Multilinear fractional integral operators on generalized weighted Morrey spaces

The classical Morrey spaces were introduced by Morrey [1] in 1938, have been studied intensively by various authors, and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [2‐4] for details. Moreover, various Morrey spaces have been defined in the process of this study. Mizuhara [5] introduced the generalized Morrey space M φ p; Komori and Shirai [6] defined the weighted Morrey spaces L p , κ ( ω ); Guliyev [7] gave the concept of generalized weighted Morrey space M φ p ( ω ), which could be viewed as an extension of both M φ p and L p , κ ( ω ). The boundedness of some operators on these Morrey spaces can be seen in [5‐9].

Let R n be the n-dimensional Euclidean space, ( R n ) m = R n × ⋯ × R n be the m-fold product space (m ∈ N), and let f → = ( f 1 , … , f m ) be a collection of m functions on R n. Given α ∈ ( 0 , m n ) and ( b 1 , … , b m ) ∈ ( B M O ) m. We consider the multilinear fractional integral operators I α , m defined by

The corresponding m-linear commutators I α , m Σ b and the iterated commutators I α , m Π b defined by, respectively,

and

As is well known, multilinear fractional integral operator was first studied by Grafakos [10], subsequently, by Kenig and Stein [11], Grafakos and Kalton [12]. In 2009, Moen [13] introduced weight function A P → , q and gave weighted inequalities for multilinear fractional integral operators; In 2013, Chen and Wu [14] obtained the weighted norm inequalities for the multilinear commutators I α , m Σ b and I α , m Π b. More results of the weighted inequalities for multilinear fractional integral and its commutators can be found in [15‐17].

The aim of the present paper is to investigate the boundedness of multilinear fractional integral operator and its commutator on the generalized weighted Morrey spaces. Our results can be formulated as follows.

Theorem 1.1 Let m ≥ 2 and let 0 < α < m n. Suppose 1 / p = ∑ i = 1 m 1 / p i, 1 / q i = 1 / p i − α / m n, and 1 / q = ∑ i = 1 m 1 / q i = 1 / p − α / n, ω → = ( ω 1 , … , ω m ) satisfy the A p → , q condition with ω 1 q 1 , … , ω m q m ∈ A ∞, and φ k → = ( φ k 1 , … , φ k m ), k = 1 , 2, satisfy the condition

where φ 2 = ∏ i = 1 m φ 2 i, ν ω → = ∏ i = 1 m ω i. If p 1 , … , p m ∈ ( 1 , ∞ ), then there exists a constant C independent of f → such that

If p 1 , … , p m ∈ [ 1 , ∞ ), and min { p 1 , … , p m } = 1, then there exists a constant C independent of f → such that

Theorem 1.2 Let m ≥ 2 and let 0 < α < m n. Suppose p 1 , … , p m ∈ ( 1 , ∞ ) with 1 / p = ∑ i = 1 m 1 / p i, 1 / q i = 1 / p i − α / m n and 1 / q = ∑ i = 1 m 1 / q i = 1 / p − α / n, ω → = ( ω 1 , … , ω m ) satisfy the A p , → q condition with ω 1 p 1 , … , ω m p m ∈ A ∞, ν ω → = ∏ i = 1 m ω i, and φ k → = ( φ k 1 , … , φ k m ), k = 1 , 2, satisfy the condition

where φ 2 = ∏ i = 1 m φ 2 i, ν ω → = ∏ i = 1 m ω i. If ( b 1 , … , b m ) ∈ ( B M O ) m, then there exists a constant C > 0 independent of f → such that

and

A weight ω is a nonnegative, locally integrable function on R n. Let B = B ( x 0 , r B ) denote the ball with the center x 0 and radius r B. For any ball B and λ > 0, λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by | E | and set weighted measure ω ( E ) = ∫ E ω ( x ) d x.

The classical A p weight theory was first introduced by Muckenhoupt in the study of weighted L p boundedness of Hardy-Littlewood maximal functions in [18]. A weight ω is said to belong to A p for 1 < p < ∞, if there exists a constant C such that for every ball B ⊂ R n,

where p ′ is the dual of p such that 1 / p + 1 / p ′ = 1. The class A 1 is defined by replacing the above inequality with

A weight ω is said to belong to A ∞ if there are positive numbers C and δ so that

for all balls B and all measurable E ⊂ B. It is well known that

We need another weight class A p , q introduced by Muckenhoupt and Wheeden in [19]. A weight function ω belongs to A p , q for 1 < p < q < ∞ if there is a constant C > 0 such that, for every ball B ⊂ R n,

When p = 1, ω is in the class A 1 , q with 1 < q < ∞ if there is a constant C > 0 such that, for every ball B ⊂ R n,

Let us recall the definition of multiple weights. For m exponents p 1 , … , p m, we write p → = ( p 1 , … , p m ). Let p 1 , … , p m ∈ [ 1 , ∞ ), 1 / p = ∑ i = 1 m 1 / p i, and let q > 0. Given ω → = ( ω 1 , … , ω m ), set ν ω → = ∏ i = 1 m ω i. We say that ω → satisfies the A p → , q condition if it satisfies

When p i = 1, ( 1 | B | ∫ B ω i ( x ) − p i ′ ( x ) d x ) 1 / p i ′ is understood as ( inf x ∈ B ω i ( x ) ) − 1.

Lemma 2.1 [13, 14]

Let 0 < α < m n, and p 1 , … , p m ∈ [ 1 , ∞ ), let 1 / p = ∑ k = 1 m 1 / p k, and let 1 / q = 1 / p − α / n. If ω → ∈ A p → , q, then

where ν ω → = ∏ i = 1 m ω i.

