Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spacesM w (l2). We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and gλ -function, and their multilinear commutators on vector-valued generalized weighted Morrey spacesM w (l2). In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on φ(x, r) without assuming any monotonicity property of φ(x, r) on r. MSC: 42B25; 42B35


Introduction
It is well-known that the commutator is an important integral operator and it plays a key role in harmonic analysis. In 1965, Calderon [2,3] studied a kind of commutators, appearing in Cauchy integral problems of Lip-line. Let K be a Calderón-Zygmund singular integral operator and b ∈ BMO(R n ). A well known result of Coifman, Rochberg and Weiss [9] states that the commutator operator [b, K]f = K(bf )−b Kf is bounded on L p (R n ) for 1 < p < ∞. The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [6]- [8], [5], [10], [11]).
The classical Morrey spaces were originally introduced by Morrey in [32] 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 [10,11,18,32]. Recently, Komori and Shirai [29] first defined the weighted Morrey spaces L p,κ (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 [21,22] introduced the generalized weighted Morrey spaces M p,ϕ w and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [25,27,28,35]).
The intrinsic square functions were first introduced by Wilson in [40,41]. They are defined as follows. For 0 < α ≤ 1, let C α be the family of functions φ : R n → R such that φ's support is contained in {x : |x| ≤ 1}, R n φ(x)dx = 0, and for x, For (y, t) ∈ R n+1 + and f ∈ L 1,loc (R n ) , set where φ t (y) = t −n φ( y t ) . Then we define the varying-aperture intrinsic square (intrinsic Lusin) function of f by the formula (A α f (t, y)) 2 dydt t n+1 1 2 , where Γ β (x) = {(y, t) ∈ R n+1 + : |x − y| < βt}. Denote G α,1 (f ) = G α (f ) . This function is independent of any particular kernel, such as Poisson kernel. It dominates pointwise the classical square function(Lusin area integral) and its real-variable generalizations. Although the function G α,β (f ) is depend of kernels with uniform compact support, there is pointwise relation between G α,β (f ) with different β: . We can see details in [40].
The intrinsic Littlewood-Paley g-function and the intrinsic g * λ function are defined respectively by When we say that f maps into l 2 , we mean that , where each f j is Lebesgue measurable and, for almost every Let f = (f 1 , f 2 , . . .) be a sequence of locally integrable functions on R n . For any x ∈ R n , Wilson [41] also defined the vector-valued intrinsic square functions of f by G α f (x) l 2 and proved the following result.
Theorem A. Let 1 ≤ p < ∞, 0 < α ≤ 1 and w ∈ A p . Then the operators G α and g * λ,α are bounded from L p w (l 2 ) into itself for p > 1 and from L 1 w (l 2 ) to W L 1 w (l 2 ). Moreover, in [31], Lerner showed sharp L p w norm inequalities for the intrinsic square functions in terms of the A p characteristic constant of w for all 1 < p < ∞. Also Huang and Liu [12] studied the boundedness of intrinsic square functions on weighted Hardy spaces. Moreover, they characterized the weighted Hardy spaces by intrinsic square functions. In [38] and [39], Wang and Liu obtained some weak type estimates on weighted Hardy spaces. In [37], Wang considered intrinsic functions and the commutators generated with BMO functions on weighted Morrey spaces. Let b be a locally integrable function on R n . Setting the kth-order commutators are defined by where b B(x,r) = 1 |B(x,r)| B(x,r) b(y)dy. By the similar argument as in [14] and [37], we can get Theorem B. Let 1 < p < ∞, 0 < α ≤ 1, w ∈ A p and b ∈ BMO(R n ). Then the kth-order commutator operators [b, G α ] k and [b, g * λ,α ] k are bounded from L p w (l 2 ) into itself.
In this paper, we will consider the boundedness of the operators G α , g α , g * λ,α and their kth-order commutators on vector-valued generalized weighted Morrey spaces. Let ϕ(x, r) be a positive measurable function on R n × R + and w be non-negative measurable function on R n . For any f ∈ L p,loc w (l 2 ) , we denote by M p,ϕ w (l 2 ) the vector-valued generalized weighted Morrey spaces, if When w ≡ 1, then M p,ϕ w (l 2 ) coincide the vector-valued generalized Morrey spaces M p,ϕ (l 2 ). There are many papers discussed the conditions on ϕ(x, r) to obtain the boundedness of operators on the generalized Morrey spaces. For example, in [17] (see, also [18]), by Guliyev the following condition was imposed on the pair (ϕ 1 , ϕ 2 ) : where C > 0 does not depend on x and r. Under the above condition, they obtained the boundedness of Calderón-Zygmund singular integral operators from M p,ϕ 1 (R n ) to M p,ϕ 2 (R n ). Also, in [1] and [20], Guliyev et. introduced a weaker condition: If 1 ≤ p < ∞, there exits a constant C > 0, such that, for any x ∈ R n and r > 0, If the pair (ϕ 1 , ϕ 2 ) satisfies condition (1.1), then (ϕ 1 , ϕ 2 ) satisfied condition (1.2). But the opposite is not true. We can see remark 4.7 in [20] for details. Recently, in [21,22] (see, also [25,28,35]), Guliyev introduced a weighted condition: If 1 ≤ p < ∞, there exits a constant C > 0, such that, for any x ∈ R n and t > 0, In this paper, we will obtain the boundedness of the vector-valued intrinsic function, the intrinsic Littlewood-Paley g function, the intrinsic g * λ function and their kth-order commutators on vector-valued generalized weighted Morrey spaces when w ∈ A p and the pair (ϕ 1 , ϕ 2 ) satisfies condition (1.3) or the following inequalities, where C does not depend on x and r. Our main results in this paper are stated as follows.
In [40], the author proved that the functions G α f and g α f are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.
Remark 1.7. Note that, in the scalar valued case the Theorems 1.1 -1.4 and Corollaries 1.5 -1.6 was proved in [26] (w ≡ 1) and [27]. Also, in the scalar valued case and w ≡ A p and ϕ 1 (x, r) = ϕ 2 (x, r) ≡ w(B(x, r)) κ−1 p , 0 < κ < 1 Theorems 1.1-1.4 and Corollaries 1.5-1.6 was proved by Wang in [37,36]. How as, if ϕ(x, r) ≡ w(B(x, r)) κ−1 p , then the vector-valued generalized weighed Morrey space M p,ϕ w (l 2 ) coincide the vector-valued weighed Morrey space L p,κ w (l 2 ) and the pair (w(B(x, r)) κ−1 p , w(B(x, r)) κ−1 p ) satisfies the both conditions (1.3) and (1.4). Indeed, by Lemma 3.1 there exists C > 0 and δ > 0 such that for all x ∈ R n and t > r: Throughout this paper, we use the notation A B to mean that there is a positive constant C independent of all essential variables such that A ≤ CB. Moreover, C may be different from place to place.

