Maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ on non-homogeneous space

Abstract

Let (X , d, µ) be a geometrically doubling metric space and assume that the measure µ satisfies the upper doubling condition. In this paper, the authors, by invoking a Cotlar type inequality, show that the maximal bilinear Calderón-Zygmund operators of type ω(t) is bounded fromMoreover, if w = (w 1 , w 2 ) belongs to the weight class A ρ p (µ), using the John-strömberg maximal operator and the John-strömberg sharp maximal operator, the authors obtain a weighted weak type estimate L p1 (w 1 ) × L p2 (w 2 ) → L p,∞ (v w ) for the maximal bilinear Calderón-Zygmund operators of type ω(t). By weakening the assumption of ω ∈ Dini(1/2) into ω ∈ Dini(1), the results obtained in this paper are substantial improvements and extensions of some known results, even on Euclidean spaces R n .

Boundedness for Commutators of Bilinearθ-Type Calderón-Zygmund Operators on Nonhomogeneous Metric Measure Spaces

“…and ( , max ∈{1,2} ( , )) ≤ ∑ 2 =1 ( , ( , )) ≤ 2 ( , max ∈{1,2} ( , )), Definition 5 in this paper is equivalent to Definition 1.4 in [20] or Definition 1.3 in [21]. Therefore, we can directly quote the result of Theorem 1.5 in [20] as Lemma 17 below in this paper.…”

“…(i) In [20,21], the term [∑ 2 =1 ( , ( , ))] −2 in (7) and (8) of this paper is substituted by min ∈{1,2} [ ( , ( , ))] −2 . In fact, as…”

“…Ri et al [18,19] researched the boundedness of -type Calderón-Zygmund operator on Hardy spaces with nondoubling measures and nonhomogeneous metric measure spaces, respectively. Zheng et al [20,21] studied the bounded properties for bilinear -type Calderón-Zygmund operator and maximal bilinear -type Calderón-Zygmund operator on nonhomogeneous metric measure spaces, respectively.…”

Bilinear Θ$\Theta$‐type Calderón–Zygmund operators and their commutators on product generalized fractional mixed Morrey spaces

“…Recently, Lu and Wang [32] introduced a new class of generalized Morrey spaces, and showed that the $$\widetilde{T}$$ and the $$[b_{1},b_{2},\widetilde{T}]$$ are bounded on generalized Morrey spaces $$\mathcal {L}^{p,\varphi,\kappa }(\mu)$$ over nonhomogeneous metric measure spaces. More research on various bilinear $$\theta$$‐type Calderón–Zygmund operators can be seen in [28–30, 46, 51].…”