j+1}B\setminus2^jB)}\bigcap{\{z:|y-z|<t\}}}\frac{\Omega(y-z)}{|y-z|^{n-1}} f(z)dz\biggr|^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}. \end{array} $

(3.2)

Without loss of generality, we can assume that for any $x\in B, $ $(y, t)\in \Gamma(x)$ and $z\in 2^{j+1}B\setminus2^{j}B, $ we have $B(x, t)\bigcap B(z, t)\neq\varnothing.$ Thus, there exists $y_0\in B(x, t)\bigcap B(z, t), $ such that

$ 2t\geq|x-y_0|+|y_0-z|\geq|x-z|\geq|z-x_0|-|x-x_0|\geq2^jr-r\geq2^{j-1}r. $

Hence

$ |\mu_{\Omega, S}f_2(x)|\leq{\biggl({\int_{2^{j-2}r}^{\infty}}\int_{|x-y|<t}\biggl|\frac{1}{t}\sum\limits_{j=1}^{\infty}\int_{2^{j+1}B\setminus2^jB} \frac{\Omega(y-z)}{|y-z|^{n-1}}f(z)dz\biggr|^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}. $

When $\Omega\in L^{\infty}(\mathbb{S}^{n-1}), $ it follows from the Hölder's inequality that

$ \begin{array}{cl} &|\mu_{\Omega, S}f_2(x)|\\ \leq&\|\Omega\|_{L^{\infty}(S^{n-1})}{\biggl(\int_{2^{j-2}r}^{\infty}\int_{|x-y|<t}\biggl[\frac{1}{t}\sum\limits_{j=1}^{\infty}\frac{1}{|2^{j+1}B|^{\frac{n-1}{n}}} \int_{2^{j+1}B\setminus2^jB}|f(z)|dz\biggr]^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}\\ \lesssim&\sum\limits_{j=1}^{\infty} \frac{1}{|2^{j+1}B|^{\frac{n-1}{n}}}\int_{2^{j+1}B}|f(z)|dz{\biggl(\int_{2^{j-2}r}^{\infty}\int_{|x-y|<t} \frac{dydt}{t^{n+3}}\biggr)}^{\frac{1}{2}}\\ \lesssim&\sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-n}\int_{2^{j+1}B}|f(z)|dz \lesssim\sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-\frac{n}{q}}\|f(z)\|_{L^q(B(x_0, 2^{j+1}r))}\\ \lesssim&\sum\limits_{j=1}^{\infty}\int_{2^{j+1}r}^{2^{j+2}r}l^{-\frac{n}{q}-1}dl\|f\|_{L^q(B(x_0, 2^{j+1}r))} \lesssim\int_{2r}^{\infty}\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}}. \end{array} $

(3.3)

When $\Omega\in L^s(\mathbb{S}^{n-1}), 1<s<\infty, $ it is obvious that

$ {\biggl(\int_{2^{j+1}B}|\Omega(y-z)|^sdz\biggr)}^{\frac{1}{s}}\lesssim{\biggl(\int_{0}^{2^{j+1}r}l^{n-1}dl\int_{S^{n-1}}|\Omega(u)|^sdu\biggr)}^{\frac{1}{s}}\\ \lesssim\|\Omega\|_{L^s(S^{n-1})}|2^{j+1}B|^{\frac{1}{s}}. $

(3.4)

