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.