Skip to content
• Research
• Open Access

# Boundedness of homogeneous fractional integral operator on Morrey space

Journal of Inequalities and Applications20162016:61

https://doi.org/10.1186/s13660-016-0999-y

• Received: 27 October 2015
• Accepted: 1 February 2016
• Published:

## Abstract

For $$0<\alpha<n$$, the homogeneous fractional integral operator $$T_{\Omega,\alpha}$$ is defined by
$$T_{\Omega,\alpha}f(x)= \int_{{\Bbb {R}}^{n}}\frac{\Omega (x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy.$$
In this paper we prove that if Ω satisfies some smoothness conditions on $$S^{n-1}$$, then $$T_{\Omega,\alpha}$$ is bounded from $$L^{\frac{\lambda}{\alpha },\lambda}({\Bbb {R}}^{n})$$ to $$\operatorname {BMO}({\Bbb {R}}^{n})$$, and from $$L^{p,\lambda}({\Bbb {R}}^{n})$$ ($$\frac{\lambda}{\alpha}< p<\infty$$) to a class of the Campanato spaces $$\mathcal{L}_{l,\lambda }({\Bbb {R}}^{n})$$, respectively.

## Keywords

• Morrey space
• Campanato space
• BMO space
• homogeneous fractional integral operator

• 42B20
• 42B25
• 47B35

## 1 Introduction

Before going into the next sections addressing details, let us agree to some conventions. The n-dimensional Euclidean space $${\Bbb {R}}^{n}$$, $$Q=Q(x_{0},d)$$ is a cube with its sides parallel to the coordinate axes and center at $$x_{0}$$, diameter $$d>0$$.

For $$1\leq l\leq\infty$$, $$-\frac{n}{l}\leq\lambda\leq1$$, we denote
$$\Vert f\Vert _{\mathcal{L}_{l,\lambda}}=\sup_{Q}\frac{1}{\vert Q\vert ^{\lambda /n}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert f(x)-f_{Q}\bigr\vert ^{l}\,dx \biggr)^{1/l},$$
where $$f_{Q}=\frac{1}{\vert Q\vert }\int_{Q}f(y)\,dy$$. Then the Campanato space $$\mathcal{L}_{l,\lambda}({\Bbb{R}}^{n})$$ is defined by
$$\mathcal{L}_{l,\lambda}\bigl({\Bbb{R}}^{n}\bigr)=\bigl\{ f\in L_{\mathrm{loc}}^{l}\bigl({\Bbb {R}}^{n}\bigr):\Vert f \Vert _{\mathcal{L}_{l,\lambda}}< \infty\bigr\} .$$
If we identify functions that differ by a constant, then $$\mathcal {L}_{l,\lambda}$$ becomes a Banach space with the norm $$\Vert \cdot \Vert _{\mathcal{L}_{l,\lambda}}$$. It is well known that
\begin{aligned} & \operatorname {Lip}_{\lambda}\bigl({\Bbb{R}}^{n}\bigr), \quad \mbox{for } 0< \lambda< 1, \\ \mathcal{L}_{l,\lambda}\bigl({\Bbb{R}}^{n}\bigr)\quad \sim\quad & \operatorname {BMO}\bigl({ \Bbb{R}}^{n}\bigr), \quad \mbox{for } \lambda=0, \\ & \mbox{Morrey space }L^{p,n+l\lambda}\bigl({\Bbb{R}}^{n}\bigr), \quad \mbox{for } -\!n/l\leq \lambda< 0. \end{aligned}
On the other properties of the spaces $$\mathcal{L}_{l,\lambda}({\Bbb {R}}^{n})$$, we refer the reader to .

The Morrey space, which was introduced by Morrey in 1938, connects with certain problems in elliptic PDE [2, 3]. Later, there were many applications of Morrey space to the Navier-Stokes equations (see ), the Schrödinger equations (see  and ) and the elliptic problems with discontinuous coefficients (see  and ).

