A Hörmander type theorem in finite fields

Dong Dong

doi:10.1016/j.ffa.2019.05.002

A Hörmander type theorem in finite fields

Research output: Contribution to journal › Article › peer-review

Abstract

Let p be a prime and given a kernel K:F_p×F_p→C, define a discrete integral operator as follows: T(f)(x)=∑y∈F_pf(y)K(x,y), where f is any complex-valued function defined on F_p. We proved that if the kernel K satisfies certain natural size condition and cancellation conditions, the l²→l²-operator norm of T is bounded by p^−γ for some positive number γ. This result can be viewed as a discrete analogue of Hörmander theorem. As an application, we recovered a power-saving estimate of certain bilinear average operator in finite fields by X. Li, Sawin and the author.

Original language	English (US)
Pages (from-to)	22-31
Number of pages	10
Journal	Finite Fields and Their Applications
Volume	59
DOIs	https://doi.org/10.1016/j.ffa.2019.05.002
State	Published - Sep 2019
Externally published	Yes

Keywords

Bilinear operator
Decay
Finite fields
Linear operator

ASJC Scopus subject areas

Theoretical Computer Science
Algebra and Number Theory
Engineering(all)
Applied Mathematics

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10.1016/j.ffa.2019.05.002

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title = "A H{\"o}rmander type theorem in finite fields",

abstract = "Let p be a prime and given a kernel K:Fp×Fp→C, define a discrete integral operator as follows: T(f)(x)=∑y∈Fpf(y)K(x,y), where f is any complex-valued function defined on Fp. We proved that if the kernel K satisfies certain natural size condition and cancellation conditions, the l2→l2-operator norm of T is bounded by p−γ for some positive number γ. This result can be viewed as a discrete analogue of H{\"o}rmander theorem. As an application, we recovered a power-saving estimate of certain bilinear average operator in finite fields by X. Li, Sawin and the author.",

keywords = "Bilinear operator, Decay, Finite fields, Linear operator",

author = "Dong Dong",

note = "Funding Information: It is a pleasure to thank Xiaochun Li for helpful discussions. The author was partially supported by LTS grant DO 0052 . Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

year = "2019",

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doi = "10.1016/j.ffa.2019.05.002",

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T1 - A Hörmander type theorem in finite fields

AU - Dong, Dong

PY - 2019/9

Y1 - 2019/9

N2 - Let p be a prime and given a kernel K:Fp×Fp→C, define a discrete integral operator as follows: T(f)(x)=∑y∈Fpf(y)K(x,y), where f is any complex-valued function defined on Fp. We proved that if the kernel K satisfies certain natural size condition and cancellation conditions, the l2→l2-operator norm of T is bounded by p−γ for some positive number γ. This result can be viewed as a discrete analogue of Hörmander theorem. As an application, we recovered a power-saving estimate of certain bilinear average operator in finite fields by X. Li, Sawin and the author.

AB - Let p be a prime and given a kernel K:Fp×Fp→C, define a discrete integral operator as follows: T(f)(x)=∑y∈Fpf(y)K(x,y), where f is any complex-valued function defined on Fp. We proved that if the kernel K satisfies certain natural size condition and cancellation conditions, the l2→l2-operator norm of T is bounded by p−γ for some positive number γ. This result can be viewed as a discrete analogue of Hörmander theorem. As an application, we recovered a power-saving estimate of certain bilinear average operator in finite fields by X. Li, Sawin and the author.

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KW - Decay

KW - Finite fields

KW - Linear operator

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JO - Finite Fields and Their Applications

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A Hörmander type theorem in finite fields

Abstract

Keywords

ASJC Scopus subject areas

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