Thus from Hölder's inequality and (3.4), we have

$ \begin{array}{cl} &|\mu_{\Omega, S}f_2(x)|\\ &\leq{\biggl(\int_{2^{j-2}r}^{\infty}\int_{|x-y|<t}\biggl|\frac{1}{t}\sum\limits_{j=1}^{\infty}\int_{2^{j+1}B\setminus2^jB}\frac{\Omega(y-z)} {|y-z|^{n-1}}f(z)dz\biggr|^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}\\ &\lesssim{\biggl(\int_{2^{j-2}r}^{\infty}\int_{|x-y|<t}\biggl[\sum\limits_{j=1}^{\infty}\frac{1}{|2^{j+1}B|^{\frac{1}{{s}^{'}} -\frac{1}{n}}}\|\Omega\|_{L^s(S^{n-1})}\biggl(\int_{2^{j+1}B\setminus2^jB}|f(z)|^{{s}^{'}}dz\biggr)^{\frac{1}{{s}^{'}}}\biggr]^2\frac{dydt}{t^{n+1}}\biggr)}^{\frac{1}{2}}\\ &\lesssim\sum\limits_{j=1}^{\infty}|2^{j+1}B|^{\frac{1}{{s}^{'}} -\frac{1}{n}}{\biggl(\int_{2^{j+1}B}|f(z)|^{{s}^{'}}dz\biggr)^{\frac{1}{{s}^{'}}}}{\biggl(\int_{2^{j-2}r}^{\infty}\int_{|x-y|<t} \frac{dydt}{t^{n+3}}\biggr)}^{\frac{1}{2}}\\ &\lesssim\sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-\frac{n}{q}}\|f\|_{L^q(B(x_0, 2^{j+1}r))}\\ &\lesssim\int_{2r}^{\infty}\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}}. \end{array} $

(3.5)

So

$ \|\mu_{\Omega, S} f_2\|_{L^q(B)} \lesssim r^{\frac{n}{q}}\int_{2r}^\infty\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}}. $

(3.6)

Therefore combining (3.1) and (3.6), we have

$ \|\mu_{\Omega, S} f\|_{L^q(B)}\lesssim r^{\frac{n}{q}}\int_{2r}^\infty\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}}. $

Thus we complete the proof of Theorem 3.1.

Theorem 3.2 Let $\Omega\in L^s(\mathbb{S}^{n-1})~ (1<s\leq\infty)$ satisfy condition (1.1) and $\max\{2, s'\}<q<\infty.$ Then, if functions $\varphi, $ $\psi:$ $\mathbb{R}^n\times(0, \infty)\rightarrow(0, +\infty)$ satisfy the inequality

$ \begin{array}{cl}\int^\infty_{r}\frac{\mbox{ess}\inf\limits_{l<\tau<\infty}\varphi(x, \tau)\tau^\frac{n}{p}}{l^{\frac{n}{p}+1}}dl\leq C\psi(x_0, r), \end{array} $

(3.7)

where $C$ does not depend on $x$ and $r, $ the operator $\mu_{\Omega, S}$ is bounded from $LM_{p, \varphi}^{\{x_0\}}$ to $LM_{p, \psi}^{\{x_0\}}.$

Proof Taking $v_1(l)=\varphi(x_0, l)^{-1}l^{-\frac{n}{q}}, $ $v_2(l)=\psi(x_0, l)^{-1}, $ $g(l)=\|f\|_{L^q(B(x_0, l))}$ and $w(l)=l^{-\frac{n}{q}-1}, $ then from Theorem 3.1, we have

$ \mbox{ess}\sup\limits_{l>0}v_2(l)\int_{l}^{\infty}\dfrac{w(s)ds}{\mbox{ess}\sup\limits_{s<\tau<\infty}v_1(\tau)}<\infty. $

Thus from Lemma 2.2, it follows that

$ \mbox{ess}\sup\limits_{l>0}v_2(l)H_wg(l)\leq C \mbox{ess}\sup\limits_{l>0}v_1(l)g(l). $

Therefore

$ \begin{array}{cl} \|\mu_{\Omega, S} f\|_{LM_{q, \psi}^{\{x_0\}}} &=\sup\limits_{r>0}\psi(x_0, r)^{-1}|B(x_0, r)|^{-\frac{1}{q}}\|\mu_{\Omega, S}{f}\|_{L^{q}(B(x_0, r))}\\ &\lesssim \sup\limits_{r>0}\psi(x_0, r)^{-1} \int^\infty_{r}\|f\|_{L^q(B(x_0, l))}\frac{dl}{l^{\frac{n}{q}+1}}\\ &\lesssim \sup\limits_{r>0}\varphi(x_0, r)^{-1} r^{-\frac{n}{q}}\|f\|_{L^q(B(x_0, r)}=\|f\|_{LM_{q, \varphi}^{\{x_0\}}}. \end{array} $

Thus we complete the proof of Theorem 3.2.

