# Some Estimates of Rough Bilinear Fractional Integral

Authors: Yun Fan; Guilian Gao
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We study the boundedness of rough bilinear fractional integralBΩ,α on Morrey spaces Lp,λ(ℝn) and modified Morrey spaces L~p,λ(ℝn) and obtain some sufficient and necessary conditions on the parameters. Furthermore, we consider the boundedness of BΩ,α on generalized central Morrey space B˙p,φ(ℝn). These extend some known results.


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## Body

## 1. Introduction

In recent years, multilinear analysis becomes a very active research topic in studying harmonic analysis. As one of the most important operators, the multilinear fractional integral has also attracted much attention. In this note, we will consider the multilinear fractional integral with rough kernel. For fixed distinct and nonzero real numbersθ1,…,θm, and 0<α<n, the m-linear fractional with rough kernel is defined by
(1.1)IΩ,α(f→)=∫ℝn‍∏i=1m‍fi(x-θiy)Ω(y)|y|n-αdy,
where Ω∈Ls(Sn-1) (s≥1) is homogeneous of degree zero on ℝn, and Sn-1 denotes the unit sphere of ℝn.WhenΩ≡1, The Lp boundedness of operator I1,α has been well studied in [1, 2]. Recently, Hendar and Idha discussed the boundedness property of I1,α on generalized Morrey space in [3].Here, without loss of generality, we will study the casem=2. More specifically, we will study the rough bilinear fractional integral:
(1.2)BΩ,α(f,g)(x)=∫ℝn‍f(x-y)g(x+y)Ω(y)|y|n-αdy,0<α<n.The study of the operatorsBΩ,α and its related operators with rough kernel Ω recently attracted many attentions. In 2002, Ding and Chin first discussed its Lp(ℝn) boundedness. The following theorem is their main result:Theorem A (see [4]).
Let0<α<n,  1≤s'<n/α and 1≤p1, p2≤∞. If
(1.3)1p1+1p2≥αn,1q=1p1+1p2-αn,
there exists a positive constant C such that for any f∈Lp1(ℝn), g∈Lp2(ℝn),
(1) whens'<min{p1,p2},
(1.4)∥BΩ,α(f,g)∥Lq(ℝn)≤C∥f∥Lp1(ℝn)∥g∥Lp2(ℝn);
(2) whens'=min{p1,p2},
(1.5)∥BΩ,α(f,g)∥Lq,∞(ℝn)≤C∥f∥Lp1(ℝn)∥g∥Lp2(ℝn).Later, whenq>n/(n-α), Chen and Fan in [5] relaxed the conditions of Ω in Theorem A using Hölder inequality. Their main result is as follows.Theorem B.
Letq>n/(n-α), 0<α<n,  p1,p2>1 and
(1.6)1q=1p1+1p2-αn.
If Ω∈Ln/(n-α)(Sn-1), then there exists a positive constant C such that
(1.7)∥BΩ,α(f,g)∥Lq(ℝn)≤C∥f∥Lp1(ℝn)∥g∥Lp2(ℝn).We note that whenq≤n/(n-α), Hölder inequality is not sufficient in Theorem B. So how to relax the index of q is left. In fact, in [6, 7] the authors have obtained the necessary and sufficient conditions on the parameters for the m-linear fractional integral operator IΩ,α with rough kernel from Lp1(ℝn)×Lp2(ℝn)×⋯Lpm(ℝn) to Lq(ℝn) by using the pointwise rearrangement estimate of the m-linear convolution.Theorem C.
Let0<α<n, Ω and be homogeneous of degree zero on ℝn, Ω∈Ln/(n-α)(Sn-1), let p be the harmonic mean of p1,p2,…,pm>1, and n/(n-α)≤p<n/α. Then the condition 1/q=1/p-α/n is necessary and sufficient for the boundedness of IΩ,α from Lp1(ℝn)×Lp2(ℝn)×⋯Lpm(ℝn) to Lq(ℝn).This paper is organized as follows: in the second part of this work we prove some boundedness properties ofBΩ,α on Morrey space and extend Theorem C to Morrey spaces; in the third part, we obtain the sufficient and necessary conditions on the parameters for the boundedness of BΩ,α on modified Morrey space; in the last part, we find the sufficient condition on the pair (φ,ν) which ensures the boundedness of the operators BΩ,α on the generalized center Morrey space. Since Morrey space, modified Morrey space and central Morrey space all can be seen as generalized Lp space.


