Abstract
WeightedLpforp∈(1,∞)and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. All results with no weight on spaces of homogeneous type are also new.
Highlights
We will be working on a space of homogeneous type
We remark that all balls defined by d satisfy the axioms of complete system of neighborhoods in X, and induce a separated topology in X, the balls B x, r for x ∈ X and r > 0 need not be open with respect to this topology
By a remarkable result of Macıas and Segovia in 2, we know that there exists another quasimetric d such that i there exists a constant C ≥ 1 such that for all x, y ∈ X, C−1d x, y ≤ d x, y ≤ Cd x, y ; ii there exist constants C > 0 and γ ∈ 0, 1 such that for all x, x, y ∈ X, d x, y − d x, y
Summary
Weighted Lp for p ∈ 1, ∞ and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. A weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. All results with no weight on spaces of homogeneous type are new.
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