For $$1\leq p<\infty$$ and $$0<\lambda\leq n$$, the Morrey space is defined by
$$L^{p,\lambda}\bigl({\Bbb {R}}^{n}\bigr)= \biggl\{ f\in L_{\mathrm{loc}}^{p}:\Vert f\Vert _{L^{p,\lambda }}= \biggl[\sup_{x\in{\Bbb {R}}^{n},\,d>0}d^{\lambda-n} \int_{Q(x,d)}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr]^{\frac {1}{p}}< \infty \biggr\} ,$$
where $$Q(x,d)$$ denotes the cube centered at x and with diameter $$d>0$$. The space $$L^{p,\lambda}({\Bbb {R}}^{n})$$ becomes a Banach space with norm $$\Vert \cdot \Vert _{L^{p,\lambda}}$$. Moreover, for $$\lambda=0$$ and $$\lambda=n$$, the Morrey spaces $$L^{p,0}({\Bbb {R}}^{n})$$ and $$L^{p,n}({\Bbb {R}}^{n})$$ coincide (with equality of norms) with the space $$L^{\infty}({\Bbb {R}}^{n})$$ and $$L^{p}({\Bbb {R}}^{n})$$, respectively.

The boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on Morrey space can be found in . It is well known that further properties and applications of the classical Morrey space have been widely studied by many authors. (For example, see [8, 1619].)

A function $$g\in \operatorname {BMO}({\Bbb {R}}^{n})$$ (see ), if there is a constant $$C>0$$ such that for any cube $$Q\in{\Bbb {R}}^{n}$$,
$$\Vert g\Vert _{\operatorname {BMO}}=\sup_{x\in{\Bbb {R}}^{n},\,r>0} \biggl( \frac{1}{\vert Q\vert }\int _{Q}\bigl\vert g(x)-g_{Q}\bigr\vert \,dx\biggr)< \infty,$$
where $$g_{Q}=\frac{1}{\vert Q\vert }\int_{Q}g(y)\,dy$$.
The Hardy-Littlewood-Sobolev theorem showed that the Riesz potential operator $$I_{\alpha}$$ is bounded from $$L^{p}( {\Bbb {R}}^{n})$$ to $$L^{q}( {\Bbb {R}}^{n})$$ for $$0<\alpha<n$$, $$1< p<\frac{n}{\alpha}$$, and $$\frac{1}{q}= \frac{1}{p}-\frac{\alpha}{n}$$. Here
$$I_{\alpha}f(x)=\frac{1}{\gamma(\alpha)} \int_{ {\Bbb {R}}^{n}}\frac {f(y)}{\vert x-y\vert ^{n-\alpha}}\,dy,\quad \mbox{and} \quad \gamma(\alpha)= \frac{\pi^{\frac{n}{2}}2^{\alpha}\Gamma (\alpha/2)}{ \Gamma(\frac{n-\alpha}{2})}.$$

In 1974, Muckenhoupt and Wheeden  gave the weighted boundedness of $$I_{\alpha}$$ from $$L^{\frac{n}{\alpha}}(w, {\Bbb {R}}^{n})$$ to $$\operatorname {BMO}_{v}( {\Bbb {R}}^{n})$$.

In 1975, Adams proved the following theorem in .

### Theorem A

(Adams) ()

Let $$\alpha\in(0,n)$$ and $$\lambda\in(0,n]$$, there is a constant $$C>0$$, such that, if $$1< p=\frac{\lambda}{\alpha}$$, then
$$\Vert I_{\alpha}f \Vert _{\operatorname {BMO}}\leq C\Vert f\Vert _{L^{p,\lambda}}.$$
On the other hand, many scholars have investigated the various map properties of the homogeneous fractional integral operator $$T_{\Omega ,\alpha}$$, which is defined by
$$T_{\Omega,\alpha}f(x)= \int_{ {\Bbb {R}}^{n}}\frac{\Omega (x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy,$$
where $$0<\alpha<n$$, Ω is homogeneous of degree zero on $${\Bbb {R}}^{n}$$ with $$\Omega\in L^{s}(S^{n-1})$$ ($$s\geq1$$) and $$S^{n-1}$$ denotes the unit sphere of $${\Bbb {R}}^{n}$$. For instance, the weighted $$(L^{p}, L^{q})$$-boundedness of $$T_{\Omega,\alpha}$$ for $$1< p<\frac{n}{\alpha}$$ had been studied in  (for power weights) and in  (for $$A(p,q)$$ weights). The weak boundedness of $$T_{\Omega,\alpha}$$ when $$p=1$$ can be found in  (unweighed) and in  (with power weights). In 2002, Ding  proved that $$T_{\Omega,\alpha}$$ is bounded from $$L ^{\frac{n}{\alpha}}( {\Bbb {R}}^{n})$$ to $$\operatorname {BMO}( {\Bbb {R}}^{n})$$ when Ω satisfies some smoothness conditions on $$S^{n-1}$$.

