Discrete Double Hilbert Transforms Along Polynomial Surfaces
Joonil Kim, Hoyoung Song
TL;DR
This work characterizes the $\ell^p$-boundedness of the discrete double Hilbert transform along polynomial graphs $P(t_1,t_2)$ via a geometric condition on the backward Newton polyhedron ${\cal N}(P,D_B)$: every vertex must have at least one even component. Using a novel two-parameter circle method, the authors decompose the discrete multiplier into major, minor, and mixed arcs and exploit two-parameter Weyl sums, averaged Gauss sums, and dual-face Newton decompositions to obtain uniform $\ell^p$-bounds for $1<p<\infty$ when the even-vertex condition holds. They also establish the necessity of this condition by constructing divergence in the presence of an odd-odd vertex, extending Garaev-type divergence phenomena to the multi-parameter discrete setting. The results connect discrete harmonic analysis with convex-geometry of Newton polyhedra and provide a robust framework for multi-parameter discrete oscillatory transforms on polynomial surfaces. Overall, the paper advances the understanding of when oscillatory discrete Radon-type transforms along polynomial graphs are bounded on $\ell^p$ spaces, with potential implications for ergodic and number-theoretic applications.
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 $1/t_2$.