Abstract
In this paper, we consider the problem of boundedness of Hausdorff operator on weighted central Morrey spaces. In particular, we obtain sharp bounds for Hausdorff operators on power weighted central Morrey spaces. Analogous results for the commutators of Hausdorff operators when the symbol functions belong to weighted central-BMO spaces are obtained as well.
Highlights
In recent years, the Hausdorff operator has gained much attention
A number of significant studies have been undertaken in this regard like for example boundedness of one and multidimensional Hausdorff operators on Hardy, Lp and BMO spaces [3,4,5,6,7,8]
Many authors have contributed a lot towards obtaining new estimates on other function spaces
Summary
The Hausdorff operator has gained much attention. This is mainly because of seminal work done by Liflyand and Móricz in [1]. In [20], the authors have obtained some size conditions on such that the commutator generated by Hausdorff operator H with Lipschitz function b is bounded on classical Morrey spaces. For the boundedness of the commutator operator on function spaces of central nature, one usually looks for a corresponding function class to which the symbol function b belongs and which has BMO-type behavior at the origin. Definition 2.2 ([28]) A weight w is said to belong to the reverse Hölder class RHr if there exists a fixed positive constant C and r > 1 such that, for every ball B ⊂ Rn, wr(x) dx It is well known for s > p that Ap ⊂ As and that if w ∈ Ap, 1 < p < ∞, w ∈ Aq for some 1 < q < p.
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