Hausdorff operators on the weighted Herz-type Hardy spaces

Abstract

Abstract. In this paper, we study the high-dimensional Hausdorff operators on the weighted Herz-type Hardy spaces and obtain some substantial extensions from the previous results in [3]. Particularly, for the Hausdorff operators, we establish their sharp boundedness on the power weighted Herz-type Hardy spaces. Our results reveal that the Housdorff operators have better performance in the Herz-type Hardy spaces HKq (R n ;w) ) than their performance in the Hardy spaces H p (R n ;w) ( h p (R n ;w) ) when 0 < p < 1 .Mathematics subject classification (2010): 42B30, 42B35.

Weighted CBMO estimates for the commutators of matrix Hausdorff operator on Heisenberg group

“…Recently, weighted boundedness of matrix Hausdorff operators and their commutators defined on different underlying spaces are established in [34,35,36,37,38,39,40,41].…”

Some results for the commutators of generalized Hausdorff operator

“…)dy. (1.3) Like one dimensional Hausdorff operator [25,26], the operators H Φ and H Φ,A also enjoy a variety of estimates on different function spaces; see [3,4,5,10,11,14,15,16,17,20,32,33] and the references therein. 1 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan…”

Some weighted inequalities for Hausdorff operators and commutators

“…On the other hand weighted norm inequalities for Hausdorff operators on function spaces have recently been reported in the literature, which includes boundedness of Hausdorff operator on power weighted Hardy spaces [ 15 , 16 ], weighted Herz-type Hardy spaces [ 17 ] and on weighted Herz space on the Heisenberg group [ 18 ].…”

“…The purpose of this study is twofold. Firstly, motivated by the work in [ 17 , 18 ], we give estimates for the Hausdorff operator on weighted central Morrey space. In addition, under some assumption on , we work out the operator norm for on power weighted central Morrey spaces.…”