A note on commutators of fractional integrals with ${\rm RBMO}(\mu)$ functions

Chen, Wengu; Sawyer, Eric T.

doi:10.1215/ijm/1258138480

Cited by 35 publications

(37 citation statements)

References 3 publications

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“…M r, (6) (T 1 f )(x). It then follows, from Lemma 3.9(ii) and Theorem 3.10(i), that, for all r ∈ (0, 1),…”

Section: Theorem 310 Letmentioning

confidence: 99%

“…Suppose that T is bounded on L 2 (μ). Then, for any s ∈ (1, ∞), there exists a positive constant C (s) , depending on s, such that, for all f ∈ L ∞ b (μ), (5) f + M s, (6) T f + T * f , (3.17) where T * denotes the maximal Calderón-Zygmund operator defined by setting, for all f ∈ L ∞ b (μ) and x ∈ X ,…”

Section: Theorem 311mentioning

confidence: 99%

“…On the other hand, in the last two decades, many classical results concerning the Calderón-Zygmund operators and function spaces have been proved still valid for metric spaces equipped with non-doubling measures; see, for example, [5,6,15,18,29,30,[38][39][40][41][42]. In particular, let μ be a non-negative Radon measure on R d which only satisfies the polynomial growth condition that there exist some positive constant C 0 and n ∈ (0, d ] such that, for all x ∈ R d and r ∈ (0, ∞), μ(B(x, r )) ≤ C 0 r n , (1.2) where B(x, r ) := {y ∈ R d : |x − y| < r }.…”

Section: Introductionmentioning

confidence: 99%

“…B, S is a useful tool in the study of commutators of fractional integrals in the setting of metric measure spaces with non-doubling measures or metric measure spaces of non-homogeneous type; see, for example, [6,10]. However, in our proof, via the Minkowski integral inequality and the Fatou lemma, we do not need to use the fractional coefficient, which is a different approach to deal with commutators of fractional integrals.…”

Section: Introductionmentioning

confidence: 99%

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Boundedness of certain commutators over non-homogeneous metric measure spaces

Lin

Yang

2016

Anal.Math.Phys.

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Let (X , d, μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and b ∈ RBMO(μ) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator T b := bT − T b generated by T and b from the atomic Hardy space H 1 (μ) with the discrete coefficient into the weak Lebesgue space L 1, ∞ (μ). From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator T b is bounded on L p (μ) for all p ∈ (1, ∞). Moreover, the boundedness of the commutator generated by the generalized fractional integral T α (α ∈ (0, 1)) and the RBMO(μ) function from H 1 (μ) into L 1/(1−α), ∞ (μ) is also presented.

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“…M r, (6) (T 1 f )(x). It then follows, from Lemma 3.9(ii) and Theorem 3.10(i), that, for all r ∈ (0, 1),…”

Section: Theorem 310 Letmentioning

confidence: 99%

Section: Theorem 311mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Boundedness of certain commutators over non-homogeneous metric measure spaces

Lin

Yang

2016

Anal.Math.Phys.

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“…In recent years, there exist several papers focus on the behavior of such fractional integrals I α ; see, for example, [1,2,4,5,11,16] and their references. The boundedness of the multilinear fractional integrals on product Lebesgue spaces with Lebesgue measures was considered by Kenig and Stein [13] as well as Grafakos and Kalton [9] .…”

Section: 2)mentioning

confidence: 99%