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 [1].

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 [4]), the Schrödinger equations (see [5] and [6]) and the elliptic problems with discontinuous coefficients (see [7–9] and [10]).

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 [11–15]. 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, 16–19].)

A function $g\in \operatorname {BMO}({\Bbb {R}}^{n})$ (see [20]), 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 [21] 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 [11].

Theorem A

(Adams) ([11])

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 [22] (for power weights) and in [23] (for $A(p,q)$ weights). The weak boundedness of $T_{\Omega,\alpha}$ when $p=1$ can be found in [24] (unweighed) and in [25] (with power weights). In 2002, Ding [26] 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) [3].

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

([26])

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.  □