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 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 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 On the other properties of the spaces \(\mathcal{L}_{l,\lambda}({\Bbb {R}}^{n})\), we refer the reader to [1].
$$\Vert f\Vert _{\mathcal{L}_{l,\lambda}}=\sup_{Q}\frac{1}{\vert Q\vert ^{\lambda /n}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert f(x)-f_{Q}\bigr\vert ^{l}\,dx \biggr)^{1/l}, $$
$$\mathcal{L}_{l,\lambda}\bigl({\Bbb{R}}^{n}\bigr)=\bigl\{ f\in L_{\mathrm{loc}}^{l}\bigl({\Bbb {R}}^{n}\bigr):\Vert f \Vert _{\mathcal{L}_{l,\lambda}}< \infty\bigr\} . $$
$$\begin{aligned} & \operatorname {Lip}_{\lambda}\bigl({\Bbb{R}}^{n}\bigr), \quad \mbox{for } 0< \lambda< 1, \\ \mathcal{L}_{l,\lambda}\bigl({\Bbb{R}}^{n}\bigr)\quad \sim\quad & \operatorname {BMO}\bigl({ \Bbb{R}}^{n}\bigr), \quad \mbox{for } \lambda=0, \\ & \mbox{Morrey space }L^{p,n+l\lambda}\bigl({\Bbb{R}}^{n}\bigr), \quad \mbox{for } -\!n/l\leq \lambda< 0. \end{aligned}$$
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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 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.
$$L^{p,\lambda}\bigl({\Bbb {R}}^{n}\bigr)= \biggl\{ f\in L_{\mathrm{loc}}^{p}:\Vert f\Vert _{L^{p,\lambda }}= \biggl[\sup_{x\in{\Bbb {R}}^{n},\,d>0}d^{\lambda-n} \int_{Q(x,d)}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr]^{\frac {1}{p}}< \infty \biggr\} , $$
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 \(Q\in{\Bbb {R}}^{n}\), where \(g_{Q}=\frac{1}{\vert Q\vert }\int_{Q}g(y)\,dy\).
$$\Vert g\Vert _{\operatorname {BMO}}=\sup_{x\in{\Bbb {R}}^{n},\,r>0} \biggl( \frac{1}{\vert Q\vert }\int _{Q}\bigl\vert g(x)-g_{Q}\bigr\vert \,dx\biggr)< \infty, $$
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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 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 \(S^{n-1}\).
$$T_{\Omega,\alpha}f(x)= \int_{ {\Bbb {R}}^{n}}\frac{\Omega (x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy, $$
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 where \(\omega_{s}(\delta)\) denotes the integral modulus of continuity of order s of Ω defined by and ρ is a rotation in \({\Bbb {R}}^{n}\) and \(\vert \rho \vert =\Vert \rho-I\Vert \).
$$\int_{0}^{1}\omega_{s}(\delta) \frac{d \delta}{\delta}< \infty, $$
$$\omega_{s}(\delta)=\sup_{\vert \rho \vert < \delta} \biggl( \int _{S^{n-1}}\bigl\vert \Omega\bigl(\rho x' \bigr)-\Omega\bigl(x'\bigr)\bigr\vert ^{s} \,dx' \biggr)^{\frac{1}{s}}, $$
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
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
then there is a
\(C>0\)
such that for
\(1\leq l\leq\lambda/(\lambda -\alpha)\),
$$\int_{0}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta }}< \infty, $$
$$ \Vert T_{\Omega,\alpha}f \Vert _{\mathcal{L}_{l,n(\frac{\alpha}{n}-\frac {1}{p}\frac{\lambda}{n})}}\leq C\Vert f\Vert _{L^{p,\lambda}}. $$
(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 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.
