Abstract
In this paper, we give necessary and sufficient conditions for the boundedness of the n-dimensional Hausdorff operators on Herz-type spaces. In addition, the sufficient condition for the boundedness of commutators generated by Lipschitz functions and the fractional Hausdorff operators on Morrey-Herz space is also provided. MSC:26D15, 42B35, 42B99.
Highlights
Recall that for a locally integrable function operator is defined by on (, ∞), the one-dimensional Hausdorff∞ (t) x h f (x) = f dt. t tThe boundedness of this operator on the real Hardy space H (R) was proved in [ ]
3 Lipschitz estimates for n-dimensional fractional Hausdorff operator we will prove that the commutator generated by Lipschitz function b and the fractional Hausdorff operator H,γ is bounded on the Morrey-Herz space
Proof In the operator Hb,γ f (x), we replace (t) = (t) = t–n+γ χ(,∞)(t), we obtain the commutator of the n-dimensional fractional Hardy operator, Hb,γ f (x) = Hγ,bf (x)
Summary
The boundedness of this operator on the real Hardy space H (R) was proved in [ ]. In [ ], the same operator was studied on product of Hardy spaces. It is easy to show that the n-dimensional fractional Hardy operator In [ ], Xaio obtained the sharp bounds for the Hardy Littlewood averaging operator on Lebesgue and BMO spaces.
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