Inspired by the $$(L^{p,\lambda}({\Bbb {R}}^{n}), \operatorname {BMO}({\Bbb {R}}^{n}))$$-boundedness of Riesz potential integral operator $$I_{\alpha}$$ for $$p=\frac{\lambda}{\alpha}$$. We will prove the $$(L^{p,\lambda }({\Bbb {R}}^{n}), \operatorname {BMO}({\Bbb {R}}^{n}))$$-boundedness of homogeneous fractional integral operator $$T_{\Omega,\alpha}$$ for $$p=\frac{\lambda}{\alpha}$$. Then we find that $$T_{\Omega,\alpha}$$ is also bounded from $$L^{p,\lambda}({\Bbb {R}}^{n})$$ ($$\frac{\lambda }{\alpha }< p<\infty$$) to a class of the Campanato spaces $$\mathcal {L}_{l,\lambda }({\Bbb {R}}^{n})$$.

We say that Ω satisfies the $$L^{s}$$-Dini condition if Ω is homogeneous of degree zero on $${\Bbb {R}}^{n}$$ with $$\Omega\in L^{s}(S^{n-1})$$ ($$s\geq1$$), and
$$\int_{0}^{1}\omega_{s}(\delta) \frac{d \delta}{\delta}< \infty,$$
where $$\omega_{s}(\delta)$$ denotes the integral modulus of continuity of order s of Ω defined by
$$\omega_{s}(\delta)=\sup_{\vert \rho \vert < \delta} \biggl( \int _{S^{n-1}}\bigl\vert \Omega\bigl(\rho x' \bigr)-\Omega\bigl(x'\bigr)\bigr\vert ^{s} \,dx' \biggr)^{\frac{1}{s}},$$
and ρ is a rotation in $${\Bbb {R}}^{n}$$ and $$\vert \rho \vert =\Vert \rho-I\Vert$$.

Now, let us formulate our result as follows.

### Theorem 1.1

Let $$0<\alpha$$, $$\lambda< n$$, if Ω satisfies the $$L^{s}$$-Dini condition ($$s>1$$), then there is a constant $$C>0$$ such that
$$\Vert T_{\Omega,\alpha}f \Vert _{\operatorname {BMO}}\leq C\Vert f\Vert _{L^{\frac{\lambda}{\alpha},\lambda}}.$$
(1.1)

### Remark 1.2

If $$\Omega\equiv1$$, $$s=\infty$$, and $$\lambda=0$$, then $$T_{\Omega,\alpha}$$ is a Riesz potential $$I_{\alpha}$$, and Theorem 1.1 becomes Theorem A (Adams) .

The following theorem shows that $$T_{\Omega,\alpha}$$ is a bounded map from $$L^{p,\lambda}({\Bbb {R}}^{n})$$ ($$\frac{\lambda}{\alpha}< p<\infty$$) to the Campanato spaces $$\mathcal{L}_{l,\lambda}({\Bbb {R}}^{n})$$ for appropriate indices $$\lambda>0$$ and $$l\geq1$$.

### Theorem 1.3

Let $$0<\alpha<1$$, $$0<\lambda<n$$, $$\lambda /\alpha< p<\infty$$, and $$s>\lambda/(\lambda-\alpha)$$. If for some $$\beta>\alpha-\lambda/p$$, the integral modulus of continuity $$\omega_{s}(\delta)$$ of order s of Ω satisfies
$$\int_{0}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta }}< \infty,$$
then there is a $$C>0$$ such that for $$1\leq l\leq\lambda/(\lambda -\alpha)$$,
$$\Vert T_{\Omega,\alpha}f \Vert _{\mathcal{L}_{l,n(\frac{\alpha}{n}-\frac {1}{p}\frac{\lambda}{n})}}\leq C\Vert f\Vert _{L^{p,\lambda}}.$$
(1.2)

### Remark 1.4

If we take $$\Omega\equiv1$$, then $$T_{\Omega ,\alpha }$$ is the Riesz potential $$I_{\alpha}$$, and Theorem 1.3 is even new for the Riesz potential $$I_{\alpha}$$.

Below the letter ‘C’ will denote a constant not necessarily the same at each occurrence.

## 2 Proof of Theorem 1.1

In this section we will give the proof of Theorem 1.1. Let us recall the following conclusion.