Vector-valued generalized weighted Morrey spaces
The classical Morrey spaces M p,λ were originally introduced by Morrey in [32] 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 [15,30].
We define the vector-valued generalized weighed Morrey spaces as follows.
Definition 2.1. Let 1 ≤ p < ∞, ϕ be a positive measurable vector-valued function on R n × (0, ∞) and w be non-negative measurable function on R n . We denote by M p,ϕ w (l 2 ) the vector-valued generalized weighted Morrey space, the space of all vector-valued functions f ∈ L p,loc w (l 2 ) with finite norm where L p w (B(x, r), l 2 ) denotes the vector-valued weighted L p -space of measurable functions f for which Furthermore, by W M p,ϕ w (l 2 ) we denote the vector-valued weak generalized weighted Morrey space of all functions f ∈ W L p,loc w (l 2 ) for which where W L p w (B(x, r), l 2 ) denotes the weak L p w -space of measurable functions f for which is the vector-valued two weighted Morrey space.

Preliminaries and some lemmas
By a weight function, briefly weight, we mean a locally integrable function on R n which takes values in (0, ∞) almost everywhere. For a weight w and a measurable set E, we define w(E) = E w(x)dx, and denote the Lebesgue measure of E by |E| and the characteristic function of E by χ E . Given a weight w, we say that w satisfies the doubling condition if there exists a constant D > 0 such that for any ball B, we have w(2B) ≤ Dw(B). When w satisfies this condition, we write brevity w ∈ ∆ 2 .
If w is a weight function, we denote by L p w (l 2 ) ≡ L p w (R n , l 2 ) the vector-valued weighted Lebesgue space defined by finiteness of the norm

We recall that a weight function w is in the Muckenhoupt's class
where the sup is taken with respect to all the balls B and 1 p + 1 p ′ = 1. Note that, for all balls B by Hölder's inequality (2) If w ∈ A ∞ , then w ∈ ∆ 2 . Moreover, for all λ > 1 w(λB) ≤ 2 λ n [w] A∞ w(B).
(3) If w ∈ A p for some 1 ≤ p ≤ ∞, then there exit C > 0 and δ > 0 such that for any ball B and a measurable set S ⊂ B, We are going to use the following result on the boundedness of the Hardy operator (Hg)(t) : where µ is a non-negative Borel measure on (0, ∞). We also need the following statement on the boundedness of the Hardy type operator where µ is a non-negative Borel measure on (0, ∞).
holds for all functions g non-negative and non-increasing on (0, ∞) if and only if and c ≈ A 1 .
Note that, Theorem 3.3 can be proved analogously to Theorem 4.3 in [19].

Proofs of main theorems
Before proving the main theorems, we need the following lemmas.
Let 0 < α ≤ 1, 1 < p < ∞ and w ∈ A p . Then any j ∈ Z + , we have This lemma is easy from the following inequality which is proved in [40].
By the similar argument as in [3], we can get the following lemma.  Then, for p > 1 the inequality holds for any ball B = B(x 0 , r) and for all f ∈ L p,loc w (l 2 ). Moreover, for p = 1 the inequality holds for any ball B = B(x 0 , r) and for all f ∈ L1locl2.
Proof. The main ideas of these proofs come from [22]. For arbitrary x ∈ R n , Then, First, let us estimate I. By Theorem A, we can obtain that On the other hand, Therefore from (4.1) and (4.2) we get Then let us estimate II.
Proof. From the definition of g * λ,α (f ), we readily see that First, let us estimate III.
Now, let us estimate IV. Thus, (4.6) By Lemma 4.3, we have In the following, we will estimate G α,2 j ( f ) L p w B,l 2 . We divide G α,2 j ( f )  For the second part.  For the second part, we divide it into two parts.
First, for A(x), we find that = b * sup = b * f M p,ϕ 1 w (l 2 ) . By using the argument as similar as the above proofs and that of Theorem 1.2, we can also show the boundedness of [b, g * λ,α ] k .