Weighted Multilinear Hardy Operators on Herz Type Spaces

This paper focuses on the bounds of weighted multilinear Hardy operators on the product Herz spaces and the product Morrey-Herz spaces, respectively. We present a sufficient condition on the weight function that guarantees weighted multilinear Hardy operators to be bounded on the product Herz spaces. And the condition is necessary under certain assumptions. Finally, we extend the obtained results to the product Morrey-Herz spaces.

With the development of analysis theory, many types of Hardy's inequalities have been discussed. For example, a quite number of papers dealt with the various generalizations, numerous variants, and applications of Hardy's inequalities in the past few years. On the detailed discussions of Hardy's inequalities, we choose to refer to [2,3].
In 1984, Carton-Lebrun and Fosset [4] gave the definition of the weighted Hardy operator to be where : [0, 1] → [0, ∞) is a measurable function and is a complex-valued measurable function on R . It is obvious that degenerates into the classical Hardy operator when ≡ 1, = 1, and is defined on R + . In addition, we call the adjoint operator of the weighted Cesàro average . And the definition of is When ≡ 1 and = 1, becomes the classical Cesàro operator It is easy to get that and satisfy when ∈ (R ), ∈ (R ), 1 < < ∞, 1/ + 1/ = 1. This means that and satisfy the commutative rule = . Under certain conditions on , Carton-Lebrun and Fosset [4] proved that maps (R ) into itself for 1 < < ∞. They also pointed out that the operator commutes with the Hilbert transform when = 1 and with certain Calderón-Zygmund singular integral operators including the Riesz 2 The Scientific World Journal transform when ≥ 2. Refer to Xiao [5] for the further extension of the results above; see also [6,7].
Since Herz space is a natural generalization of weighted Lebesgue spaces with power weights, researchers are also interested in studying the boundedness of on Herz spaces. To make the description more clear below, we review the definition of the Herz spaces now. In the following definitions, = , = \ −1 , and = { ∈ R : | | ≤ 2 }, for ∈ Z,̃= ,̃0 = 0 , and̃= , for ∈ N, and is the characteristic function of a set .
In the past few years, the properties of multilinear operators have also been extensively studied by researchers. There are two reasons for this. First, the multilinear operators are the generalization of the linear ones, and its study makes the research contents of analysis theory more rich. Second, the multilinear operators naturally appear in analysis. The study of multilinear operators is traced to the multilinear singular integral operator theory (see [14]). For more detailed studies on multilinear operators, the readers refer to [15][16][17][18] and the references therein. Recently, we have studied the boundedness of weighted multilinear Hardy operators H on the product of Lebesgue spaces and central Morrey spaces in [19]. Based on these results and inspired by the results of [12,13], this paper further concerns the boundedness of H on the product Herz spaces and the product Morrey-Herz spaces. We first recall the definition of H .
be an integrable function. The weighted multilinear Hardy operator H is defined by In accordance with the case of , we also recall the definition of the weighted multilinear Cesàro operator G that is the adjoint operator of H .
where are measurable complex-valued functions on R , 1 ≤ ≤ .
Note that H and G do not satisfy the following commutative rule: This is different from the case of and . The paper is organized as follows. In Section 2, we present the estimate of the boundedness of H on the product Herz spaces. In Section 3, we give the estimates of the boundedness of H on the product Morrey-Herz spaces.

Boundedness of H on the Product of Herz Spaces
We give the first main result of this paper.
Proof of Theorem 5. In order to simplify the proof, we only consider the case of = 2. Actually, a similar procedure works for all ∈ N.

Boundedness of H on the Product of Morrey-Herz Spaces
Lets give the second main result of this paper.
For the operator G , we have a corresponding result.
Since the proof of Theorem 10 is similar to the proof of Theorem 9, we give the proof of Theorem 9 only.