In this paper, we introduce a type of generalized Hausdorff operators and characterize the boundedness of these operators on Lebesgue spaces and central Morrey spaces. Moreover, we obtain the operator norms on these spaces. We also obtain sufficient and necessary conditions which ensure the boundedness of their commutators on Lebesgue spaces and central Morrey spaces with symbols in central BMO spaces. As applications, we give a new method to obtain sharp bounds for weighted Hardy operators and weighted Cesàro operators on Lebesgue spaces and central Morrey spaces.
The aim of this paper is to study the compactness for the commutators of intrinsic square functions, including the intrinsic g * λ-function and the intrinsic Littlewood-Paley g-function. Using a weighted version of the Frechét-Kolmogorov-Riesz theorem, the compactness for their commutators generated with the CMO functions is obtained on the weighted Lebesgue spaces.
In this paper, the authors study the fractional Carleson type maximal operators T * β which is defined bywhere 0 < β < n and Ω satisfies the L q -Dini conditions with 1 < q < ∞ . The authors prove the L p → L p boundedness of T * β under certain conditions.
In this paper, we give sufficient and necessary conditions for the endpoint estimates of the commutators generated by the strongly singular integrals and the BMO function on the extreme case.
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