# Boundedness of homogeneous fractional integral operator on Morrey space

## 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.

## 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

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.  □

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## 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).

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Correspondence to Yanping Chen.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

MS carried out the boundedness of the homogeneous fractional integral operator on Morrey space studies and drafted the manuscript. YC participated in the study of fractional integral operator and helped to check the manuscript. All authors read and approved the final manuscript.

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