Abstract
We prove decoupling inequalities for mixed-homogeneous bivariate polynomials, which partially answers a conjecture of Bourgain, Demeter and Kemp.
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Notes
For a formal definition, see the appendix.
Strictly, P may not be a rectangle if \(s\ne r\); however, this technicality is easily solved since P is always a parallelogram, which can be slightly enlarged to a rectangle that is equivalent to it (see Sect. 2).
Technically, the properties over a slightly enlarged parallelogram are required. We omit the details here. Readers may refer to Sect. 5.
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Acknowledgements
Jianhui Li would like to thank Betsy Stovall for her advice and help throughout the project. Tongou Yang would like to thank his advisor Malabika Pramanik for suggesting a prototype of this problem at an early stage. The authors would also like to thank the anonymous referee for their thoughtful comments. Jianhui Li was supported by NSF DMS-1653264.
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Communicated by Loukas Grafakos.
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Li, J., Yang, T. Decoupling for mixed-homogeneous polynomials in \({\mathbb {R}}^3\). Math. Ann. 383, 1319–1351 (2022). https://doi.org/10.1007/s00208-021-02273-9
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DOI: https://doi.org/10.1007/s00208-021-02273-9