### Lemma 2.1

()

Suppose that $$0<\alpha<n$$, $$s>1$$, Ω satisfies the $$L^{s}$$-Dini condition. There is a constant $$0< a_{0}<\frac{1}{2}$$ such that if $$\vert x\vert < a_{0}R$$, then
\begin{aligned} &\biggl( \int_{R< \vert y\vert < 2R}\biggl\vert \frac{\Omega(y-x)}{|y-x|^{n-\alpha }}-\frac {\Omega(y)}{ |y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac{1}{s}}\\ &\quad \leq CR^{n/s-(n-\alpha )} \biggl\{ \frac{\vert x\vert }{R}+ \int_{\vert x\vert /2R< \delta< \vert x\vert /R}\omega_{s}(\delta)\frac{d \delta }{\delta} \biggr\} . \end{aligned}

### Proof of Theorem 1.1

Fix a cube $$Q\subset{\Bbb {R}}^{n}$$, we denote the center and the diameter of Q by $$x_{0}$$ and d, respectively. We write
\begin{aligned} T_{\Omega,\alpha}f & = \int_{B}\frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha }}f(y)\,dy+ \int_{R^{n}\backslash B}\frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy \\ & :=T_{1}f(x)+T_{2}f(x), \end{aligned}
where $$B=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert < d\}$$. It is sufficient to prove (1.1) for $$T_{1}f(x)$$ and $$T_{2}f(x)$$, respectively.
First let us consider $$T_{1}f(x)$$. We have
\begin{aligned}[b] \frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert \,dx \leq{}&\frac {1}{\vert Q\vert } \int _{Q} \int_{B}\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\bigl\vert f(y)\bigr\vert \,dy \,dx \\ &{}+ \frac{1}{\vert Q\vert } \int_{Q} \biggl(\frac{1}{\vert Q\vert } \int_{Q} \int_{B}\frac {\vert \Omega(z-y)\vert }{\vert z-y\vert ^{n-\alpha}}\bigl\vert f(y)\bigr\vert \,dy \,dz \biggr)\,dx \\ \leq{}&\frac{2}{\vert Q\vert } \int_{B}\bigl\vert f(y)\bigr\vert \int_{Q}\frac{\vert \Omega (x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\,dx\,dy \\ \leq{}&\frac{2}{\vert Q\vert } \int_{B}\bigl\vert f(y)\bigr\vert \int_{\vert x-y\vert < 2d}\frac{\vert \Omega (x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\,dx\,dy. \end{aligned}
(2.1)
Note that $$\Omega(x')\in L^{s}(S^{n-1})$$, $$\Vert \Omega \Vert _{L^{s}(S^{n-1})}=(\int_{S^{n-1}}\vert \Omega(y')\vert ^{s}\,d\sigma(y'))^{\frac {1}{s}}$$, we get
\begin{aligned}[b] \int_{\vert x-y\vert < 2d}\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\,dx & \leq Cd^{\alpha} \Vert \Omega \Vert _{L^{s}(S^{n-1})} \\ & \leq C\vert Q\vert ^{\frac{\alpha}{n}} \Vert \Omega \Vert _{L^{s}(S^{n-1})}. \end{aligned}
(2.2)
On the other hand, since $$p'<\frac{1}{\frac{1}{p}(\frac{\lambda }{n}-1)-\frac{\alpha}{n}}$$, by using the Hölder inequality, we get
\begin{aligned} \int_{B}\bigl\vert f(y)\bigr\vert \,dy & \leq \biggl( \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{B}1^{p'}\,dy \biggr)^{\frac{1}{p'}} \\ &\leq \biggl( \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \biggl( \int _{B}1^{\frac{1}{\frac{1}{p}(\frac{\lambda}{n}-1) -\frac{\alpha}{n}}}\,dy \biggr)^{{\frac{1}{p}(\frac{\lambda}{n}-1)-\frac{\alpha}{n}}} \\ & =\vert B\vert ^{{\frac{1}{p}(\frac{\lambda}{n}-1)-\frac{\alpha}{n}}} \biggl( \int _{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \\ & =\vert B\vert ^{{\frac{1}{p}(\frac{\lambda}{n}-1)-\frac{\alpha }{n}}}\vert B\vert ^{{\frac {1}{p}(1-\frac{\lambda}{n})}} \biggl(d^{\lambda-n} \int _{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr) ^{\frac{1}{p}} \\ & = \vert B\vert ^{-\frac{\alpha}{n}} \Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
(2.3)
Here and below we denote $$p=\frac{\lambda}{\alpha}$$ in the proof of Theorem 1.1. Plugging (2.2) and (2.3) into (2.1), we obtain
