Weighted estimates for the Calderón commutator. Issue 1 (23rd February 2020)

• Record Type: Journal Article
• Authors: Chen, Jiecheng; Hu, Guoen
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 63:Issue 1(2020)
• Page Start: 169
• Page End: 192
• Publication Date: 2020-02-23
• Subjects: Calderón commutator, -- weighted inequality, -- multilinear singular integral operator, -- sparse operator, -- multiple weight; 42B20; Mathematics -- Periodicals; 510
• DOI: 10.1017/S001309151900021X
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: In this paper the authors consider the weighted estimates for the Calderón commutator defined by \mathcal{C}_{m+1, A}(a_1, \ldots, a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y, with P 2 ( A ; x, y ) = A ( x ) − A ( y ) − A ′( y )( x − y ) and A ′ ∈ BMO(ℝ). Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\mathbb {R}, w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R}, w_{m+1})$ to $L^{p}(\mathbb {R}, \nu _{\vec {\kern 1pt w}})$, with p 1, …, p m +1 ∈ (1, ∞), 1/ p = 1/ p 1 + · · · + 1/ p m +1, and $\vec {\kern 1pt w}=(w_1, \ldots, w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$ . The authors also obtain the weighted weak type endpoint estimates for $\mathcal {C}_{m+1, A}$ .