## 2.   The Boundedness ofBΩ,α on Morrey Space

The classical Morrey spacesLp,λ(ℝn) were originally introduced by Morrey in [8] to study the local behavior of solutions to second-order elliptic partial differential equations. The reader can find more details in [9].Forx∈ℝn and t>0, let B(x,t) denotes the open ball centered at x of radius t, and |B(x,t)| is the Lebesgue measure of the ball B(x,t). When 1≤p<∞ and λ≥0, Morrey space Lp,λ(ℝn) is defined by
(2.1)Lp,λ(ℝn)={f∈Llocp(ℝn):∥f∥Lp,λ(ℝn)<∞},
where
(2.2)∥f∥Lp,λ(ℝn)=supx∈ℝn,t>0(1tλ∫B(x,t)‍|f(x)|pdx)1/p.If1≤p<∞, then Lp,0(ℝn)=Lp(ℝn) and Lp,n(ℝn)=L∞(ℝn). When λ>n, Lp,λ(ℝn)={0}. So we only consider the case 0<λ<n.Since Morrey space can be seen as the generalizedLp space, we will be interested in the boundedness of BΩ,α on Morry space Lp,λ(ℝn). In order to prove our results, we need the following bilinear maximal function:
(2.3)M(f,g)(x)=supr>01rn∫|y|<r‍|f(x-y)||g(x+y)|dy.Lemma 2.1.
Letp>1, 0<λ<n and  1/p=1/p1+1/p2. If
(2.4)λp=λ1p1+λ2p2,0<λ1,λ2<n,
then there exists a positive constant C such that
(2.5)∥M(f,g)∥Lp,λ(ℝn)≤C∥f∥Lp1,λ(ℝn)∥g∥Lp2,λ(ℝn).Proof.
In [10], Fefferman and Stein have proved that for every p, 1<p<∞, there is a constant Cp>0 such that for any measurable functions f on ℝn and φ≥0, the following inequality holds,
(2.6)∫ℝn‍(Mf(x))pφ(x)dx≤Cp∫ℝn‍|f(x)|pMφ(x)dx,
where M is the Hardy-LittleWood maximal function. Set φ(x) be the characteristic function χ(x), when 1≤δ<p, by the above inequality, we can get
(2.7)∫ℝn‍(Mδf(x))pχ(x)dx≤Cp∫ℝn‍|f(x)|pMχ(x)dx,
where Mδf(x)=(Mfδ)1/δ(x).
Takingf∈Lp,λ(ℝn), 0<λ<n, χ(x)  is  the characteristic function of a ball B(x0,r) in ℝn, by simple calculating,
(2.8)∫B(x0,r)‍(Mδf(x))pdx≤C∥f∥Lp,λ(ℝn)prλ,
that is, ∥Mδf∥Lp,λ(ℝn)≤C∥f∥Lp,λ(ℝn). For More details, see [11] about the boundedness of Hardy-Littlewood maximal function on Morrey space.
So whenp>1,  1/p=1/p1+1/p2,   λ/p=λ1/p1+λ2/p2, we have
(2.9)∥M(f,g)∥Lp,λ(ℝn)≤∥Mp1/p(f)Mp2/p(g)∥Lp,λ(ℝn)≤∥Mp1/p(f)∥Lp1,λ1(ℝn)∥Mp2/p(g)∥Lp2,λ2(ℝn)≤C∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).Theorem 2.2.
Suppose0<α<n, and let Ω∈Ls(Sn-1) be homogeneous of degree zero on ℝn, let p be the harmonic mean of p1 and p2, 1<p<n/α, 0<λ<n-αp and s'<p. If
(2.10)1q=1p-αn-λ,λp=λ1p1+λ2p2,0<λ1,λ2<n,
then there exists a positive constant C such that
(2.11)∥BΩ,α(f,g)∥Lq,λ(ℝn)≤C∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).Proof.
Letf∈Lp1,λ1(ℝn), g∈Lp2,λ2(ℝn), σ=(n-αs'+λ)/2, for s'<p and 0<λ<n-αp, we can get λ<σ<n-αs', (n-λ)/p>α>(n-σ)/s'. First, |BΩ,α(f,g)(x)| is decomposed by
(2.12)|BΩ,α(f,g)(x)|=(∫|y|≤ε‍+∫|y|≥ε‍)f(x-y)g(x+y)Ω(y)|y|n-α=:I1(x)+I2(x).
