Necessary and sufficient conditions for generalized Hausdorff operators and commutators

Abstract

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.

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

“…on function spaces when the symbol function b are either from Lipschitz space or central BM O space. However, when the matrix A(y) is diagonal, we get the commutators of H Φ which were studied in [15,16] and [17]. For detailed history and other developments regarding Hausdorff operators, we refer the interested readers to the review articles [18,19].…”

Weighted Morrey estimates for Hausdorff operator and its commutator on the Heisenberg group

“…For the Lebesgue space L p (p ≥ 1) and the Hardy space H 1 , the boundedness of H Φ (even H Φ,A ) are well established (see [4,7,19,20,23,24,27,30,35]). Besides spaces L p and H 1 , the boundedness of H Φ on other function spaces was recently also studied by many authors (see, for instance, [5,15,22,26,31,32,36,37] and the references therein).…”

Estimates for the Commutators of p-Adic Hausdorff Operator on Herz-Morrey Spaces

“…The boundedness of the analog of H b Φ,A on R n and its special cases when A(t) = diag[1/|t|, 1/|t|, ..., 1/|t|] were discussed in [31][32][33][34][35][36][37]. However, this topic still needs further considerations in the sense of its boundedness on p-adic function spaces.…”