$$\begin{aligned} T_{\Omega,\alpha}f & = \int_{B}\frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha }}f(y)\,dy+ \int_{R^{n}\backslash B}\frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy \\ & :=T_{1}f(x)+T_{2}f(x), \end{aligned}$$
First let us consider \(T_{1}f(x)\). We have 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 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 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 Now, let us turn to the estimate for \(T_{2}f(x)\). In this case we have By Hölder’s inequality, we get Since we have Let us give the estimates of \(J_{1}\) and \(J_{2}\), respectively. We write \(J_{1}\) as 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 By \(z\in Q\) and using a similar method, we have Since \(p=\frac{\lambda}{\alpha}\) and \(\frac{n}{s}-(n-\alpha )<-\frac {n}{s'(p/s')'}\), we get where \(2^{j+1}\sqrt{n}Q\) denote the cube with the center at \(x_{0}\) and the diameter \(2^{j+1}\sqrt{n}d\).
$$ \begin{aligned}[b] \frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert \,dx \leq{}&\frac {1}{\vert Q\vert } \int _{Q} \int_{B}\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\bigl\vert f(y)\bigr\vert \,dy \,dx \\ &{}+ \frac{1}{\vert Q\vert } \int_{Q} \biggl(\frac{1}{\vert Q\vert } \int_{Q} \int_{B}\frac {\vert \Omega(z-y)\vert }{\vert z-y\vert ^{n-\alpha}}\bigl\vert f(y)\bigr\vert \,dy \,dz \biggr)\,dx \\ \leq{}&\frac{2}{\vert Q\vert } \int_{B}\bigl\vert f(y)\bigr\vert \int_{Q}\frac{\vert \Omega (x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\,dx\,dy \\ \leq{}&\frac{2}{\vert Q\vert } \int_{B}\bigl\vert f(y)\bigr\vert \int_{\vert x-y\vert < 2d}\frac{\vert \Omega (x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\,dx\,dy. \end{aligned} $$
(2.1)
$$ \begin{aligned}[b] \int_{\vert x-y\vert < 2d}\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha}}\,dx & \leq Cd^{\alpha} \Vert \Omega \Vert _{L^{s}(S^{n-1})} \\ & \leq C\vert Q\vert ^{\frac{\alpha}{n}} \Vert \Omega \Vert _{L^{s}(S^{n-1})}. \end{aligned} $$
(2.2)
$$\begin{aligned} \int_{B}\bigl\vert f(y)\bigr\vert \,dy & \leq \biggl( \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{B}1^{p'}\,dy \biggr)^{\frac{1}{p'}} \\ &\leq \biggl( \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \biggl( \int _{B}1^{\frac{1}{\frac{1}{p}(\frac{\lambda}{n}-1) -\frac{\alpha}{n}}}\,dy \biggr)^{{\frac{1}{p}(\frac{\lambda}{n}-1)-\frac{\alpha}{n}}} \\ & =\vert B\vert ^{{\frac{1}{p}(\frac{\lambda}{n}-1)-\frac{\alpha}{n}}} \biggl( \int _{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \\ & =\vert B\vert ^{{\frac{1}{p}(\frac{\lambda}{n}-1)-\frac{\alpha }{n}}}\vert B\vert ^{{\frac {1}{p}(1-\frac{\lambda}{n})}} \biggl(d^{\lambda-n} \int _{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr) ^{\frac{1}{p}} \\ & = \vert B\vert ^{-\frac{\alpha}{n}} \Vert f\Vert _{L^{p,\lambda}}. \end{aligned}$$
(2.3)
$$ \begin{aligned}[b] \frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert \,dx &\leq C\vert Q\vert ^{\frac{\alpha}{n}} \vert B\vert ^{-\frac{\alpha}{n}} \Vert f \Vert _{L^{p,\lambda }} \\ &\leq C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned} $$
(2.4)
$$\begin{aligned} &\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert \,dx \\ &\quad =\frac{1}{|Q|} \int _{Q}\biggl\vert \frac{1}{|Q|} \int_{Q} \biggl\{ \int_{|y-x_{0}|\geq d}f(y) \biggl[\frac{\Omega(x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr]\,dy \biggr\} \,dz\biggr\vert \,dx \\ &\quad \leq\frac{1}{\vert Q\vert } \int_{Q}\frac{1}{\vert Q\vert } \int_{Q} \Biggl\{ \sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha }} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \,dy \Biggr\} \,dz\,dx. \end{aligned}$$