Estimate ofI1(x) is
(2.13)I1(x)=∑m=1∞‍∫|y|~ε2-m‍|f(x-y)g(x+y)||Ω(y)||y|n-αdy≤∑m=1∞‍(ε2-m)α-n∫|y|~ε2-m‍|f(x-y)g(x+y)||Ω(y)|dy≤∑m=1∞‍(ε2-m)αM(fs′,gs′)1/s′(x)≤CεαM(fs′,gs′)1/s′(x)=:CεαMs′(f,g)(x),
and estimate of  I2(x) is
(2.14)I2(x)≤(∫|y|≥ε‍fs′(x-y)gs′(x+y)|y|σdy)1/s′(∫|y|≥ε‍|y|(σ/s′+α-n)s|Ω(y)|sdy)1/s≤Cε(σ/s′+α-n)+n/s(∫|y|≥ε‍fs′(x-y)gs′(x+y)|y|σdy)1/s′=:Cε(σ/s′+α-n)+n/sFσ(f,g)(x).
For Fσ(f,g)(x), we have the following estimates:
(2.15)Fσ(f,g)(x)≤(∑k=0∞‍∫|y|~ε2k‍|fs′(x-y)gs′(x+y)||y|σdy)1/s′≤∑k=0∞‍(∫|y|~ε2k‍|fs′(x-y)gs′(x+y)||y|σdy)1/s′≤∑k=0∞‍(ε2k)-σ/s′(∫|y|~ε2k‍|fs′(x-y)gs′(x+y)|dy)1/s′≤∑k=0∞‍(ε2k)(n-σ)/s′-n/p(∫|y|~ε2k‍|fp1(x-y)|dy)1/p1(∫|y|~ε2k‍|gp2(x-y)|dy)1/p2≤∑k=0∞‍(ε2k)(n-σ)/s′-n/p+λ1/p1+λ2/p2∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn)≤C(ε)(n-σ)/s′-(n-λ)/p∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).
Combining the above estimates, we have
(2.16)|BΩ,α(f,g)(x)|≤CεαMs'(f,g)(x)+Cε(λ-n)/p+α∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).
Let εαMs′(f,g)(x)=ε((λ-n)/p)+α∥f∥p1,λ1∥g∥p2,λ2, then
(⋆)|BΩ,α(f,g)(x)|≤C(Ms′(f,g)(x))p/q∥f∥Lp1,λ1(ℝn)1-p/q∥g∥Lp2,λ2(ℝn)1-p/q.
By computation, we get
(2.17)(1rλ∫B(x,r)‍(Ms′(f,g)(x))(p/q)×qdx)1/q=(1rλ∫B(x,r)‍(M(fs′,gs′)(x))p/s′dx)1/p×p/q≤(1rλ1∫B(x,r)‍f(x)p1dx)1/p1×p/q(1rλ2∫B(x,r)‍g(x)p2dx)1/p2×p/q≤∥f∥Lp1,λ1(ℝn)p/q∥g∥Lp2,λ2(ℝn)p/q.
Taking the supremum of r, we have
(2.18)∥(Ms′(f,g))p/q∥Lq,λ(ℝn)≤∥f∥Lp1,λ1(ℝn)p/q∥g∥Lp2,λ2(ℝn)p/q.
Hence(2.19)∥BΩ,α(f,g)∥Lq,λ(ℝn)≤C∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).Theorem 2.3.
Suppose0<α<n, and let Ω∈Ls(Sn-1) be homogeneous of degree zero on ℝn, let p be the harmonic mean of p1 and p2, 1<p<n/α, 0<λ<n-αp, s′<p and λ/p=λ1/p1+λ2/p2, 0<λ1,λ2<n, then the condition 1/q=1/p-α/(n-λ) is necessary and sufficient for the boundedness of BΩ,α from Lp1,λ1(ℝn)×Lp2,λ2(ℝn) to Lq,λ(ℝn).Proof.
Sufficiency part of Theorem2.3 is proved in Theorem 2.2.
Necessity. Let 1<p<n/α and f∈Lp1,λ1(ℝn), g∈Lp2,λ2(ℝn). Denote ft(x)=:f(tx) and gt(x)=:g(tx). Then we have
(2.20)∥ft∥Lp1,λ1(ℝn)=t-n/p1+λ1/p1∥f∥Lp1,λ1(ℝn),∥gt∥Lp2,λ2(ℝn)=t-n/p2+λ2/p2∥g∥Lp2,λ2(ℝn),BΩ,α(ft,gt)(x)=t-αBΩ,α(f,g)(tx),∥BΩ,α(ft,gt)∥Lq,λ(ℝn)=t-α-n/q+λ/q∥BΩ,α(f,g)∥Lq,λ(ℝn).
