Discrete Double Hilbert Transforms Along Polynomial Surfaces

Joonil Kim; Hoyoung Song

Discrete Double Hilbert Transforms Along Polynomial Surfaces

Classical Analysis and ODEs 2025-10-01 v2

Authors: Joonil Kim , Hoyoung Song

Abstract

We obtain a necessary and sufficient condition on a polynomial $P(t_1,t_2)$ for the $\ell^{p}$ boundedness of the discrete double Hilbert transforms associated with $P(t)$ for $1 < p < \infty$ . The proof is based on the multi-parameter circle method treating the cases of $|t_1|\not\approx |t_2|$ arising from $1/t_1$ and $1/t_2$ .

Keywords

hilbert functions lp spaces and inequalities operator theory and elliptic operators

Cite

@article{arxiv.2209.08072,
  title  = {Discrete Double Hilbert Transforms Along Polynomial Surfaces},
  author = {Joonil Kim and Hoyoung Song},
  journal= {arXiv preprint arXiv:2209.08072},
  year   = {2025}
}

Comments

In this revised version, we have carefully rewritten the main theorems to ensure full accuracy, and we have provided complete and detailed proofs for clarity

Discrete Double Hilbert Transforms Along Polynomial Surfaces

Abstract

Keywords

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