(2.5)
$$ \begin{aligned}[b] & \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \,dy \\ &\quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}} \\ & \qquad {}\times \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac {\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac{1}{s}}. \end{aligned} $$
(2.6)
$$\begin{aligned} \biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert & =\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{o}|^{n-\alpha}}+\frac{\Omega (y-x_{0})}{|y-x_{o}|^{n-\alpha}} - \frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \\ & \leq\biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{o}|^{n-\alpha}}\biggr\vert +\biggl\vert \frac{\Omega(y-x_{0})}{|y-x_{o}|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert , \end{aligned}$$
$$\begin{aligned} & \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ &\quad \leq \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{0}|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ & \qquad {}+ \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (z-y)}{|z-y|^{n-\alpha}} -\frac{\Omega(y-x_{0})}{|y-x_{0}|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ & \quad :=J_{1}+J_{2}. \end{aligned}$$
(2.7)
$$J_{1}= \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega ((x-x_{0})-y)}{|x-x_{0}-y|^{n-\alpha}} -\frac{\Omega(y)}{|y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac{1}{s}}. $$
$$ \begin{aligned}[b] J_{1}& \leq C\bigl(2^{j}d \bigr)^{n/s-(n-\alpha)} \biggl\{ \frac{\vert x-x_{0}\vert }{2^{j}d}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta )\frac{d \delta}{\delta} \biggr\} \\ & \leq C\bigl(2^{j}d\bigr)^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j+1}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta )\frac{d \delta}{\delta} \biggr\} . \end{aligned} $$
(2.8)
$$ J_{2}\leq C\bigl(2^{j}d\bigr)^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j+1}}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta )\frac{d \delta}{\delta} \biggr\} . $$
(2.9)
$$\bigl(2^{j}d\bigr)^{n/s-(n-\alpha)}\leq C\bigl\vert 2^{j+1} \sqrt{n}Q\bigr\vert ^{-\frac{1}{s'(p/s')'}}, $$
Thus, plugging (2.8) and (2.9) into (2.7), we have 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 where \(B_{1}=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert <2^{j+1}d\}\).
$$ \begin{aligned}[b] & \biggl( \int_{ 2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}} \biggr\vert ^{s}\,dy \biggr)^{\frac {1}{s}} \\ &\quad \leq C\bigl\vert 2^{j+1}\sqrt{n}Q\bigr\vert ^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j}} + \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \\ & \qquad {}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \biggr\} \\ &\quad \leq C\bigl\vert 2^{j+1}\sqrt{n}Q\bigr\vert ^{-\frac{1}{s'(p/s')'}} \biggl\{ \frac {1}{2^{j}} + \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta< \vert x-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \\ & \qquad {}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \biggr\} . \end{aligned} $$
(2.10)
