Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# L2 Boundedness of Discrete Double Hilbert Transform Along polynomials

Version 1 : Received: 11 October 2021 / Approved: 13 October 2021 / Online: 13 October 2021 (13:16:34 CEST)

How to cite: Song, H. L2 Boundedness of Discrete Double Hilbert Transform Along polynomials. Preprints 2021, 2021100204 (doi: 10.20944/preprints202110.0204.v1). Song, H. L2 Boundedness of Discrete Double Hilbert Transform Along polynomials. Preprints 2021, 2021100204 (doi: 10.20944/preprints202110.0204.v1).

## Abstract

We will show $L^{2}$ boundedness of Discrete Double Hilbert Transform along polynomials satisfying some conditions. Double Hilbert exponential sum along polynomials:$\mu(\xi)$ is Fourier multiplier of discrete double Hilbert transform along polynomials. In chapter 1, we define the reverse Newton diagram. In chapter 2, We make approximation formula for the multiplier of one valuable discrete Hilbert transform by study circle method. In chapter 3, We obtain result that $\mu(\xi)$ is bounded by constants if $|D|\geq2$ or all $(m,n)$ are not on one line passing through the origin. We study property of $1/(qt^{n})$ and use circle method (Propsotion 2.1) to calculate sums. We also envision combinatoric thinking about $\mathbb{N}^{2}$ lattice points in j-k plane for some estimates. Finally, we use geometric property of some inequalities about $(m,n)\in\Lambda$ to prove Theorem 3.3. In chapter 4, We obtain the fact that $\mu(\xi)$ is bounded by sums which are related to $\log_{2}({\xi_{1}-a_{1}\slash {q}})$ and $\log_{2}({\xi_{2}-a_{2}\slash {q}})$ and the boundedness of double Hilbert exponential sum for even polynomials with torsion without conditions in Theorem 3.3. We also use $\mathbb{N}^{2}$ lattice points in j-k plane and Proposition 2.1 which are shown in chapter 2 and some estimates to show that Fourier multiplier of discrete double Hilbert transform is bounded by terms about $\log$ and integral this with torsion is bounded by constants.

## Keywords

Discrete; Double Hilbert transform; Circle method; exponential sums; discrete double Hilbert transform; discrete double exponential sums; Newton diagram

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