Theorem 4.1 Let $\Omega\in L^s(\mathbb{S}^{n-1})~ (1<s\leq\infty)$ satisfy condition (1.1) and $\max\{2, s'\}<q<\infty.$ Let $1<p, q_1, q_2, \cdots, q_m<\infty, $ such that $\frac{1}{q}=\frac{1}{p}+\frac{1}{q_1}+\frac{1}{q_2}+\cdots+\frac{1}{q_m}, $ and $b_i\in LC_{q_i, \lambda_i}^{\{x_0\}}$ for $0\leq\lambda_i<\frac{1}{n}, i=1, 2, \cdots, m.$ Then the inequality

$ \|\mu_{\Omega, S}^{\vec{b}}f\|_{L^q(B(x_0, r))}\lesssim \prod\limits_{i=1}^{m}\|b_i\|_{LC^{\{x_0\}}_{p_i, \lambda_i}}r^\frac{n}{q}\int^{\infty}_{2r}\biggl(1+\ln\frac{l}{r}\biggr)^m\|f\|_{L^p(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-n\lambda}} $

holds for any ball $B(x_0, r), $ where $\lambda=\lambda_1+\lambda_2+\cdots+\lambda_m.$

Proof Without loss of generality, it is sufficient for us to show that the conclusion holds for $m=2.$

Let $B=B(x_0, r).$ And we write $f=f_1+f_2, $ where $f_1=f\chi_{2B}, $ $f_2=f\chi_{{(2B)}^c}.$ Thus we have

$ \begin{array}{cl} &\|\mu_{\Omega, S}^{(b_1, b_2)}f\|_{L^q(B)}\leq\|\mu_{\Omega, S}^{(b_1, b_2)}f_1\|_{L^q(B)}+\|\mu_{\Omega, S}^{(b_1, b_2)}f_2\|_{L^q(B)} =:I+II. \end{array} $

Let us estimate $I$ and $II, $ respectively. It is obvious that

$ \begin{array}{cl} &\|\mu_{\Omega, S}^{(b_1, b_2)}f_1\|_{L^{q}(B)}\\ =&\|(b_1-(b_1)_{B})(b_2-(b_2)_{B})\mu_{\Omega, S}f_1\|_{L^{q}(B)}+\|(b_1-(b_1)_{B})\mu_{\Omega, S}(b_2-(b_2)_{B})f_1\|_{L^{q}(B)}\\ &+\|(b_2-(b_2)_{B})\mu_{\Omega, S}(b_1-(b_1)_{B})f_1\|_{L^{q}(B)}+\|\mu_{\Omega, S}(b_1-(b_1)_{B})(b_2-(b_2)_{B})f_1\|_{L^{q}(B)}\\ =:&I_1+I_2+I_3+I_4. \end{array} $

From Lemma 2.1, it is easy to see that

$ \|b_i-(b_i)_{B}\|_{L^{q_i}(B)}\leq Cr^{\frac{n}{p_{i}}+n\lambda_i}\|b_i\|_{LC_{p_i, \lambda_i}^{\{x_0\}}}\;\mbox{for}\;i=1, 2. $

(4.1)

Since $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}+\frac{1}{p}$ and $\max\{2, s'\}<q<\infty.$ It is obvious that $\max\{2, s'\}<p<\infty.$ Thus using Hölder's inequality, Theorem 3.1 and (4.1), we have

$ \begin{array}{cl} I_1&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}\|\mu_{\Omega, S}f_1\|_{{L^{p}}(B)}\\ &\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}r^{\frac{n}{p}}\int^\infty_{2r}\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1}}\\ &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}} \int^{\infty}_{2r}{\biggl(1+\ln\frac{l}{r}\biggr)}^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

(4.2)

Moreover, from Lemma 2.1, it is easy to see that

$ \|b_i-(b_i)_{B}\|_{L^{p_i}(2B)}\leq Cr^{\frac{n}{p_{i}}+n\lambda_i}\|b_i\|_{LC_{p_i, \lambda_i}^{\{x_0\}}}~~\mbox{for}\;i=1, 2. $

(4.3)