$$\begin{aligned} & \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac {1}{s'}} \\ & \quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d}1^{s'(\frac{p}{s'})}{'}\,dy \biggr)^{\frac {1}{(p/s'){'}s'}} \\ &\quad \leq C \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d}1^{\frac{1}{\frac{1}{p}(\frac{\lambda }{n}-1)+\frac{1}{(p/s'){'}s'}}}\,dy \biggr)^{\frac{1}{p}(\frac{\lambda }{n}-1)+\frac{1}{(p/s'){'}s'}} \\ &\quad \leq C\vert B_{1}\vert ^{\frac{1}{p}(\frac{\lambda}{n}-1)+\frac {1}{(p/s'){'}s'}}\vert B_{1} \vert ^{\frac{1}{p}(1-\frac{\lambda}{n})} \biggl(\bigl(2^{j+1}d\bigr)^{\lambda-n} \int_{\vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \\ & \quad \leq C\vert B_{1}\vert ^{\frac{1}{(p/s'){'}s'}}\Vert f\Vert _{L^{p,\lambda}}, \end{aligned}$$
(2.11)
Plugging (2.10) and (2.11) into (2.6) we obtain Therefore, applying (2.12) into (2.5) we obtain Combining (2.4) and (2.13), we get Thus, we complete the proof of Theorem 1.1. □
$$\begin{aligned} &\sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega(x-y)}{|x-y|^{n-\alpha}}-\frac{\Omega (z-y)}{|z-y|^{n-\alpha }}\biggr\vert \,dy \\ &\quad \leq C \sum_{j=0}^{\infty} \Vert f\Vert _{L^{p,\lambda}} \vert B_{1}\vert ^{\frac {1}{(p/s'){'}s'}}\bigl\vert 2^{j+1}\sqrt{n}Q\bigr\vert ^{n/s-(n-\alpha)} \biggl\{ \frac{1}{2^{j}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta < \vert x-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \\ &\qquad {} + \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \biggr\} \\ & \quad \leq C\Vert f\Vert _{L^{p,\lambda}}\sum_{j=0}^{\infty} \vert B_{1}\vert ^{\frac {1}{(p/s'){'}s'}}\vert B_{1}\vert ^{-\frac{1}{(p/s'){'}s'}} \biggl\{ \frac{1}{2^{j}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta < \vert x-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \\ & \qquad {}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \biggr\} \\ & \quad \leq C\Vert f\Vert _{L^{p,\lambda}} \biggl\{ 2+2 \int_{0}^{1}\omega_{s}(\delta ) \frac {d\delta}{\delta} \biggr\} \\ & \quad \leq C \Vert f\Vert _{L^{p,\lambda}}. \end{aligned}$$
(2.12)
$$ \frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert \,dx\leq C\Vert f\Vert _{L^{p,\lambda}}. $$
(2.13)
$$\begin{aligned} \Vert T_{\Omega,\alpha}f \Vert _{\operatorname {BMO}} & =\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{\Omega ,\alpha}f(y) -(T_{\Omega,\alpha}f)_{Q}\bigr\vert \,dy \\ & \leq\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(y) -(T_{1}f)_{Q}\bigr\vert \,dy+\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(y) -(T_{2}f)_{Q}\bigr\vert \,dy \\ & \leq C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}$$
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 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 On the other hand, by Hölder’s inequality, Plugging (3.2) and (3.3) into (3.1) we get Now, let us turn to the estimate for \(T_{2}f(x)\). In this case we have By Hölder’s inequality and the proof of Theorem 1.1, \(s'<\frac {\lambda}{\alpha}<p\), where \(B_{1}=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert <2^{j+1}d\}\).
$$\begin{aligned} &\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad \leq\frac{2}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)\bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad =\frac{2}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q} \biggl\vert \int_{B} \frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy \biggr\vert ^{l}\,dx \biggr)^{\frac {1}{l}} \\ &\quad \leq\frac{2}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}}\frac {1}{\vert Q\vert ^{\frac{1}{l}}} \int_{B}\bigl\vert f(y)\bigr\vert \biggl( \int_{\vert y-x\vert < 2d} \biggl(\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha }} \biggr)^{l}\,dx \biggr)^{\frac{1}{l}}\,dy. \end{aligned}$$
(3.1)
