Abstract
Let \((\mathcal{X},d,\mu)\) be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytonen. In this paper, the authors establish the boundedness of the commutator generated by the \(\operatorname {RBMO}(\mu)\) function and the Marcinkiewicz integral with kernel satisfying a Hormander-type condition, respectively, from \(L^{p}(\mu)\) with \(1< p<\infty\) to itself.
Highlights
1 Introduction In, Marcinkiewicz [ ] introduced the integral on one-dimensional Euclidean space R, which is today called the Marcinkiewicz integral, and conjectured that it is bounded on Lp([, π]), < p < ∞
The main purpose of this paper is to establish the bound of the commutator generated by the Marcinkiewicz integral and the RBMO(μ) function on the non-homogeneous metric measure spaces
We show that the commutator Mb, associating with b ∈ RBMO(μ) and M, which is defined by
Summary
In , Marcinkiewicz [ ] introduced the integral on one-dimensional Euclidean space R, which is today called the Marcinkiewicz integral, and conjectured that it is bounded on Lp([ , π]), < p < ∞. The main purpose of this paper is to establish the bound of the commutator generated by the Marcinkiewicz integral and the RBMO(μ) function on the non-homogeneous metric measure spaces. A metric measure space (X , d, μ) is said to be upper doubling if μ is a Borel measure on X and there exist a dominating function λ : X × ( , ∞) → ( , ∞) and a positive constant cλ such that, for each x ∈ X , r → λ(x, r) is non-decreasing and μ B(x, r) ≤ λ(x, r) ≤ cλλ(x, r/ ), for all x ∈ X , r >. A function f ∈ L loc(μ) is said to be in the space RBMO(μ) if there exist a positive constant c and, for any ball B ⊂ X , a number fB such that μ(ρB).
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