And let $\frac{1}{\bar{q}}=\frac{1}{q_2}+\frac{1}{p}.$ Then it is easy to see that $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{\bar{q}}$ and $\max\{s', 2\}<\bar{q}<\infty.$ Then similarly to the estimate of (4.2), we have

$ \begin{array}{cl} I_2&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|\mu_{\Omega, S}(b_2-(b_2)_{B})f_1\|_{L^{\bar{q}}(B)}\\ &\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(2B)}\|f\|_{L^{p}(2B)}\\ &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int^\infty_{2r} \biggl(1+\ln\dfrac{l}{r}\biggr)^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

Similarly,

$ \begin{array}{cl} I_3 &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int^\infty_{2r} \biggl(1+\ln\dfrac{l}{r}\biggr)^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

Similarly, since $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}+\frac{1}{p}.$ Then, by Lemma 2.3, Hölder's inequality and (4.3), we obtain

$ \begin{array}{cl} I_4&=\|\mu_{\Omega, S}(b_1-(b_1))(b_2-(b_2)_{B})f_1\|_{L^{q}(B)}\\ &\lesssim\|(b_1-(b_1)_{B})(b_2-(b_2)_{B})f\|_{L^{q}(2B)}\\ &\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}\|f\|_{L^{p}(2B)}\\ &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}} \int^\infty_{2r}\biggl(1+\ln\dfrac{l}{r}\biggr)^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

Therefore combining the estimates of $I_1, I_2, I_3$ and $I_4, $ we have

$ \begin{array}{cl}I &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}} r^{\frac{n}{q}}\int^\infty_{2r}\biggl(1+\ln\dfrac{l}{r}\biggr)^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

Let us estimate $II.$

$ \begin{array}{cl} &\|\mu^{(b_1, b_2)}_{\Omega, S}f_2\|_{L^{q}(B)}\\ =&\|(b_1-(b_1)_{B})(b_2-(b_2)_{B})\mu_{\Omega, S}f_2\|_{L^{q}(B)}+\|(b_1-(b_1)_{B})\mu_{\Omega, S}(b_2-(b_2)_{B})f_2\|_{L^{q}(B)}\\ &+\|(b_2-(b_2)_{B})\mu_{\Omega, S}(b_1-(b_1)_{B})f_2\|_{L^{q}(B)}+\|\mu_{\Omega, S}(b_1-(b_1)_{B})(b_2-(b_2)_{B})f_2\|_{L^{q}(B)}\\ =:&II_1+II_2+II_3+II_4. \end{array} $

Since $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}+\frac{1}{p}.$ Then using Hölder's inequality and (3.6), we have

$ \begin{array}{cl} II_1&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}\|\mu_{\Omega, S}f_2\|_{L^{{p}}(B)}\\ &\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}r^{\frac{n}{p}}\int_{2r}^\infty\|f\|_{L^p(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1}}\\ &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int^\infty_{2r}{\biggl(1+\ln\dfrac{l}{r}\biggr)}^2 \|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

In the following, let us estimate $II_2.$ For $x\in B, $ when $\Omega\in L^{\infty}(\mathbb{S}^{n-1}), $ from Lemma 2.1 and estimate of (3.3), we have

$ \begin{array}{cl} &|\mu_{\Omega, S}(b_2-(b_2)_B)f_2(x)|\\ &\lesssim \sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-n}\int_{2^{j+1}B}|b_2(z)-(b_2)_{B(x_0, r)}||f(z)|dz\\ &\lesssim \sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-n}\|f\|_{L^{p}(B(x_0, 2^{j+1}r))}\|b_2-(b _2)_{B(x_0, r)}\|_{L^{q_2}(B(x_0, 2^{j+1}r))}|2^{j+1}B|^{1-\frac{1}{p}-\frac{1}{q_2}}\\ &\lesssim\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-n\lambda_2}}. \end{array} $

(4.4)

For $x\in B, $ when $\Omega\in L^s(\mathbb{S}^{n-1}), 1<s<\infty, $ from Lemma 2.1 and the estimate of (3.5), it follows that

