1 Introduction and results
Multilinear harmonic analysis is an active area of research that is still developing. Multilinear operators appear also as technical tools in the study of linear singular integral (through the method of rotations), the analysis of nonlinear operators (through power series and similar expansions), and the resolution of many linear and nonlinear partial differential equations [4, 5, 15-17, 41].
It is well known that in 1967, Bajsanski and Coifman  proved the boundedness of the multilinear operator associated with the commutators of singular integrals considered by Calderon. In 1981, Cohen  studied the Lp boundedness of the multilinear integral operator TA defined by where Ω is homogeneous of degree zero on ℝn with mean value zero on Sn-1. Moreover, Rm (A;x, y) denotes the m-th (m ≥ 2) remainder of the Taylor series of A at x about y; more precisely,
Using the method of good — λ inequality, in 1986, Cohen and Gosselin  proved that if Ω ∈ Lip1 (Sn-1) and DγA ∈ BMO(ℝn), then where the constant C > 0 is independent of f and A.
In 1994, for m = 2, Hofmann  proved that the multilinear operator TA is a bounded operator on Lp,w when Ω ∈ L∞(Sn-1) and w ∈ Ap.
It is natural to ask whether the multilinear fractional integral operator with a rough kernel has the mapping properties similar to those of The purpose of  is to study this problem. Let us give the definition of the multilinear fractional integral operator as follows:
where Ω is homogeneous of degree zero and is as above.
When k = 1 and m = 1, then is just the commutator of the fractional integral TΩ,αf with function A,
When mj = 1 and Aj = A for j = 1,..., k, then is just the higher-order commutator given in ,
When mj ≥ 2, f is a non-trivial generalization of the above commutator.
The classical Morrey spaces were originally introduced by Morrey in  to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [17, 18, 23, 36, 41]. Mizuhara  introduced generalized Morrey spaces. Later, in  Guliyev defined the generalized Morrey spaces Mp,ϕ with normalized norm. Recently, Komori and Shirai  considered the weighted Morrey spaces Lp,k (w) and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev  introduced the generalized weighted Morrey spaces Mp,ϕ(w) and studied the boundedness of the classical operators and its commutators in these spaces Mp,ϕ(w), see also [24, 29, 33, 40]. In  the author gave a concept of generalized weighted Morrey space Mp,ϕ(w) which could be viewed as extension of both generalized Morrey space Mp,ϕ and weighted Morrey space Lp,K(w).
The following theorem was proved by Ding and Lu in .
(). Let and let Ω be homogeneous of degree zero with Ω ∈ Ls(Sn-1). Moreover,for 1 ≤ j ≤ k, |γj| = mj - 1, mj ≥ 2 and Then there exists a constant C, independent of Aj,1 ≤ j ≤ k and f, such that
Here and in the sequel, we always denote by p' the conjugate index of any p > 1, that is 1=p + 1=p' = 1, and by C a constant which is independent of the main parameters and may vary from line to line. We define the generalized weighed Morrey spaces as follows.
Let 1 ≤ ρ < ∞, ϕ be a positive measurable function on ℝ × (0, ∞) and w be non-negative measurable function on ℝn. We denote by Mp,ϕ(w) the generalized weighted Morrey space, the space of all functions with finite norm
where Lp,w(B(x, r)) denotes the weighted Lp-space of measurable functions f for which
Furthermore, by WMp,ϕ(w) we denote the weak generalized weighted Morrey space of all functions for which
where WLp,w(B(x, r)) denotes the weak Lp,w-space of measurable functions f for which
(1) If w ≡ 1, then Mp,ϕ(1) = Mp,ϕ is the generalized Morrey space.
(2) If is the weighted Morrey space.
(3) If is the two weighted Morrey space.
(4) If is the classical Morrey space and WMρ,ϕ(w) = WLp,λ(ℝn) is the weak Morrey space.
(5) If is the weighted Lebesgue space.
The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [15-17, 26, 27, 41]. In the recent development of commutators, Pérez and Trujillo-González  generalized these multilinear commutators and proved the weighted Lebesgue estimates. Ye and Zhu in  obtained the boundedness of the multilinear commutators in weighted Morrey spaces Lp,k(w) for 1 < p < ∞ and 0 < κ < 1, where the symbol belongs to bounded mean oscillation (BMO)n. Furthermore, the weighted weak type estimate of these operators in weighted Morrey spaces is Lp, k(w) for p = 1 and 0 < κ < 1. The following statement was proved by Guliyev in .
