Commutators of weighted Hardy operators on Herz-type spaces

Multilinear Hardy–Cesàro operator and commutator on the product of Morrey–Herz spaces

“…Lu [5] Notice that there are several results on commutator for other type of Hardy operators or even for some class of Hausdorff operators (for example see [1,7,8,6,10,11] and references therein). But there is only partial such results for the commutator operator [b, U ψ ], with symbols are Lipschitz, in Morrey-Herz space by Tang-Xue-Zhou [16]. In this section, we will improve this result and extend it to the operator U …”

“…, s m such that U m,n, b ψ, s (see definition in Section 4) is bounded on the product of Morrey-Herz spaces. When reduce to the case of U ψ , we will show that our sufficient condition is weaker than one obtained in [16].…”

“…On the other hand, recently Tang, Xue, and Zhou [16] find a sufficient condition on weights so that U ψ is bounded on Morrey-Herz spaces. In this paper we make use of another approach which allows us to obtain a sufficient condition in terms of a finite integrals on ψ and s 1 , .…”

“…Xiao's results has been followed by a vast amount of research geared towards understanding the weighted Hardy-Littlewood average operators U ψ , see for examples [1,5,6,8,7,9,13,16] and the references therein. These includes the work of Chuong and Hung in [1], where they introduced a class of integral transforms as follows.…”