$ \begin{array}{cl} &|\mu_{\Omega, S}(b_2-(b_2)_B)f_2(x)|\\ &\lesssim \sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-\frac{n}{s^{'}}}{\biggl(\int_{B(x_0, 2^{j+1}r)}(|b_2(z)-(b_2)_{B(x_0, r)}||f(z)|)^{s^{'}}dz\biggr)}^{\frac{1}{s^{'}}}\\ &\lesssim \sum\limits_{j=1}^{\infty}\|f\|_{L^{p}(B(x_0, 2^{j+1}r))}\|b_2-(b_2)_{{B(x_0, r)}}\|_{L^{q_2}(B(x_0, 2^{j+1}r))}|2^{j+1}B|^{-\frac{1}{p}-\frac{1}{q_2}}\\ &\lesssim\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-n\lambda_2}}. \end{array} $

(4.5)

Let $1<\tilde{q}<\infty$ such that $\frac{1}{\tilde{q}}=\frac{1}{p}+\frac{1}{q_2}, $ then $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{\tilde{q}}$ and $\max\{2, s'\}<\tilde{q}<\infty.$ Thus, from Hölder's inequality, (4.4) and (4.5), we obtain

$ \begin{array}{cl} II_2&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|\mu_{\Omega, S}(b_2-(b_2)_{B})f_2\|_{L^{\tilde{q}}(B)}\\ &\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{\tilde{q}}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-n\lambda_2}}\\ &\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int^\infty_{2r} \biggl(1+\ln\dfrac{l}{r}\biggr)^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

(4.6)

Similarly,

$ II_3\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int^\infty_{2r} \biggl(1+\ln\dfrac{l}{r}\biggr)^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. $

Let us estimate $II_4.$ It is analogue to the estimates of (4.4), (4.5) and (4.6), we have the following estimates.

When $x\in B, $ $\Omega\in L^{\infty}(\mathbb{S}^{n-1}), $ we have

$ \begin{array}{cl} &|\mu_{\Omega, S}(b_1-(b_1)_B)(b_2-(b_2)_B)f_2(x)|\\ \lesssim& \sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-n}\int_{2^{j+1}B}|b_1(z)-(b_1)_{B(x_0, r)}||b_2(z)-(b_2)_{B(x_0, r)}||f(z)|dz\\ \lesssim& \sum\limits_{j=1}^{\infty}\|f\|_{L^{p}(B(x_0, 2^{j+1}r))}\|b_1-(b _1)_{B(x_0, r)}\|_{L^{q_1}(B(x_0, 2^{j+1}r))} \|b_2-(b _2)_{B(x_0, r)}\|_{L^{q_2}(B(x_0, 2^{j+1}r))}\\ &|2^{j+1}B|^{-\frac{1}{q}}\\ \lesssim&\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

(4.7)

When $x\in B, $ $\Omega\in L^s(\mathbb{S}^{n-1}), 1<s<\infty, $ we have

$ \begin{array}{cl} &|\mu_{\Omega, S}(b_1-(b_1)_B)(b_2-(b_2)_B)f_2(x)|\\ \lesssim&\sum\limits_{j=1}^{\infty}(2^{j+1}r)^{-\frac{n}{s^{'}}}{\biggl(\int_{B(x_0, 2^{j+1}r)}(|b_1(z)-(b_1)_{B(x_0, r)}||b_2(z)-(b_2)_{B(x_0, r)}||f(z)|)^{s^{'}}dz\biggr)}^{\frac{1}{s^{'}}}\\ \lesssim&\sum\limits_{j=1}^{\infty}\|f\|_{L^{p}(B(x_0, 2^{j+1}r))}\|b_1-(b_1)_{{B(x_0, r)}}\|_{L^{q_1}(B(x_0, 2^{j+1}r))}\\ &\times\|b_2-(b_2)_{{B(x_0, r)}}\|_{L^{q_2}(B(x_0, 2^{j+1}r))}|2^{j+1}B|^{-\frac{1}{p}-\frac{1}{q_1}-\frac{1}{q_2}}\\ \lesssim&\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

(4.8)

Therefore from (4.7) and (4.8), we have

$ II_4\lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. $

So from the estimates of $II_1, II_2, II_3$ and $II_4, $ it follows that

$ II\lesssim \|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. $

Therefore from the estimates of $I$ and $II, $ we deduced that

$ \begin{array}{cl}\|\mu_{\Omega, S}^{(b_1, b_2)}f\|_{L^q(B(x_0, r))} \lesssim\|b_1\|_{LC_{q_1, \lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2, \lambda_2}^{\{x_0\}}}r^{\frac{n}{q}}\int_{2r}^{\infty}{\biggl(1+\ln\frac{l}{r}\biggr)}^2\|f\|_{L^{p}(B(x_0, l))}\frac{dl}{l^{\frac{n}{p}+1-(\lambda_1+\lambda_2)n}}. \end{array} $

Thus the proof of Theorem 4.1 is completed.

