Explicit ℓ2 Decoupling in Rn, Part I: Bounds for Constants in an Alternate Formulation

Bian, Guomengchao; Chen, Feifei; Wu, Senlin

doi:10.3390/axioms15050349

Open AccessArticle

Explicit ℓ² Decoupling in $R^{n}$ , Part I: Bounds for Constants in an Alternate Formulation

by

Guomengchao Bian

,

Feifei Chen

and

Senlin Wu

^*

School of Mathematics, North University of China, Taiyuan 030051, China

^*

Author to whom correspondence should be addressed.

Axioms 2026, 15(5), 349; https://doi.org/10.3390/axioms15050349

Submission received: 6 April 2026 / Revised: 1 May 2026 / Accepted: 5 May 2026 / Published: 8 May 2026

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Abstract

Z.K. Li established the first explicit decoupling estimate for the parabola in the range

4 < p < 6

. Such explicit bounds are essential in number theory applications. Following the approach of J. Bourgain and C. Demeter, we derive an explicit bound for the implicit constant in an alternate form of decoupling in J. Bourgain and C. Demeter’s work. This is the first part of our three-part project to extend Z.K. Li’s result on the explicit bound for the decoupling constant to the n-dimensional case.

Keywords: ℓ² decoupling; paraboloid; explicit constants; harmonic analysis; decoupling inequalities; high dimensions

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MDPI and ACS Style

Bian, G.; Chen, F.; Wu, S. Explicit ℓ² Decoupling in $R^{n}$ , Part I: Bounds for Constants in an Alternate Formulation. Axioms 2026, 15, 349. https://doi.org/10.3390/axioms15050349

AMA Style

Bian G, Chen F, Wu S. Explicit ℓ² Decoupling in $R^{n}$ , Part I: Bounds for Constants in an Alternate Formulation. Axioms. 2026; 15(5):349. https://doi.org/10.3390/axioms15050349

Chicago/Turabian Style

Bian, Guomengchao, Feifei Chen, and Senlin Wu. 2026. "Explicit ℓ² Decoupling in $R^{n}$ , Part I: Bounds for Constants in an Alternate Formulation" Axioms 15, no. 5: 349. https://doi.org/10.3390/axioms15050349

APA Style

Bian, G., Chen, F., & Wu, S. (2026). Explicit ℓ² Decoupling in $R^{n}$ , Part I: Bounds for Constants in an Alternate Formulation. Axioms, 15(5), 349. https://doi.org/10.3390/axioms15050349

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Explicit ℓ² Decoupling in $R^{n}$ , Part I: Bounds for Constants in an Alternate Formulation

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