$$ \begin{aligned}[b] \biggl( \int_{\vert y-x\vert < 2d} \biggl(\frac{\vert \Omega(x-y)\vert }{\vert x-y\vert ^{n-\alpha }} \biggr)^{l}\,dx \biggr)^{\frac{1}{l}} & \leq Cd^{\frac{n}{l}-(n-\alpha)}\Vert \Omega \Vert _{L^{s}(S^{n-1})} \\ &\leq C\vert Q\vert ^{\frac{1}{l}-(1-\frac{\alpha}{n})}\Vert \Omega \Vert _{L^{s}(S^{n-1})}. \end{aligned} $$
(3.2)
$$\begin{aligned} \int_{B}\bigl\vert f(y)\bigr\vert \,dy \leq {}&\biggl( \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{B}1^{p'}\,dy \biggr)^{\frac{1}{p'}} \\ ={}&\vert B\vert ^{\frac{1}{p}(1-\frac{\lambda}{n})} \biggl(\vert B\vert ^{\frac{\lambda }{n}-1} \int_{B}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac{1}{p}} \\ &{}\times\biggl( \int_{B}1^{\frac{1}{1-\frac{\lambda}{n}(1-\frac {1}{p'})+\frac {1}{p}(\frac{\lambda}{n}-1)}}\,dy \biggr)^{ 1-\frac{\lambda}{n}(1-\frac{1}{p'})+\frac{1}{p}(\frac{\lambda }{n}-1)} \\ \leq{}&\vert B\vert ^{1-\frac{\lambda}{n}(1-\frac{1}{p'})+\frac{1}{p}(\frac {\lambda }{n}-1)}\vert B\vert ^{\frac{1}{p}(1-\frac{\lambda}{n})} \Vert f \Vert _{L^{p,\lambda}} \\ ={}&\vert B\vert ^{1-\frac{\lambda}{n}(1-\frac{1}{p'})}\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}$$
(3.3)
$$ \begin{aligned}[b] &\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad \leq C\vert Q\vert ^{\frac{\lambda}{n}\frac{1}{p}-\frac{\alpha}{n}-\frac {1}{l}+1-\frac{\lambda}{n}(1-\frac{1}{p'})+\frac{1}{l}-(1-\frac {\alpha}{n})} \Vert \Omega \Vert _{L^{s}(S^{n-1})} \Vert f\Vert _{L^{p,\lambda}} \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned} $$
(3.4)
$$ \begin{aligned}[b] &\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &\quad =\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \Biggl(\frac{1}{\vert Q\vert } \int_{Q} \Biggl\vert \frac{1}{|Q|} \int_{Q} \Biggl\{ \sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}f(y)\\ &\qquad {}\times \biggl[\frac{\Omega (x-y)}{\vert x-y\vert ^{n-\alpha }} -\frac{\Omega(z-y)}{\vert z-y\vert ^{n-\alpha}} \biggr]\,dy \Biggr\} \,dz\Biggr\vert ^{l}\,dx \Biggr)^{\frac{1}{l}}. \end{aligned} $$
(3.5)
$$\begin{aligned} & \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert \biggl\vert \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} -\frac{\Omega(z-y)}{|z-y|^{n-\alpha}}\biggr\vert \,dy \\ &\quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac {1}{s'}}(J_{1}+J_{2}) \\ &\quad \leq \biggl( \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d} 1^{s'(p/s')'}\,dy \biggr)^{\frac{1}{s'(p/s')'}}(J_{1}+J_{2}) \\ &\quad =\bigl(2^{j+1}d\bigr)^{-\frac{\lambda-n}{p}} \biggl[\bigl(2^{j+1}d \bigr)^{\lambda-n} \int _{\vert y-x_{0}\vert < 2^{j+1}d}\bigl\vert f(y)\bigr\vert ^{p}\,dy \biggr]^{\frac{1}{p}} \\ &\qquad {}\times \biggl( \int_{\vert y-x_{0}\vert < 2^{j+1}d} 1^{\frac{1}{\frac{1}{s'(p/s')'}+\frac{\lambda-n}{np}+\frac {1}{p}(1-\frac {\lambda}{n})}}\,dy \biggr)^{\frac{1}{s'(p/s')'} +\frac{\lambda-n}{np}+\frac{1}{p}(1-\frac{\lambda }{n})}(J_{1}+J_{2}) \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}} \vert B_{1}\vert ^{-\frac{\lambda -n}{np}}\vert B_{1}\vert ^{\frac{1}{s'(p/s')'} +\frac{\lambda-n}{np}+\frac{1}{p}(1-\frac{\lambda }{n})}(J_{1}+J_{2}) \\ &\quad =C\Vert f\Vert _{L^{p,\lambda}}\bigl(2^{j}d\bigr)^{\frac{n}{s'(p/s')'} +\frac{n}{p}(1-\frac{\lambda}{n})}(J_{1}+J_{2}) \\ &\quad =C\Vert f\Vert _{L^{p,\lambda}}\bigl(2^{j}d\bigr)^{\frac{n}{s'(p/s')'}}2^{j(\frac {n}{p}-\frac{\lambda}{p})} \vert Q\vert ^{\frac{1}{p}-\frac{1}{p} \frac{\lambda}{n}}(J_{1}+J_{2}), \end{aligned}$$
(3.6)
