Some Estimates of Rough Bilinear Fractional Integral

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 numbers θ1, . . . , θm, and 0 < α < n, the m-linear fractional with rough kernel is defined by


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 numbers θ 1 , . . ., θ m , and 0 < α < n, the m-linear fractional with rough kernel is defined by where Ω ∈ L s S n−1 s ≥ 1 is homogeneous of degree zero on R n , and S n−1 denotes the unit sphere of R n .
When Ω ≡ 1, The L p boundedness of operator I 1,α has been well studied in 1, 2 .Recently, Hendar and Idha discussed the boundedness property of I 1,α on generalized Morrey space in 3 .
The study of the operators B Ω,α and its related operators with rough kernel Ω recently attracted many attentions.In 2002, Ding and Chin first discussed its L p R n boundedness.The following theorem is their main result: there exists a positive constant C such that for any f ∈ L p 1 R n , g ∈ L p 2 R n , (1) when s < min{p 1 , p 2 }, (2) when s min{p 1 , p 2 }, Later, when q > 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. Let q > n/ n − α , 0 < α < n, p 1 , p 2 > 1 and If Ω ∈ L n/ n−α S n−1 , then there exists a positive constant C such that We note that when q ≤ 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 I Ω,α with rough kernel from by using the pointwise rearrangement estimate of the m-linear convolution.
Theorem C. Let 0 < α < n, Ω and be homogeneous of degree zero on R n , Ω ∈ L n/ n−α S n−1 , let p be the harmonic mean of p 1 , p 2 , . . ., p m > 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 This paper is organized as follows: in the second part of this work we prove some boundedness properties of B Ω,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 B Ω,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 B Ω,α on the generalized center Morrey space.Since Morrey space, modified Morrey space and central Morrey space all can be seen as generalized L p space.

The Boundedness of B Ω,α on Morrey Space
The classical Morrey spaces L p,λ R n 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 .
For x ∈ R n 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 L p,λ R n is defined by where So we only consider the case 0 < λ < n.
Since Morrey space can be seen as the generalized L p space, we will be interested in the boundedness of B Ω,α on Morry space L p,λ R n .In order to prove our results, we need the following bilinear maximal function: then there exists a positive constant C such that Proof.In 10 , Fefferman and Stein have proved that for every p, 1 < p < ∞, there is a constant C p > 0 such that for any measurable functions f on R n and ϕ ≥ 0, the following inequality holds, 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 where

2.9
Theorem 2.2.Suppose 0 < α < n, and let Ω ∈ L s S n−1 be homogeneous of degree zero on R n , let p be the harmonic mean of p 1 and p 2 , then there exists a positive constant C such that

2.12
Estimate of I 1 x is and estimate of I 2 x is

2.14
For F σ f, g x , we have the following estimates: Journal of Function Spaces and Applications

2.15
Combining the above estimates, we have Let By computation, we get

2.17
Taking the supremum of r, we have
Corollary 2.4.Let 0 < α < n, Ω ∈ L s S n−1 be homogeneous of degree zero on R n , p be the harmonic mean of p 1 and p 2 , 1 < p < n/α, 0 < λ < n − αp, and s < p.If then there exists a positive constant C such that Proof.By H ölder inequality, it is easy to know when t n − λ q/ n − μ , we have L t,λ R n ⊆ L q,μ R n , through the given condition, 1/t 1/p − α/ n − λ .Applying Theorem 2.2, we get 2.24 From the inequality and Theorem 2.2, we obtain an Olsen inequality involving a multiplication operator.

2.25
One has In 21 , the authors discussed the boundedness of maximal function in modified Morrey spaces L p,λ R n and obtained the following generalized Hardy-Littlewood-Sobolev inequalities in modified Morrey spaces.
/ n−λ is necessary and sufficient for the boundedness of the operator I α from L p,λ R n to L q,λ R n .
We also can extend Theorem D to the multilinear case.Lemma 3.2.Let p > 1, 0 < λ < n and 1/p 1/p 1 1/p 2 .If then there exists a positive constant C such that Proof.When 1 ≤ δ < p, the following inequality: holds, where M is the Hardy-littlewood maximal function and Hence, with the same arguments in Lemma 2.1, we complete the proof of Lemma 3.2.
Theorem 3.3.Suppose 0 < α < n, Ω ∈ L s S n−1 and let be homogeneous of degree zero on R n , let p be the harmonic mean of p 1 and p 2 , 1 < p < n/α, 0 < λ < n − αp, s < p and λ/p λ 1 /p 1 λ 2 /p 2 , 0 < λ 1 , λ 2 < n.Then the condition α/n ≤ 1/p − 1/q ≤ α/ n − λ is necessary and sufficient for the boundedness of B Ω,α from Do the same decomposition of B Ω,α f, g x in the proof of Theorem 2.2, then we only need to estimate F σ f, g x .We can easily obtain

Journal of Function Spaces and Applications
For 0 < ε < 1/2, we get

3.7
While ε ≥ 1/2, we obtain Thus, we obtain Hence, by the boundedness of M f, g x in Lemma 3.2, we prove that B Ω,α is bounded from 2 Necessity.Let 1 < p < n/α and f ∈ L p 1 ,λ 1 R n , g ∈ L p 2 ,λ 2 R n .Denote f t x : f tx , g t x : g tx , and t 1, max{1, t}.Then from 21 , we have B Ω,α f, g L q,λ R n .

3.12
By the boundedness of B Ω,α , we have

The Boundedness of B Ω,α on Generalized Center Morrey Space
Definition 4.1.Let ϕ r be a positive measurable function on R and 1 ≤ p < ∞.We denote by Ḃp,ϕ R n the generalized central Morrey space, the space of all functions f ∈ L loc p R n with finite quasinorm 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 spaces Ḃp,λ R n under the choice ϕ r r nλ .About the Ḃp,λ R n space, the readers can refer to 22 , In fact, we can easily check that Ḃp,λ R n is a Banach space, Ḃp,λ R n reduce to {0} when λ < −1/p, Ḃp, −1/p R n L p R n and Ḃp,0 R n Ḃp R n .There are many papers that discussed the conditions on ϕ 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 ϕ 1 , ϕ 2 : ∞ r ess inf t<s<∞ ϕ 1 s s n/p t n/q 1 ≤ Cϕ 2 r 4.2 for the fractional integral I α , where 1/q 1/p − α/n and C > 0 does not depend on r. Theorem

4.5
In this section we are going to discuss the boundedness of B Ω,α on generalized central Morrey space.Lemma 4.2.Suppose 0 < α < n, 1/p 1/p 1 1/p 2 , 1/q 1/p − α/n, and s ≥ p , then for 1 < p < n/α, the inequality holds for any ball B 0, r and for all f ∈ L loc p 1 R n and g ∈ L loc p 2 R n .

4.8
Journal of Function Spaces and Applications 13 Since B Ω,α is bounded from L p 1 × L p 2 to L q , we have 4.9 where the constant C > 0 is independent of f and g.
To estimate B Ω,α f 1 , g 2 , it follows that

4.11
By the same estimating, we also can obtain

1 g 2 p 2
B 0,t −q/n dt p/p 1 × ν 2 r −q/n −p 2 /p r 0 /p L p 2 B 0,t −q/n dt p/p 2 16Journal of Function Spaces and Applications 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 R n in 21 .