(). Let 0 < α < n, 1 < p < n/α and 1/q = 1/p — α/n, Ω ∈ L∞(𝕊n-1), w ∈ Ap,q, A ∈ BMO(ℝn), and (ϕ1, ϕ2) satisfies the condition (1)
where C does not depend on χ and r. Then the operator is bounded from
It has been proved by many authors that most of the operators which are bounded on a weighted (unweighted) Lebesgue space are also bounded in an appropriate weighted (unweighted) Morrey space, see [8, 44]. As far as we know, there is no research regarding boundedness of the fractional multilinear integral operator on Morrey space. In this paper, we are going to prove that these results are valid for the rough fractional multilinear integral operator on generalized weighted Morrey space. Our main results can be formulated as follows.
Let 0 < α < n, 1 ≤ s' < p < n/α and 1/q = 1/ p — α/n. Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1) and (ϕ1,ϕ2) satisfy the condition (1). Let also, for 1 ≤ j ≤ k, |γj| = mj - 1, mj ≥ 2 and Suppose then the operator is bounded from Moreover, then there is a constant C > 0 independent of f and A1, A2, ... , Ak such that
(). Let 0 < a < n, 1 ≤ s' < p < n/α and 1/q = 1/p — α/n. Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1), and (ϕ1, ϕ2) satisfy the condition
where C0 does not depend on x and r. Let also, for 1 ≤ j ≤ k, |γj| = mj - 1, mj ≥ 2 and Then the operator is bounded from Moreover, there is a constant C > 0 independent of f such that
Let and wq ∈ A∞(ℝn). Then (ϕ1, ϕ2) satisfies the condition (1).
In fact, from (2.8) in Section 2 we have constant δ > 0 such that
Since 0 < κ < p/q, then Thus
Let 0 < α < n, let 1 ≤ s' < p < n/α, and let 1/q = 1/p — α/n. Let also, for 1 ≤ j ≤ k, |γj| = mj - 1, mj ≥ 2 and Suppose is bounded from Lp,k(wp,wq)(ℝn) to Lq,k q/p(wq,ℝn) and
where the constant C > 0 is independent of f and A1, A2 ,..., Ak.
Note that, in  the Nikolskii-Morrey type spaces were introduced and the authors studied some embedding theorems. In the next paper, we shall introduce the generalized weighted Nikolskii-Morrey spaces and will study some embedding theorems. We will also investigate the boundedness of fractional multilinear integral operators with rough kernels on the generalized weighted Nikolskii-Morrey spaces, see for example, . These results may be applicable to some problems of partial differential equations; see for example [6, 7, 19, 20, 26, 28, 30, 43].
2 Some preliminaries
We begin with some properties of Ap weights which play a great role in the proofs of our main results. A weight w is a nonnegative, locally integrable function on ℝn. Let B = B(x0,rB) denote the ball with the center x0 and radius rB. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by | E | and set weighted measure For any given weight function w on ℝn, Ω ⊆ ℝn and 0 < p < ∞, denote by Lp,w (Ω) the space of all function ƒ satisfying
A weight w is said to belong to Ap for 1 < p < ∞, if there exists a constant
where p' is the dual of ρ such that The class α1 is defined by
A weight w is said to belong to A∞(ℝn) if there are positive numbers C and δ so that
for all balls B and all measurable E C B. It is well known that
The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp -boundedness of Hardy-Littlewood maximal function in .
(i) For any 1 ≤ p < ∞, there is a positive number C such that
(ii) For any 1 ≤ p < ∞, there is a positive number C and S such that (2)
(iii) For any 1 < p < ∞, one has
We also need another weight class Ap,q introduced by Muckenhoupt and Wheeden in  to study weighted boundedness of fractional integral operators.
Given 1 ≤ p ≤ q < ∞. We say that ω ∈ Ap,q if there exists a constant C such that for every ball B ⊂ ℝn, the inequality (3) holds when 1 < p < ∞, and for every ball B ⊂ ℝn the inequality
holds when p = 1.
By (3), we have (4)
Given 1 ≤ p ≤ q < ∞.