Lemma 2.2 [20]

Let m ≥ 2, q 1 , … , q m ∈ [ 1 , ∞ ) and q ∈ ( 0 , ∞ ) with 1 / q = ∑ i = 1 m 1 / q i. Assume that ω 1 q 1 , … , ω m q m ∈ A ∞ and ν ω → = ∏ i = 1 m ω i. Then for any ball B, there exists a constant C > 0 such that

Let 1 ≤ p < ∞, let φ be a positive measurable function on R n × ( 0 , ∞ ), and let ω be a nonnegative measurable function on R n. Following [7], we denote by M φ p ( ω ) the generalized weighted Morrey space and the space of all functions f ∈ L loc p ( ω ) with finite norm

where

Furthermore, by W M φ p ( ω ) we denote the weak generalized weighted Morrey space of all function f ∈ W M φ p ( ω ) for which

where

Let us recall the definition and some properties of B M O. A locally integrable function b is said to be in B M O if

where b B = | B | − 1 ∫ B b ( y ) d y.

Lemma 2.3 (John-Nirenberg inequality; see [21])

Let b ∈ B M O. Then for any ball B ⊂ R n, there exist positive constants C 1 and C 2 such that for all λ > 0,

By Lemma 2.3, it is easy to get the following.

Lemma 2.4 Suppose ω ∈ A ∞ and b ∈ B M O. Then for any p ≥ 1 we have

Lemma 2.5 [22]

Let b ∈ B M O, 1 ≤ p < ∞, and r 1 , r 2 > 0. Then

where C > 0 is independent of f, x 0, r 1, and r 2.

By Lemma 2.4 and Lemma 2.5, it is easily to prove the following results.

Lemma 2.6 Suppose ω ∈ A ∞ and b ∈ B M O. Then for any 1 ≤ p < ∞ and r 1 , r 2 > 0, we have

We also need the following result.

Lemma 2.7 [23]

Let f be a real-valued nonnegative function and measurable on E. Then

At the end of this section, we list some known results about weighted norm inequalities for the multilinear fractional integrals and their commutators.

Lemma 2.8 [13]

Let m ≥ 2 and let 0 < α < m n. Suppose 1 / p = 1 / p 1 + ⋯ + 1 / p m, 1 / q = 1 / p − α / n, ω → = ( ω 1 , … , ω m ) satisfies the A p → , q condition. If p 1 , … , p m ∈ ( 1 , ∞ ), then there exists a constant C independent of f → = ( f 1 , … , f m ) such that

If p 1 , … , p m ∈ [ 1 , ∞ ), and min { p 1 , … , p m } = 1, then there exists a constant C independent of f → such that

where ν ω → = ∏ i = 1 m ω i.

Lemma 2.9 [14]

Let m ≥ 2, let 0 < α < m n and let ( b 1 , … , b m ) ∈ ( B M O ) m. For 1 < p 1 , … , p m < ∞, 1 / p = 1 / p 1 + ⋯ + 1 / p m, and 1 / q = 1 / p − α / n, if ω → ∈ A p → , q, then there exists a constant C > 0 such that

and

where ν ω → = ∏ i = 1 m ω i.

We first prove the following conclusions.

Theorem 3.1 Let m ≥ 2 and let 0 < α < m n. Suppose 1 / p = ∑ i = 1 m 1 / p i, 1 / q i = 1 / p i − α / m n, and 1 / q = ∑ i = 1 m 1 / q i = 1 / p − α / n, ω → = ( ω 1 , … , ω m ) satisfy the A P → , q condition with ω 1 q 1 , … , ω m q m ∈ A ∞. If p 1 , … , p m ∈ ( 1 , ∞ ), then there exists a constant C independent of f → such that

If p 1 , … , p m ∈ [ 1 , ∞ ), and min { p 1 , … , p m } = 1, then there exists a constant C independent of f → such that

where ν ω → = ∏ i = 1 m ω i.

Proof We represent f i as f i = f i 0 + f i ∞, where f i 0 = f i χ B ( x 0 , 2 s ), i = 1 , … , m, and χ B ( x 0 , 2 s ) denotes the characteristic function of B ( x 0 , 2 s ). Then

where each term of Σ ′ contains at least one α i ≠ 0. Since I α , m is an m-linear operator,

and

Then by (2.17), if 1 < p i < ∞, i = 1 , … , m, we get

By (2.18), if min { p 1 , … , p m } = 1, then

Applying Hölder’s inequality, for 1 ≤ p i ≤ q i < ∞, i = 1 , … , m, we have

for any ball B ⊂ R n. Then

Thus, for 1 ≤ p i < ∞,

From (2.7) and Lemma 2.2 we get

Using Hölder’s inequality,

Note that 1 / q i = 1 / p i − α / m n, then

Then for 1 ≤ p i < ∞, i = 1 , … , m,

This gives J 0 , … , 0 and K 0 , … , 0 are majored by

For the other term, let us first consider the case when α 1 = ⋯ = α m = ∞. For any x ∈ B ( x 0 , s ), y ∈ B ( x 0 , 2 j + 1 s ) ∖ B ( x 0 , 2 j s ), we have | x − y i | ≈ | x − y j | for i ≠ j. Then

Applying Hölder’s inequality, it can be found that sup x ∈ B ( x 0 , s ) | I α , m ( f 1 ∞ , … , f m ∞ ) ( x ) | is less than

Hence,