\begin{aligned}[b] \frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert \,dx &\leq C\vert Q\vert ^{\frac{\alpha}{n}} \vert B\vert ^{-\frac{\alpha}{n}} \Vert f \Vert _{L^{p,\lambda }} \\ &\leq C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
(2.4)
Now, let us turn to the estimate for $$T_{2}f(x)$$. In this case we have
\begin{aligned} &\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert \,dx \\ &\quad =\frac{1}{|Q|} \int _{Q}\biggl\vert \frac{1}{|Q|} \int_{Q} \biggl\{ \int_{|y-x_{0}|\geq d}f(y) \biggl[\frac{\Omega(x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr]\,dy \biggr\} \,dz\biggr\vert \,dx \\ &\quad \leq\frac{1}{\vert Q\vert } \int_{Q}\frac{1}{\vert Q\vert } \int_{Q} \Biggl\{ \sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha }} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \,dy \Biggr\} \,dz\,dx. \end{aligned}
(2.5)
By Hölder’s inequality, we get
\begin{aligned}[b] & \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \,dy \\ &\quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}} \\ & \qquad {}\times \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac {\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac{1}{s}}. \end{aligned}
(2.6)
Since
\begin{aligned} \biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert & =\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{o}|^{n-\alpha}}+\frac{\Omega (y-x_{0})}{|y-x_{o}|^{n-\alpha}} - \frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \\ & \leq\biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{o}|^{n-\alpha}}\biggr\vert +\biggl\vert \frac{\Omega(y-x_{0})}{|y-x_{o}|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert , \end{aligned}
we have
\begin{aligned} & \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ &\quad \leq \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{0}|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ & \qquad {}+ \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (z-y)}{|z-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{0}|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ & \quad :=J_{1}+J_{2}. \end{aligned}
(2.7)
Let us give the estimates of $$J_{1}$$ and $$J_{2}$$, respectively. We write $$J_{1}$$ as
$$J_{1}= \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega ((x-x_{0})-y)}{|x-x_{0}-y|^{n-\alpha}} -\frac{\Omega(y)}{|y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac{1}{s}}.$$
Note that $$x\in Q$$, if taking $$R=2^{j}d$$, then $$\vert x-x_{0}\vert <\frac{1}{2^{j+1}}R$$. Applying Lemma 2.1 to $$J_{1}$$, we get
\begin{aligned}[b] J_{1}& \leq C\bigl(2^{j}d \bigr)^{n/s-(n-\alpha)} \biggl\{ \frac{\vert x-x_{0}\vert }{2^{j}d}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta )\frac{d \delta}{\delta} \biggr\} \\ & \leq C\bigl(2^{j}d\bigr)^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j+1}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta )\frac{d \delta}{\delta} \biggr\} . \end{aligned}
(2.8)
By $$z\in Q$$ and using a similar method, we have
$$J_{2}\leq C\bigl(2^{j}d\bigr)^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j+1}}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta )\frac{d \delta}{\delta} \biggr\} .$$
(2.9)
Since $$p=\frac{\lambda}{\alpha}$$ and $$\frac{n}{s}-(n-\alpha )<-\frac {n}{s'(p/s')'}$$, we get
$$\bigl(2^{j}d\bigr)^{n/s-(n-\alpha)}\leq C\bigl\vert 2^{j+1} \sqrt{n}Q\bigr\vert ^{-\frac{1}{s'(p/s')'}},$$
where $$2^{j+1}\sqrt{n}Q$$ denote the cube with the center at $$x_{0}$$ and the diameter $$2^{j+1}\sqrt{n}d$$.
Thus, plugging (2.8) and (2.9) into (2.7), we have