Since the integral modulus of continuity \(\omega_{s}(\delta)\) of order s of Ω satisfies (1.2) and we know that Ω satisfies also the \(L^{s}\)-Dini condition. From Lemma 2.1 and the proof of Theorem 1.1, we get Note that Moreover, 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 Plugging (3.10) into (3.5), we obtain Then by (3.4) and (3.11) we get Thus, we complete the proof of Theorem 1.3. □
$$\int_{o}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta}< \int _{o}^{1}\omega _{s}(\delta) \frac{d\delta}{\delta^{1+\beta}} < \infty, $$
$$ \begin{aligned}[b] J_{1}+J_{2}\leq{}& C \bigl(2^{j}d\bigr)^{n/s-(n-\alpha)}\times \biggl\{ \frac{1}{2^{j}}+ \int_{\vert x-x_{0}\vert /2^{j+1}d< \delta < \vert x-x_{0}\vert /2^{j}d}\omega _{s}(\delta) \frac{d\delta}{\delta} \\ &{}+ \int_{\vert z-x_{0}\vert /2^{j+1}d< \delta< \vert z-x_{0}\vert /2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} \biggr\} . \end{aligned} $$
(3.7)
$$ \begin{aligned}[b] \bigl(2^{j}d\bigr)^{n/[s'(p/s')']+n/s-(n-\alpha)}2^{j(\frac{n}{p}-\frac {\lambda }{p})}&= \bigl(2^{j}d\bigr)^{n(\frac{\alpha}{n}-\frac{1}{p})} 2^{j(\frac{n}{p}-\frac{\lambda}{p})} \\ &\leq C\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}}2^{jn(\frac{\alpha }{n}-\frac {1}{p})+j(\frac{n}{p}-\frac{\lambda}{p})} \\ &=C\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}}2^{jn(\frac{\alpha}{n}-\frac {1}{p}\frac{\lambda}{n})}. \end{aligned} $$
(3.8)
$$\begin{aligned} 2^{jn(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})} \int _{\vert x-x_{0}\vert /2^{j+1}d}^{\vert x-x_{0}\vert / 2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta} &\leq2^{jn(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n})}\bigl(\vert x-x_{0}\vert /2^{j}d \bigr)^{\beta} \int_{\vert x-x_{0}\vert /2^{j+1}d}^{\vert x-x_{0}\vert / 2^{j}d}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta}} \\ &\leq C2^{j[n(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})-\beta ]} \int _{0}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta}}. \end{aligned}$$
(3.9)
$$\begin{aligned} &\sum_{j=0}^{\infty} \int_{2^{j}d\leq \vert y-x_{0}\vert < 2^{j+1}d}f(y) \biggl[\frac {\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}} -\frac{\Omega(z-y)}{\vert z-y\vert ^{n-\alpha}} \biggr]\,dy \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}} \vert Q\vert ^{\frac{1}{p}-\frac{1}{p}\frac {\lambda }{n}}\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}} \sum_{j=0}^{\infty} \biggl\{ 2^{j[n(\frac{\alpha}{n}-\frac {1}{p}\frac {\lambda}{n})-1]} +C2^{j[n(\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda}{n})-\beta ]} \int _{0}^{1}\omega_{s}(\delta) \frac{d\delta}{\delta^{1+\beta}} \biggr\} \\ &\quad \leq C\Vert f\Vert _{L^{p,\lambda}} \vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac {\lambda}{n}}. \end{aligned}$$
(3.10)
$$ \frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac {1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}}\leq C\Vert f\Vert _{L^{p,\lambda}}. $$
(3.11)
$$\begin{aligned} \Vert T_{\Omega,\alpha}f \Vert _{\mathcal{L}_{l,n(\frac{\alpha}{n}-\frac {1}{p}\frac{\lambda}{n})}} \leq{}&\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{1}f(x)-(T_{1}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ &{}+\frac{1}{\vert Q\vert ^{\frac{\alpha}{n}-\frac{1}{p}\frac{\lambda }{n}}} \biggl(\frac{1}{\vert Q\vert } \int_{Q}\bigl\vert T_{2}f(x)-(T_{2}f)_{Q} \bigr\vert ^{l}\,dx \biggr)^{\frac{1}{l}} \\ \leq{}& C\Vert f\Vert _{L^{p,\lambda}}. \end{aligned}$$
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.
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.