SinceBΩ,α is bounded from Lp1,λ1(ℝn)×Lp2,λ2(ℝn) to Lq,λ(ℝn), it is true that
(2.21)∥BΩ,α(f,g)∥Lq,λ(ℝn)=tα+n/q-λ/q∥BΩ,α(ft,gt)∥Lq,λ(ℝn)≤Ctα+n/q-λ/q∥ft∥Lp1,λ1(ℝn)∥gt∥Lp2,λ2(ℝn)≤Ctα+n/q-λ/q-n/p+λ/p∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn),
where C depends only on p,  q,  λ, and n.
If1/q<1/p-α/(n-λ), then in the case t→0, for all f∈Lp1,λ1(ℝn), g∈Lp2,λ2(ℝn), we have ∥BΩ,α(f,g)∥Lq,λ(ℝn)=0.
If1/q>1/p-α/(n-λ), then in the case t→∞, for all f∈Lp1,λ1(ℝn), g∈Lp2,λ2(ℝn), we have ∥BΩ,α(f,g)∥Lq,λ(ℝn)=0.
Therefore, we get1/q=1/p-α/(n-λ).Corollary 2.4.
Let0<α<n, Ω∈Ls(Sn-1) be homogeneous of degree zero on ℝn, p be the harmonic mean of p1 and p2, 1<p<n/α, 0<λ<n-αp, and s'<p. If
(2.22)1q=1p-αn;μq=λp=λ1p1+λ2p2,0<λ1,λ2<n,
then there exists a positive constant C such that
(2.23)∥BΩ,α(f,g)∥Lq,μ(ℝn)≤C∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).Proof.
By Hölder inequality, it is easy to know whent=(n-λ)q/(n-μ), we have Lt,λ(ℝn)⊆Lq,μ(ℝn), through the given condition, 1/t=1/p-α/(n-λ). Applying Theorem 2.2, we get
(2.24)∥BΩ,α(f,g)∥Lq,μ(ℝn)≤∥BΩ,α(f,g)∥Lt,λ(ℝn)≤C∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).
From the inequality(⋆) and Theorem 2.2, we obtain an Olsen inequality involving a multiplication operator.Corollary 2.5.
Suppose0<α<n, and let Ω∈Ls(Sn-1) be homogeneous of degree zero on ℝn, let p be the harmonic mean of p1 and p2, 1<p<n/α, 0<λ<n-αp,  s'<p, and W∈L(n-λ)/α,λ(ℝn). If
(2.25)1q=1p-αn-λ;λp=λ1p1+λ2p2,0<λ1,λ2<n.
One has
(2.26)∥W·BΩ,α(f,g)∥Lp,λ(ℝn)≤C∥W∥L(n-λ)/α,λ(ℝn)∥f∥Lp1,λ1(ℝn)∥g∥Lp2,λ2(ℝn).


## 3. The Boundedness ofBΩ,α on Modified Morrey Space

After studying Morrey spaces in detail, people are led to considering the local and global counterpart. There are many famous work by V. I. Burenkov, H. V. Guliyev and V. S. Guliyev, and so forth and (see [12–20]). Recently, Guliyev et al. have considered the following modified Morrey spaces L~p,λ(ℝn) in [21].Definition 3.1.
Let1≤p<∞, 0≤λ≤n and [t]1=min{1,t}. L~p,λ(ℝn) is defined as the set of all functions f∈Lploc(ℝn), with the finite norms
(3.1)∥f∥L~p,λ=supx∈ℝn,t>0(1[t]1λ∫B(x,t)‍|f(y)|pdy)1/p.
Note that
(3.2)L~p,0(ℝn)=Lp,0(ℝn)=Lp(ℝn),L~p,λ(ℝn)⊂≻Lp,λ(ℝn)∩Lp(ℝn),max{∥f∥Lp,λ,∥f∥Lp}≤∥f∥L~p,λ,
and if λ<0 or λ>n, then L~p,λ(ℝn)=Lp,λ(ℝn)={0}.