\begin{aligned}[b] & \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ &\quad \leq C\bigl\vert 2^{j+1}\sqrt{n}Q\bigr\vert ^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j}} + \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \\ & \qquad {}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \biggr\} \\ &\quad \leq C\bigl\vert 2^{j+1}\sqrt{n}Q\bigr\vert ^{-\frac{1}{s'(p/s')'}} \biggl\{ \frac {1}{2^{j}} + \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \\ & \qquad {}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \biggr\} . \end{aligned}
(2.10)
On the other hand, we estimate $$(\int_{2^{j}d\leq \vert y-x_{0}\vert <2^{j+1}d}\vert f(y)\vert ^{s'}\,dy )^{\frac{1}{s'}}$$, since $$p'< s'(\frac{p}{s'})'$$ and $$s'(\frac{p}{s'})'<\frac{1}{\frac {1}{p}(\frac {\lambda}{n}-1)+\frac{1}{(p/s'){'}s'}}$$, by using Hölder’s inequality again, we get
\begin{aligned} & \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac {1}{s'}} \\ & \quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d}1^{s'(\frac{p}{s'})}{'}\,dy \biggr)^{\frac {1}{(p/s'){'}s'}} \\ &\quad \leq C \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d}1^{\frac{1}{\frac{1}{p}(\frac{\lambda }{n}-1)+\frac{1}{(p/s'){'}s'}}}\,dy \biggr)^{\frac{1}{p}(\frac{\lambda }{n}-1)+\frac{1}{(p/s'){'}s'}} \\ &\quad \leq C\vert B_{1}\vert ^{\frac{1}{p}(\frac{\lambda}{n}-1)+\frac {1}{(p/s'){'}s'}}\vert B_{1} \vert ^{\frac{1}{p}(1-\frac{\lambda}{n})} \biggl(\bigl(2^{j+1}d\bigr)^{\lambda-n} \int_{\vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \\ & \quad \leq C\vert B_{1}\vert ^{\frac{1}{(p/s'){'}s'}}\Vert f\Vert _{L^{p,\lambda}}, \end{aligned}
(2.11)
where $$B_{1}=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert <2^{j+1}d\}$$.
Plugging (2.10) and (2.11) into (2.6) we obtain
\begin{aligned} &\sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha}}-\frac{\Omega (z-y)}{|z-y|^{n-\alpha }}\biggr\vert \,dy \\ &\quad \leq C \sum_{j=0}^{\infty} \Vert f\Vert _{L^{p,\lambda}} \vert B_{1}\vert ^{\frac {1}{(p/s'){'}s'}}\bigl\vert 2^{j+1}\sqrt{n}Q\bigr\vert ^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta < \vert x-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \\ &\qquad {} + \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \biggr\} \\ & \quad \leq C\Vert f\Vert _{L^{p,\lambda}}\sum_{j=0}^{\infty} \vert B_{1}\vert ^{\frac {1}{(p/s'){'}s'}}\vert B_{1}\vert ^{-\frac{1}{(p/s'){'}s'}} \biggl\{ \frac{1}{2^{j}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta < \vert x-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \\ & \qquad {}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \biggr\} \\ & \quad \leq C\Vert f\Vert _{L^{p,\lambda}} \biggl\{ 2+2 \int_{0}^{1}\omega_{s}(\delta ) \frac {d\delta}{\delta} \biggr\} \\ & \quad \leq C \Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
(2.12)
Therefore, applying (2.12) into (2.5) we obtain
$$\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert \,dx\leq C\Vert f\Vert _{L^{p,\lambda}}.$$
(2.13)
Combining (2.4) and (2.13), we get
\begin{aligned} \Vert T_{\Omega,\alpha}f \Vert _{\operatorname {BMO}} & =\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{\Omega ,\alpha}f(y) -(T_{\Omega,\alpha}f)_{Q}\bigr\vert \,dy \\ & \leq\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(y) -(T_{1}f)_{Q}\bigr\vert \,dy+\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(y) -(T_{2}f)_{Q}\bigr\vert \,dy \\ & \leq C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
Thus, we complete the proof of Theorem 1.1. □

