A Revisit on Commutators of linear and bilinear Fractional Integral Operator

2016/04/24 by Mingming Cao, Qingying Xue, Cao, Mingming +1
Mathematics · #42B25 #47G10 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1604.06992

openalex publication_date 2016/04/24 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

Abstract

Let I_α be the linear and I_α be the bilinear fractional integral operators. In the linear setting, it is known that the two-weight inequality holds for the first order commutators of I_α. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of I_α. In this paper, we first give an alternative proof for the first order commutators of I_α. This new approach allows us to consider the higher order commutators. This was done by showing that the commutator [b,I_α] can be represented as a finite linear combination of some paraproducts. Then, by using the Cauchy integral theorem, we show that the two-weight inequality holds for the higher order commutators of I_α. In the bilinear setting, we present a dyadic proof for the characterization between BMO and the boundedness of [b,I_α]. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of [b,I_α].

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