In [21], the authors discussed the boundedness of maximal function in modified Morrey spaces L~p,λ(ℝn) and obtained the following generalized Hardy-Littlewood-Sobolev inequalities in modified Morrey spaces.Theorem D.
Let0<α<n and 0≤λ<n-α. If 1<p<(n-λ)/α, then condition α/n≤1/p-1/q≤α/(n-λ) is necessary and sufficient for the boundedness of the operator Iα from L~p,λ(ℝn) to L~q,λ(ℝn).We also can extend Theorem D to the multilinear case.Lemma 3.2.
Letp>1, 0<λ<n and 1/p=1/p1+1/p2. If
(3.3)λp=λ1p1+λ2p2,0<λ1,λ2<n,
then there exists a positive constant C such that
(3.4)∥M(f,g)∥L~p,λ(ℝn)≤C∥f∥L~p1,λ(ℝn)∥g∥L~p2,λ(ℝn).Proof.
When1≤δ<p, the following inequality:
(3.5)∫ℝn‍(Mδf(x))pχ(x)dx≤Cp∫ℝn‍|f(x)|pMχ(x)dx
holds, where M is the Hardy-littlewood maximal function and Mδf(x)=(Mfδ)1/δ(x).
Takingf∈L~p,λ(ℝn), 0<λ<n. Using the method in [21], we get ∥Mδf∥L~p,λ(ℝn)≤C∥f∥L~p,λ(ℝn).
Hence, with the same arguments in Lemma2.1, we complete the proof of Lemma 3.2.Theorem 3.3.
Suppose0<α<n, Ω∈Ls(Sn-1) and let be homogeneous of degree zero on ℝn, let p be the harmonic mean of p1 and p2, 1<p<n/α, 0<λ<n-αp, s'<p and λ/p=λ1/p1+λ2/p2, 0<λ1,λ2<n. Then the condition α/n≤1/p-1/q≤α/(n-λ) is necessary and sufficient for the boundedness of BΩ,α from L~p1,λ1(ℝn)×L~p2,λ2(ℝn) to L~q,λ(ℝn).Proof.
(1)Sufficiency. Let f∈L~p1,λ1(ℝn), g∈L~p2,λ2(ℝn), σ=(n-αs'+λ)/2, since s'<p and 0<λ<n-αp, we can get λ<σ<n-αs', (n-λ)/p>α>(n-σ)/s' and λ<n-((n-σ)/s′)p<n-αp.
Do the same decomposition ofBΩ,α(f,g)(x) in the proof of Theorem 2.2, then we only need to estimate Fσ(f,g)(x). We can easily obtain
(3.6)Fσ(f,g)(x)≤(∑k=0∞‍∫|y|~ε2k‍|fs′(x-y)gs′(x+y)||y|σdy)1/s′≤∑k=0∞‍(ε2k)(n-σ)/s′-n/p(∫|y|~ε2k‍|fp1(x-y)|dy)1/p1×(∫|y|~ε2k‍|gp2(x-y)|dy)1/p2≤(ε)(n-σ)/s′-n/p∑k=0∞‍(2k)(n-σ)/s′-n/p[ε2k]1λ/p∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn).
For0<ε<1/2, we get
(3.7)∑k=0∞‍(2k)(n-σ)/s′-n/p[ε2k]1λ/p=∑k=0[log2(1/2ε)]‍ελ/p(2k)(n-σ)/s′-n/p+λ/p+∑k=[log2(1/2ε)]+1∞‍(2k)(n-σ)/s′-n/p≤C(ελ/p+ε(n-σ)/s′-n/p)≤Cελ/p.
Whileε≥1/2, we obtain
(3.8)∑k=0∞‍(2k)(n-σ)/s′-n/p[ε2k]1λ/p=∑k=0∞‍(2k)(n-σ)/s′-n/p≤C.
Thus, we obtain(3.9)Fσ(f,g)(x)≤C(ε)((n-σ)/s′)-(n/p)[2ε]1λ/p∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn),|BΩ,α(f,g)(x)|≤C(εαMs′(f,g)(x)+εα-(n/p)[ε]1λ/p∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn))≤Cmin{εαMs′(f,g)(x)+εα-n/p∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn),εαMs′(f,g)(x)+εα-(n-λ)/p∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn)}.