## 3 Proof of Theorem 1.3

Similarly to the proof of Theorem 1.1. We need only to prove (1.2) for $$T_{1}$$ and $$T_{2}$$, respectively. First let us consider $$T_{1}f(x)$$. We have
\begin{aligned} &\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad \leq\frac{2}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)\bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad =\frac{2}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q} \biggl\vert \int_{B} \frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy \biggr\vert ^{l}\,dx \biggr)^{\frac {1}{l}} \\ &\quad \leq\frac{2}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}}\frac {1}{\vert Q\vert ^{\frac{1}{l}}} \int_{B}\bigl\vert f(y)\bigr\vert \biggl( \int_{\vert y-x\vert < 2d} \biggl(\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha }} \biggr)^{l}\,dx \biggr)^{\frac{1}{l}}\,dy. \end{aligned}
(3.1)
Note that $$\Omega(x')\in L^{s}(S^{n-1})$$, $$\Vert \Omega \Vert _{L^{s}(S^{n-1})}=(\int_{S^{n-1}}\vert \Omega(y')\vert ^{s}\,d\sigma(y'))^{\frac {1}{s}}$$, and $$s>\frac{\lambda}{\lambda-\alpha}\geq l$$, hence
\begin{aligned}[b] \biggl( \int_{\vert y-x\vert < 2d} \biggl(\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha }} \biggr)^{l}\,dx \biggr)^{\frac{1}{l}} & \leq Cd^{\frac{n}{l}-(n-\alpha)}\Vert \Omega \Vert _{L^{s}(S^{n-1})} \\ &\leq C\vert Q\vert ^{\frac{1}{l}-(1-\frac{\alpha}{n})}\Vert \Omega \Vert _{L^{s}(S^{n-1})}. \end{aligned}
(3.2)
On the other hand, by Hölder’s inequality,
\begin{aligned} \int_{B}\bigl\vert f(y)\bigr\vert \,dy \leq {}&\biggl( \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{B}1^{p'}\,dy \biggr)^{\frac{1}{p'}} \\ ={}&\vert B\vert ^{\frac{1}{p}(1-\frac{\lambda}{n})} \biggl(\vert B\vert ^{\frac{\lambda }{n}-1} \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \\ &{}\times\biggl( \int_{B}1^{\frac{1}{1-\frac{\lambda}{n}(1-\frac {1}{p'})+\frac {1}{p}(\frac{\lambda}{n}-1)}}\,dy \biggr)^{ 1-\frac{\lambda}{n}(1-\frac{1}{p'})+\frac{1}{p}(\frac{\lambda }{n}-1)} \\ \leq{}&\vert B\vert ^{1-\frac{\lambda}{n}(1-\frac{1}{p'})+\frac{1}{p}(\frac {\lambda }{n}-1)}\vert B\vert ^{\frac{1}{p}(1-\frac{\lambda}{n})} \Vert f \Vert _{L^{p,\lambda}} \\ ={}&\vert B\vert ^{1-\frac{\lambda}{n}(1-\frac{1}{p'})}\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
(3.3)
Plugging (3.2) and (3.3) into (3.1) we get
\begin{aligned}[b] &\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad \leq C\vert Q\vert ^{\frac{\lambda}{n}\frac{1}{p}-\frac{\alpha}{n}-\frac {1}{l}+1-\frac{\lambda}{n}(1-\frac{1}{p'})+\frac{1}{l}-(1-\frac {\alpha}{n})} \Vert \Omega \Vert _{L^{s}(S^{n-1})} \Vert f\Vert _{L^{p,\lambda}} \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
(3.4)
Now, let us turn to the estimate for $$T_{2}f(x)$$. In this case we have
\begin{aligned}[b] &\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad =\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \Biggl(\frac{1}{\vert Q\vert } \int_{Q} \Biggl\vert \frac{1}{|Q|} \int_{Q} \Biggl\{ \sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}f(y)\\ &\qquad {}\times \biggl[\frac{\Omega (x-y)}{\vert x-y\vert ^{n-\alpha }} -\frac{\Omega(z-y)}{\vert z-y\vert ^{n-\alpha}} \biggr]\,dy \Biggr\} \,dz\Biggr\vert ^{l}\,dx \Biggr)^{\frac{1}{l}}. \end{aligned}
(3.5)
By Hölder’s inequality and the proof of Theorem 1.1, $$s'<\frac {\lambda}{\alpha}<p$$,
\begin{aligned} & \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \,dy \\ &\quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac {1}{s'}}(J_{1}+J_{2}) \\ &\quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d} 1^{s'(p/s')'}\,dy \biggr)^{\frac{1}{s'(p/s')'}}(J_{1}+J_{2}) \\ &\quad =\bigl(2^{j+1}d\bigr)^{-\frac{\lambda-n}{p}} \biggl[\bigl(2^{j+1}d \bigr)^{\lambda-n} \int _{\vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr]^{\frac{1}{p}} \\ &\qquad {}\times \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d} 1^{\frac{1}{\frac{1}{s'(p/s')'}+\frac{\lambda-n}{np}+\frac {1}{p}(1-\frac {\lambda}{n})}}\,dy \biggr)^{\frac{1}{s'(p/s')'} +\frac{\lambda-n}{np}+\frac{1}{p}(1-\frac{\lambda }{n})}(J_{1}+J_{2}) \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}} \vert B_{1}\vert ^{-\frac{\lambda -n}{np}}\vert B_{1}\vert ^{\frac{1}{s'(p/s')'} +\frac{\lambda-n}{np}+\frac{1}{p}(1-\frac{\lambda }{n})}(J_{1}+J_{2}) \\ &\quad =C\Vert f\Vert _{L^{p,\lambda}}\bigl(2^{j}d\bigr)^{\frac{n}{s'(p/s')'} +\frac{n}{p}(1-\frac{\lambda}{n})}(J_{1}+J_{2}) \\ &\quad =C\Vert f\Vert _{L^{p,\lambda}}\bigl(2^{j}d\bigr)^{\frac{n}{s'(p/s')'}}2^{j(\frac {n}{p}-\frac{\lambda}{p})} \vert Q\vert ^{\frac{1}{p}-\frac{1}{p} \frac{\lambda}{n}}(J_{1}+J_{2}), \end{aligned}
