Initial $L^2\times \cdots \times L^2$ bounds for multilinear operators
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- by Loukas Grafakos, Danqing He, Petr Honzík and Bae Jun Park;
- Trans. Amer. Math. Soc. 376 (2023), 3445-3472
- DOI: https://doi.org/10.1090/tran/8877
- Published electronically: February 16, 2023
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Abstract:
The $L^p$ boundedness theory of convolution operators is based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel’s identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we obtain initial $L^2\times \cdots \times L^2\to L^{2/m}$ estimates for three types of important multilinear operators: rough singular integrals, multipliers of Hörmander type, and multipliers whose derivatives satisfy qualitative estimates. These estimates lay the foundation for the derivation of other $L^p$ estimates for such operators.References
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Bibliographic Information
- Loukas Grafakos
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 288678
- ORCID: 0000-0001-7094-9201
- Email: grafakosl@missouri.edu
- Danqing He
- Affiliation: School of Mathematical Sciences, Fudan University, People’s Republic of China
- MR Author ID: 1059054
- Email: hedanqing@fudan.edu.cn
- Petr Honzík
- Affiliation: Department of Mathematics, Charles University, 116 36 Praha 1, Czech Republic
- ORCID: 0000-0001-6545-6461
- Email: honzik@gmail.com
- Bae Jun Park
- Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
- MR Author ID: 1252594
- Email: bpark43@skku.edu
- Received by editor(s): February 5, 2021
- Received by editor(s) in revised form: August 27, 2022
- Published electronically: February 16, 2023
- Additional Notes: Danqing He is the corresponding author
The second author was supported by National Key R&D Program of China (No. 2021YFA1002500), and NNSF of China (No. 11701583, No. 12161141014). The first author would like to acknowledge the support of the Simons Foundation grant 624733 and of the Simons Fellows Program Award # 819503. The third author was supported by GAČR P201/18-07996S. The fourth author was supported in part by NRF grant 2022R1F1A1063637 and was supported in part by a KIAS Individual Grant MG070001 at the Korea Institute for Advanced Study - © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 3445-3472
- MSC (2020): Primary 42B15, 42B25
- DOI: https://doi.org/10.1090/tran/8877
- MathSciNet review: 4577337