Set
(3.10)ε=(Ms'(f,g)(x)-1∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn))p/(n-λ),ε=(Ms'(f,g)(x)-1∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn))p/n,
we have
(3.11)|BΩ,α(f,g)(x)|≤Cmin{(Ms′(f,g)(x)∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn))1-pα/(n-λ),(Ms′(f,g)(x)∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn))1-pα/n}×∥f∥L~p1,λ1(ℝn)∥g∥L~p2,λ2(ℝn)≤C(Ms′(f,g)(x))p/q∥f∥L~p1,λ1(ℝn)1-p/q∥g∥L~p2,λ2(ℝn)1-p/q.
Hence, by the boundedness ofM(f,g)(x) in Lemma 3.2, we prove that BΩ,α is bounded from L~p1,λ1(ℝn)×L~p2,λ2(ℝn) to L~q,λ(ℝn).
(2)Necessity. Let 1<p<n/α and f∈L~p1,λ1(ℝn), g∈L~p2,λ2(ℝn). Denote ft(x)=:f(tx), gt(x)=:g(tx), and [t]1,+=max{1,t}. Then from [21], we have
(3.12)∥ft∥L~p1,λ1(ℝn)=t-n/p1[t]1,+λ1/p1∥f∥L~p1,λ1(ℝn),∥gt∥L~p2,λ2(ℝn)=t-n/p2[t]1,+λ2/p2∥f∥L~p2,λ2(ℝn),BΩ,α(ft,gt)(x)=t-αBΩ,α(f,g)(tx),∥BΩ,α(ft,gt)∥L~q,λ(ℝn)=t-α-n/q[t]1,+λ/q∥BΩ,α(f,g)∥L~q,λ(ℝn).
By the boundedness ofBΩ,α, we have
(3.13)∥BΩ,α(f,g)∥L~q,λ(ℝn)=tα+n/q[t]1,+-λ/q∥BΩ,α(ft,gt)∥L~q,λ(ℝn)≤Ctα+n/q[t]1,+-λ/q∥ft∥L~p1,λ1(ℝn)∥gt∥L~p2,λ2(ℝn)≤Ctα+n/q-(n/p)[t]1,+λ/p-λ/q∥ft∥L~p1,λ1(ℝn)∥gt∥L~p2,λ2(ℝn).
If1/q>1/p-α/(n-λ), then in the case t→0, for all f∈L~p1,λ1(ℝn), g∈L~p2,λ2(ℝn), we have ∥BΩ,α(f,g)∥L~q,λ(ℝn)=0.
If1/q<1/p-α/(n-λ), then in the case t→∞, for all f∈L~p1,λ1(ℝn), g∈L~p2,λ2(ℝn), we have ∥BΩ,α(f,g)∥Lq,λ(ℝn)=0.
Thereforα/n≤1/p-1/q≤α/(n-λ).


## 4. The Boundedness ofBΩ,α on Generalized Center Morrey Space

Definition 4.1.
Letφ(r) be a positive measurable function on ℝ+ and 1≤p<∞. We denote by B˙p,φ(ℝn) the generalized central Morrey space, the space of all functions f∈Lploc(ℝn) with finite quasinorm
(4.1)∥f∥B˙p,φ(ℝn)=supr>0φ(r)-1|B(0,r)|-1/p∥f∥Lp(B(0,r)),
where B(0,r) denotes a ball centered at 0 with side length r and |B(0,r)| is the Lebesgue measure of the ball B(0,r).
According to this definition, we recover the spacesB˙p,λ(ℝn) under the choice φ(r)=rnλ. About the B˙p,λ(ℝn) space, the readers can refer to [22], In fact, we can easily check that B˙p,λ(ℝn) is a Banach space, B˙p,λ(ℝn) reduce to {0} when λ<-1/p, B˙p,(-1/p)(ℝn)=Lp(ℝn) and B˙p,0(ℝn)=B˙p(ℝn).
There are many papers that discussed the conditions onφ to obtain the boundedness of fractional integral on the generalized Morrey spaces, see [23, 24]. In [25] the following condition was imposed on the pair (φ1,φ2):
(4.2)∫r∞‍essinft<s<∞φ1(s)sn/ptn/q+1≤Cφ2(r)
for the fractional integral Iα, where 1/q=1/p-α/n and C (>0) does not depend on r.Theorem E (see [26]).
