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Compactness of Riesz transform commutator associated with Bessel operators

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Abstract

Let λ > 0 and

$${\Delta _\lambda }: = - \frac{{{d^2}}}{{d{x^2}}} - \frac{{2\lambda }}{x}\frac{d}{{dx}}$$

be the Bessel operator on R+:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R+, x2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function b ∈ BMO(R+, x2λdx) is in CMO(R+, x2λdx) if and only if the Riesz transform commutator xxxx is compact on Lp(R+, x2λdx) for all p ∈ (1,∞).

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Authors and Affiliations

  1. Department of Mathematics, Macquarie University, NSW, 2109, Australia

    Xuan Thinh Duong & Ji Li

  2. School of Sciences, Nanchang Institute of Technology, Nanchang, 330099, China

    Suzhen Mao

  3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Huoxiong Wu & Dongyong Yang

Authors
  1. Xuan Thinh Duong
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  2. Ji Li
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  3. Suzhen Mao
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  4. Huoxiong Wu
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  5. Dongyong Yang
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Corresponding author

Correspondence to Dongyong Yang.

Additional information

Supported by ARC DP 140100649.

Supported by ARC DP 160100153 and Macquarie University New Staff Grant.

Supported by the Fund of Jiangxi Provincial Department of Education (No. GJJ171014) and the NSF of Jiangxi Province of China (No. 20171BAB211003.)

Supported by the NNSF of China (Grant Nos. 11371295, 11471041) and the NSF of Fujian Province of China (No. 2015J01025).

Supported by the NNSF of China (Grant No. 11571289) and the State Scholarship Fund of China (No. 201406315078).

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Duong, X.T., Li, J., Mao, S. et al. Compactness of Riesz transform commutator associated with Bessel operators. JAMA 135, 639–673 (2018). https://doi.org/10.1007/s11854-018-0048-5

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  • DOI: https://doi.org/10.1007/s11854-018-0048-5

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