Weighted variable Morrey–Herz estimates for fractional Hardy operators

The present article discusses the boundedness criteria for the fractional Hardy operators on weighted variable exponent Morrey–Herz spaces ${M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(w)}$ M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

Generalized Hausdorff Operators on K ̇ α , q β , p ℝ and H K ̇ α , q β , p , N ℝ in the Dunkl Settings

In the present paper, we obtain some new results, and we generalize some known results for the Hausdorff operators. We have studied the generalized Hausdorff operators H α , φ on the Dunkl-type homogeneous weighted Herz spaces K ̇ α , q β , p ℝ and Dunkl Herz-type Hardy spaces H K ̇ α , q β , p , N ℝ . We have determined simple sufficient conditions for these operators to be bounded on these spaces. As applications, we provide necessary and sufficient conditions for generalized Cesàro operator to be bounded on K ̇ α , q β , p ℝ and Hardy inequality for K ̇ α , q β , p ℝ .

Extrapolation to product Herz spaces and some applications

This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

Weighted Central BMO Spaces and Their Applications

In this paper, the central BMO spaces with Muckenhoupt A p weight is introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on weighted Lebesgue spaces. The boundedness of vector-valued commutators on weighted Herz spaces is also considered.