The inequality(4.3)esssupt>0ω(t)Hg(t)≤cesssupt>0ν(t)g(t)
holds for all nonnegative and nonincreasing g on (0,∞) if and only if
(4.4)A:=supt>0ω(t)t∫0t‍dresssupt>0ν(s)<∞,
and c≈A, where the H is the Hardy operator
(4.5)Hg(t):=1t∫0t‍g(r)dr,0<t<∞.In this section we are going to discuss the boundedness ofBΩ,α on generalized central Morrey space.Lemma 4.2.
Suppose0<α<n,  1/p=1/p1+1/p2, 1/q=1/p-α/n, and s≥p′, then for 1<p<n/α, the inequality
(4.6)∥BΩ,α(f,g)∥Lq(B(0,r))≤Crn/q(∫2r∞∥f∥Lp1(B(0,t))p1/p‍dttn/q+1)p/p1(∫2r∞‍∥g∥Lp2(B(0,t))p2/pdttn/q+1)p/p2
holds for any ball B(0,r) and for all f∈Lp1loc(ℝn) and g∈Lp2loc(ℝn).Proof.
Let1<p<n/α, 1/p=1/p1+1/p2, 1/q=1/p-α/n and s≥p′. For any r>0, set B=B(0,r), we write
(4.7)f(x)=f(x)χ3B(x)+f(x)χ(3B)c(x):=f1(x)+f2(x),g(x)=g(x)χ3B(x)+g(x)χ(3B)c(x):=g1(x)+g2(x).
Hence(4.8)∥BΩ,α(f,g)∥Lq(B)≤∥BΩ,α(f1,g1)∥Lq(B)+∥BΩ,α(f1,g2)∥Lq(B)+∥BΩ,α(f2,g1)∥Lq(B)+∥BΩ,α(f2,g1)∥Lq(B).
SinceBΩ,α is bounded from Lp1×Lp2 to Lq, we have
(4.9)∥BΩ,α(f1,g1)∥Lq(B)≤∥BΩ,α(f1,g1)∥Lq(ℝn)≤C∥f1∥Lp1(ℝn)∥g1∥Lp2(ℝn)≤C∥f∥Lp1(3B)∥g∥Lp2(3B),
where the constant C>0 is independent of f and g.
To estimateBΩ,α(f1,g2), it follows that
(4.10)|BΩ,α(f1,g2)|=|∫ℝn‍f1(x-y)g2(x+y)Ω(y)|y|n-αdy|≤(∫ℝn‍|f1p1/p(x-y)Ω(y)|dy)p/p1(∫ℝn‍|g2p2/p(x-y)Ω(y)||y|(n-α)p2/pdy)p/p2≤(∫4B‍|fp1(y)|dy)1/p1(∫4B‍|Ωp′(x-y)|dy)p/p1p′×(∫(2B)c‍|gp2/p(y)Ω(x-y)||y|(n-α)p2/pdy)p/p2≤Crpn/p1p′(rn/q∥f∥Lp1(4B)p1/p∫4r∞‍dttn/q+1)p/p1×(∫(2B)c‍|gp2/p(y)Ω(x-y)|∫|y|∞‍dt|t|(n-α)p2/p+1dy)p/p2≤Crpn/p1p′+np/qp1(∫4r∞‍∥f∥Lp1(B(0,t))p1/pdttn/q+1)p/p1×(∫2r∞‍∫2r≤|y|<t∞‍|gp2/p(y)Ω(x-y)|dydt|t|(n-α)p2/p+1)p/p2≤Cr(p/p1)(n-α)(∫2r∞‍∥f∥Lp1(B(0,t))p1/pdttn/q+1)p/p1×(∫2r∞‍∥g∥Lp2(B(0,t))p2/pdt|t|(n-α)p2/p+1-(n/p′))p/p2≤C(∫2r∞‍∥f∥Lp1(B(0,t))p1/pdttn/q+1)p/p1×(∫2r∞‍|t|(p/p1)(n-α)∥g∥Lp2(B(0,t))p2/pdt|t|(n-α)p2/p+1-(n/p′))p/p2≤C(∫2r∞‍∥f∥Lp1(B(0,t))p1/pdttn/q+1)p/p1×(∫2r∞‍∥g∥Lp2(B(0,t))p2/pdt|t|n/q+1)p/p2.
So
(4.11)∥BΩ,α(f1,g2)∥Lq(B(0,r))≤Crn/q(∫2r∞∥f∥‍Lp1(B(0,t))p1/pdtt(n/q)+1)p/p1(∫2r∞‍∥g∥Lp2(B(0,t))p2/pdtt(n/q)+1)p/p2.