Theorem 4.2 Let $\Omega\in L^s(\mathbb{S}^{n-1})~ (1<s\leq\infty)$ satisfy condition (1.1) and $\max\{2, s'\}<q<\infty.$ Let $1<p, q_1, q_2, \cdots, p_m<\infty, $ such that $\frac{1}{q}=\frac{1}{p}+\frac{1}{q_1}+\frac{1}{q_2}+\cdots+\frac{1}{q_m}, $ and $b\in LC_{q_i, \lambda_i}^{\{x_0\}}$ for $0\leq\lambda_i<\frac{1}{n}, i=1, 2, \cdots, m.$ Then, if functions $\varphi, $ $\psi:$ $\mathbb{R}^n\times(0, \infty)\rightarrow(0, +\infty)$ satisfy the inequality

$ \int_{r}^{\infty}\biggl(1+\ln\frac{l}{r}\biggr)^{m}\frac{\mbox{ess}\inf\limits_{l<\tau<\infty}\varphi(x_0, \tau)\tau^{\frac{n}{p}}} {l^{\frac{n}{p}+1-n\lambda}}dl\leq C\psi(x_0, r), $

where $\lambda=\sum\limits_{i=1}^{m}\lambda_i$ and the constant $C>0$ doesn't depend on $r.$ Then $\mu_{\Omega, S}^{\vec{b}}$ is bounded from $LM_{p, \varphi}^{\{x_0\}}$ to $LM_{q, \psi}^{\{x_0\}}.$

Proof Taking $v_1(l)=\varphi(x_0, l)^{-1}l^{-\frac{n}{p}}, $ $v_2(l)=\psi(x_0, l)^{-1}, $ $g(l)=\|f\|_{L^{q}(B(x_0, l))}$ and $w(l)=(1+\ln\frac{l}{r})^{m}l^{n\lambda-\frac{n}{p}-1}.$ It is easy to see that

$ \mbox{ess}\sup\limits_{l>0}v_2(l)\int_{l}^{\infty}\dfrac{w(s)ds}{\mbox{ess}\sup\limits_{s<\tau<\infty}v_1(\tau)}<\infty. $

Thus by Lemma 2.2, we have

$ \mbox{ess}\sup\limits_{l>0}v_2(l)H_wg(l)\leq C \mbox{ess}\sup\limits_{l>0}v_1(l)g(l). $

So

$ \begin{array}{cl} &\|\mu_{\Omega, S}^{\vec{b}}{f}\|_{LM_{q, \psi}^{\{x_0\}}}\\ =&\sup\limits_{r>0}\psi(x_0, r)^{-1}|B(x_0, r)|^{-\frac{1}{q}}\|\mu_{\Omega, S}^{\vec{b}}f\|_{L^{q}(B(x_0, r))}\\ \lesssim&\prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i, \lambda_i}^{\{x_0\}}}\sup\limits_{r>0}\psi(x_0, r)^{-1} \int^{\infty}_{2r}\biggl(1+\ln\frac{l}{r}\biggr)^m l^{n\lambda-\frac{n}{p}-1}\|f\|_{L^{p}(B(x_0, l))}dl\\ \lesssim&\prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i, \lambda_i}^{\{x_0\}}}\sup\limits_{r>0}\varphi(x_0, r)^{-1}r^{-\frac{n}{p}}\|f\|_{L^{p}(B(x_0, r))}\\ =&\prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i, \lambda_i}^{\{x_0\}}}\|f\|_{LM_{p, \varphi}^{\{x_0\}}}. \end{array} $

Thus the proof of Theorem 4.2 is finished.