Abstract
For \(0<\alpha<n\), the homogeneous fractional integral operator \(T_{\Omega,\alpha}\) is defined by
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.
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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].
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.
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 \(g_{Q}=\frac{1}{\vert Q\vert }\int_{Q}g(y)\,dy\).
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
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
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 \(S^{n-1}\).
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 \(\vert \rho \vert =\Vert \rho-I\Vert \).
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
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
then there is a \(C>0\) such that for \(1\leq l\leq\lambda/(\lambda -\alpha)\),
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
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.
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 \(2^{j+1}\sqrt{n}d\).
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 \(B_{1}=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert <2^{j+1}d\}\).
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. □
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 \(B_{1}=\{y\in{\Bbb {R}}^{n};\vert y-x_{0}\vert <2^{j+1}d\}\).
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. □
References
Peeter, J: On the theory of \(\mathcal{L}_{p,\lambda}\) spaces. J. Funct. Anal. 4, 71-87 (1969)
Morrey, C: On the solutions of quasi linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126-166 (1938)
Adams, DR, Xiao, J: Morry spaces in harmonic analysis. Arch. Math. 50(2), 201-230 (2012)
Mazzucato, A: Besov-Morrey spaces: functions space theory and applications to non-linear PDE. Trans. Am. Math. Soc. 355, 1297-1364 (2002)
Ruiz, A, Vega, L: On local regularity of Schrödinger equations. Int. Math. Res. Not. 1, 13-27 (1993)
Shen, Z: Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains. Am. J. Math. 125, 1079-1115 (2003)
Caffarelli, L: Elliptic second order equations. Rend. Semin. Mat. Fis. Milano 58, 253-284 (1990)
Di Fazio, G, Palagachev, D, Ragusa, M: Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J. Funct. Anal. 166, 179-196 (1999)
Huang, Q: Estimates on the generalized Morrey spaces \(L_{\varphi }^{2,\lambda}\) and BMO for linear elliptic systems. Indiana Univ. Math. J. 45, 397-439 (1996)
Palagachev, D, Softova, L: Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal. 20, 237-263 (2004)
Adams, DR: A note on Riesz potentials. Duke Math. J. 42, 765-778 (1975)
Adams, DR: Lectures on \(L^{p}\)-potential theory, vol. 2. Department of Mathematics, University of Umeå (1981)
Chiarenza, F, Frasca, M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl. 7, 273-279 (1987)
Chen, Y, Ding, Y, Li, R: The boundedness for commutator of fractional integral operator with rough variable kernel. Potential Anal. 38(1), 119-142 (2013)
Chen, Y, Wu, X, Liu, H: Vector-valued inequalities for the commutators of fractional integrals with rough kernels. Stud. Math. 222(2), 97-122 (2014)
Ding, Y, Lu, S: The \(L^{p_{1}}\times L^{p_{2}}\times\cdots\times L^{p_{k}}\) boundedness for some rough operators. J. Math. Anal. Appl. 203, 166-186 (1996)
Fan, D, Lu, S, Yang, D: Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients. Georgian Math. J. 5, 425-440 (1998)
Di Fazio, G, Ragusa, MA: Interior estimates in Morrey spaces for strongly solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112, 241-256 (1993)
Chen, Y, Ding, Y, Wang, X: Compactness of commutators for singular integrals on Morrey spaces. Can. J. Math. 64(2), 257-281 (2012)
John, F, Nirenberg, L: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415-426 (1961)
Muckenhoupt, B, Wheeden, RL: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261-274 (1974)
Muckenhoupt, B, Wheeden, RL: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161, 249-258 (1971)
Ding, Y, Lu, S: Weighted norm inequalities for fractional integral operators with rough kernel. Can. J. Math. 50, 29-39 (1998)
Chanillo, S, Watson, D, Wheeden, RL: Some integral and maximal operators related to star-like. Stud. Math. 107, 223-255 (1993)
Ding, Y: Weak type bounds for a class of rough operators with power weights. Proc. Am. Math. Soc. 125, 2939-2942 (1997)
Ding, Y, Lu, S: Boundedness of homogeneous fractional integrals on \(L^{p}\) for \(n/\alpha\leq p\leq\infty\). Nagoya Math. J. 167, 17-33 (2002)
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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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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Meng, S., Chen, Y. Boundedness of homogeneous fractional integral operator on Morrey space. J Inequal Appl 2016, 61 (2016). https://doi.org/10.1186/s13660-016-0999-y
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DOI: https://doi.org/10.1186/s13660-016-0999-y