By the same estimating, we also can obtain(4.12)∥BΩ,α(f2,g1)∥Lq(B(0,r))≤Crn/q(∫2r∞‍∥f∥Lp1(B(0,t))p1/pdtt(n/q)+1)p/p1(∫2r∞‍∥g∥Lp2(B(0,t))p2/pdtt(n/q)+1)p/p2.
To estimateBΩ,α(f2,g2), we get
(4.13)|BΩ,α(f2,g2)|=|∫ℝn‍f1(x-y)g2(x+y)Ω(y)|y|n-αdy|≤(∫ℝn‍|f2p1/p(x-y)Ω(y)||y|n-αdy)p/p2×(∫ℝn‍|g2p2/p(x-y)Ω(y)||y|n-αdy)p/p2≤(∫(2B)c‍|fp1/p(y)Ω(x-y)||y|n-αdy)p/p1×(∫(2B)c‍|gp2/p(y)Ω(x-y)||y|n-αdy)p/p2≤C(∫(2B)c‍|fp1/p(y)Ω(x-y)|∫|y|∞‍dt|t|n-α+1dy)p/p1×(∫(2B)c‍|gp2/p(y)Ω(x-y)|∫|y|∞‍dt|t|n-α+1dy)p/p2≤C(∫2r∞‍∫2r≤|y|<t∞‍|fp1/p(y)Ω(x-y)|dydt|t|n-α+1)p/p1×(∫2r∞‍∫2r≤|y|<t∞‍|gp2/p(y)Ω(x-y)|dydt|t|n-α+1)p/p2≤C(∫2r∞‍∥f∥Lp1(B(0,t))p1/pdttn/q+1)p/p1(∫2r∞‍∥g2∥Lp2(B(0,t))p2/pdt|t|n/q+1)p/p2.
Combining the above estimates, we end the proof of Lemma4.2.Theorem 4.3.
Suppose0<α<n, 1/p=1/p1+1/p2, 1<p<n/α,  1/q=1/p-α/n, and s≥p′. If (φ1,ν1) satisfies the condition
(4.14)∫r∞‍essinft<s<∞φ1p1/p(s)sn/ptn/q+1≤Cν1p1/p(r),
and (φ2,ν2) satisfies the condition
(4.15)∫r∞‍essinft<s<∞φ2p1/p(s)sn/ptn/q+1≤Cν2p1/p(r),
where the constant C>0 does not depend on r. Let φ=ν1ν2, then BΩ,α is bounded from B˙p1,φ1×B˙p2,φ2 to B˙q,φ.Proof.
By Theorem E and Lemma4.2, we have
(4.16)∥BΩ,α(f,g)∥B˙q,φ(ℝn)≤Csupr>0φ(r)-1(∫r∞∥f∥Lp1(B(0,t))p1/p‍dttn/q+1)p/p1×(∫r∞‍∥g2∥Lp2(B(0,t))p2/pdt|t|n/q+1)p/p2=Csupr>0(ν1(r)-p1/p∫0r-n/q‍∥f∥Lp1(B(0,t-q/n))p1/pdt)p/p1×(ν2(r)-p2/p∫0r-n/q‍∥g2∥Lp2(B(0,t-q/n))p2/pdt)p/p2=Csupr>0(ν1(r-q/n)-p1/p∫0r‍∥f∥Lp1(B(0,t-q/n))p1/pdt)p/p1×(ν2(r-q/n)-p2/p∫0r‍∥g2∥Lp2(B(0,t-q/n))p2/pdt)p/p2≤Csupr>0(φ1(r-q/n)-p1/prq/p∥f∥Lp1(B(0,r-q/n))p1/p)p/p1×supr>0(φ2(r-q/n)-p2/prq/p∥g∥Lp2(B(0,r-q/n))p2/p)p/p2≤C∥f∥B˙p1,φ1(ℝn)∥g∥B˙p2,φ2(ℝn).Corollary 4.4.
Suppose0<α<n, 1/p=1/p1+1/p2, 1<p<n/α, 1/q=1/p-α/n, s≥p′, λ1<-αp/np1, λ2<-αp/np2, and λ<λ1+λ2+α/n, then BΩ,α is bounded from B˙p1,λ1×B˙p2,λ2 to B˙q,λ.Remark 4.5.
Although we worked on the bilinear case. Applying same ideas in the argument, we may obtain similar extension ofIΩ,α(f→).



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*Source: 406540-2012-11-20.xml*