(i) w ∈ Ap,q if and only if
(ii) w ∈ Ap,q if and only if
(iii) w ∈ Ap,p if and only if wp ∈ Ap;
(iv) If p1 < p2 and q2 > q1, then
In this paper, we need the following statement on the boundedness of the Hardy type operator
where μ is a non-negative Borei measure on (0, ∞).
holds for all non-negative and non-increasing g on (0, ∞) if and only if
and c ≈ A1.
([39, Theorem 5, p. 236]). Let w ∈ A∞. Then the norm of BMO(ℝn) is equivalent to the norm of BMO(w), where
(1) The John-Nirenberg inequality : there are constants C1,C2 > 0, such that for all b ∈ BMO(ℝn) and β > 0
(2) For 1 < p < ∞ the John-Nirenberg inequality implies that
and for 1 < ρ < ∞ and w ∈ A∞
The following lemma was proved by Guliyev in .
i) Let w ∈ A∞ and b be a function in BMO(ℝn). Let also 1 ≤ p < ∞, x ∈ ℝn, and r1,r2 > 0. Then
where C > 0 is independent of f, x, r1 and r2.
ii) Let w ∈ Ap and b be a function in BMO(ℝn). Let also 1 < p < ∞, x ∈ ℝn, and r1,r2 > 0. Then
where C > 0 is independent of f, x, r1 and r2.
Below we present some conclusions about Rm(A;x, y).
(). Suppose b is a function on ℝn with the m-th derivatives in Lq(ℝn), q > n. Then
The following property is valid.
Let x ∈ B(x0, r), y ∈ B(x0, 2j+1r)\B(x0, 2jr). Assume that A has derivatives of order m - 1 in BMO(ℝn). Then there exists a constant C, independent of A, such that (5)
For fixed x ∈ ℝn, let
From Lemma 2.7 we have, (7)
When x ∈ B(x0,r), y ∈ B(x0, 2j+1r)\B(x0, 2jr), then 2j-1 r ≤ |x - y| ≤ 2j+2 r. Thus, we have
Combining with (6), (7) and (8), then (5) is proved.
Finally, we present a relationship between essential supremum and essential infimum.
(). Let f be a real-valued nonnegative function and measurable on E. Then
3 A local weighted Guliyev type estimates
Let 1 ≤ s' <p <n/α, and let 1/q = 1/p – α/n. Let also, for 1 ≤ j ≤ k, |γj | = mj – 1, mj ≥ 2 and Aj ∈ BMO. (ℝn) Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1), then for any r > 0, there is a constant C independent of f such that (9)
We write f as f = f1 + f2, where denotes the characteristic function of B(x0,2r) Then
Since , by the boundedness of from (Theorem 1.1) we get
Note that q > p > 1 and , then by Hölder’s inequality,
Since by (4), for all r > 0 we get (10)
To simplify process of Theorem 3.1, in the following discussion we consider only the case k = 2. The method can be used to deal with the case k >2 without any essential difficulty.
Let N = m1 + m2 – 2, Δi = (B(x0, 2i+1 r)) \ (B(x0, 2i r)), and let x ∈ B(x0, r)) By Lemma 2.8,
By Hölder’s inequalities,
When x ∈ B(x0, s) and y ∈ Δi, then by a direct calculation, we can see that 2i-1 r ≤|y–x| < 2i-1 r. Hence (11)
We also note that if x ∈ B(x0, r), y ∈ B(x0, 2r)c, then |y – x| ≈ |y – x0|. Consequently (12)
Since s' <p, it follows from Hölder’s inequality that
From (10) we know (13)
On the other hand, by Hölder’s inequality and (11), (12), we have
Applying Hölder’s inequality we get
Similarly to the estimates for I2, we have
Finally, we come to estimate I4.
By Hölder’s inequality and (11), (12), we have
Applying Hölder’s inequality we get
Since , then from the Lemma2.6 and the inequality (13) we have (14)
Then from (14) we have
Combining with the estimates of I1, I2, I3 and I4, we have
Then we get
This completes the proof of Theorem 3.1.
4 Proof of Theorem 1.5
The authors would like to thank the referees for careful reading the paper and useful comments.
The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A3.16.023).
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Published Online: 2016-12-17
Published in Print: 2016-01-01