Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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SHARP Lp→Lr ESTIMATES OF RESTRICTED AVERAGING OPERATORS OVER CURVES ON PLANES IN FINITE FIELDS

  • Koh, Doowon (Department of Mathematics Chungbuk National University)
  • Received : 2015.01.10
  • Accepted : 2015.04.24
  • Published : 2015.05.15
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Abstract

Let $\mathbb{F}^d_q$ be a d-dimensional vector space over a finite field $\mathbb{F}^d_q$ with q elements. We endow the space $\mathbb{F}^d_q$ with a normalized counting measure dx. Let ${\sigma}$ be a normalized surface measure on an algebraic variety V contained in the space ($\mathbb{F}^d_q$, dx). We define the restricted averaging operator AV by $A_Vf(X)=f*{\sigma}(x)$ for $x{\in}V$, where $f:(\mathbb{F}^d_q,dx){\rightarrow}\mathbb{C}$: In this paper, we initially investigate $L^p{\rightarrow}L^r$ estimates of the restricted averaging operator AV. As a main result, we obtain the optimal results on this problem in the case when the varieties V are any nondegenerate algebraic curves in two dimensional vector spaces over finite fields. The Fourier restriction estimates for curves on $\mathbb{F}^2_q$ play a crucial role in proving our results.

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References

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  2. RESTRICTED AVERAGING OPERATORS IN THE FINITE FIELD SETTING vol.30, pp.2, 2015, https://doi.org/10.14403/jcms.2017.30.2.259
  3. Restricted averaging operators to cones over finite fields vol.30, pp.6, 2018, https://doi.org/10.1515/forum-2017-0042
  4. Restricted averaging operators to cones over finite fields vol.30, pp.6, 2018, https://doi.org/10.1515/forum-2017-0042