Discrete Double Hilbert Transforms Along Polynomial Surfaces

TL;DR

This paper establishes a precise criterion for the boundedness of discrete double Hilbert transforms along polynomial surfaces in p spaces, using advanced multi-parameter harmonic analysis techniques.

Contribution

It provides a necessary and sufficient condition for p boundedness of these transforms, advancing understanding of discrete harmonic analysis along polynomial surfaces.

Findings

Derived a complete characterization of polynomial conditions for boundedness.

Applied multi-parameter circle method to analyze different cases.

Extended the theory of discrete Hilbert transforms to polynomial surfaces.

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 $1/t_2$.