(3.6)
where $$B_{1}=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert <2^{j+1}d\}$$.
Since the integral modulus of continuity $$\omega_{s}(\delta)$$ of order s of Ω satisfies (1.2) and
$$\int_{o}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta}< \int _{o}^{1}\omega _{s}(\delta) \frac{d\delta}{\delta^{1+\beta}} < \infty,$$
we know that Ω satisfies also the $$L^{s}$$-Dini condition. From Lemma 2.1 and the proof of Theorem 1.1, we get
\begin{aligned}[b] J_{1}+J_{2}\leq{}& C \bigl(2^{j}d\bigr)^{n/s-(n-\alpha)}\times \biggl\{ \frac{1}{2^{j}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta < \vert x-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \\ &{}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \biggr\} . \end{aligned}
(3.7)
Note that
\begin{aligned}[b] \bigl(2^{j}d\bigr)^{n/[s'(p/s')']+n/s-(n-\alpha)}2^{j(\frac{n}{p}-\frac {\lambda }{p})}&= \bigl(2^{j}d\bigr)^{n(\frac{\alpha}{n}-\frac{1}{p})} 2^{j(\frac{n}{p}-\frac{\lambda}{p})} \\ &\leq C\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}}2^{jn(\frac{\alpha }{n}-\frac {1}{p})+j(\frac{n}{p}-\frac{\lambda}{p})} \\ &=C\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}}2^{jn(\frac{\alpha}{n}-\frac {1}{p}\frac{\lambda}{n})}. \end{aligned}
(3.8)
Moreover,
\begin{aligned} 2^{jn(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})} \int _{\vert x-x_{0}\vert /2^{j+1}d}^{\vert x-x_{0}\vert / 2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} &\leq2^{jn(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n})}\bigl(\vert x-x_{0}\vert /2^{j}d \bigr)^{\beta} \int_{\vert x-x_{0}\vert /2^{j+1}d}^{\vert x-x_{0}\vert / 2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta}} \\ &\leq C2^{j[n(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})-\beta ]} \int _{0}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta}}. \end{aligned}
(3.9)
By $$0<\alpha<1$$ and $$\beta>\alpha-\frac{\lambda}{p}$$, we have $$n(\frac {\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})-1<0$$ and $$n(\frac{\alpha }{n}-\frac{1}{p}\frac{\lambda}{n})-\beta<0$$, respectively. Applying (3.7), (3.8), and (3.9) to (3.6) we obtain
\begin{aligned} &\sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}f(y) \biggl[\frac {\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}} -\frac{\Omega(z-y)}{\vert z-y\vert ^{n-\alpha}} \biggr]\,dy \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}} \vert Q\vert ^{\frac{1}{p}-\frac{1}{p}\frac {\lambda }{n}}\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}} \sum_{j=0}^{\infty} \biggl\{ 2^{j[n(\frac{\alpha}{n}-\frac {1}{p}\frac {\lambda}{n})-1]} +C2^{j[n(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})-\beta ]} \int _{0}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta}} \biggr\} \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}} \vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac {\lambda}{n}}. \end{aligned}
(3.10)
Plugging (3.10) into (3.5), we obtain
$$\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac {1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}}\leq C\Vert f\Vert _{L^{p,\lambda}}.$$
(3.11)
Then by (3.4) and (3.11) we get
\begin{aligned} \Vert T_{\Omega,\alpha}f \Vert _{\mathcal{L}_{l,n(\frac{\alpha}{n}-\frac {1}{p}\frac{\lambda}{n})}} \leq{}&\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &{}+\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ \leq{}& C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}
Thus, we complete the proof of Theorem 1.3.  □

## Declarations

### Acknowledgements

The authors would like to express their deep gratitude to the referee for giving many valuable suggestions. The research was supported by NSF of China (Grant: 11471033), NCET of China (Grant: NCET-11-0574), the Fundamental Research Funds for the Central Universities (FRF-TP-12-006B).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

(1)
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

## References

Advertisement 