A Hörmander type theorem in finite fields

Research output: Contribution to journalArticlepeer-review

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 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 languageEnglish (US)
Pages (from-to)22-31
Number of pages10
JournalFinite Fields and Their Applications
Volume59
DOIs
StatePublished - Sep 2019
Externally publishedYes

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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