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Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces

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Abstract

Denote by T and \(\mathcal {I}_{\alpha }\) the bilinear Calderón-Zygmund operator and bilinear fractional integrals, respectively. In this paper we give the boundedness and compactness of the commutators [T,b] i , maximal operator T ∗,b,i and \([\mathcal {I}_{\alpha },b]_{i}\) on Morrey spaces. More precisely, we prove that [T,b] i , T ∗,b,i and \([\mathcal {I}_{\alpha },b]_{i}\) are all the bounded operators (if b∈B M O) and compact operators (if b ∈ C M O, the BMO-closure of \(C_{c}^{\infty }\)) from \(L^{p_{1},\lambda _{1}}\times L^{p_{2},\lambda _{2}}\) to L p, λ for some suitable indexes λ, λ 1, λ 2 and p, p 1, p 2. As an application of our results, we give also the boundedness and compactness of the commutators formed by the bilinear pseudodifferential operators on Morrey spaces.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Yong Ding & Ting Mei

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  1. Yong Ding
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  2. Ting Mei
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Correspondence to Yong Ding.

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The work is supported by NSFC (No.11371057, 11471033), SRFDP (No.20130003110003) and Fundamental Research Funds for the Central Universities (No.2014KJJCA10).

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Ding, Y., Mei, T. Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces. Potential Anal 42, 717–748 (2015). https://doi.org/10.1007/s11118-014-9455-0

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