Sparse domination of Hilbert transforms along curves

TL;DR

This paper establishes sharp sparse bounds for Hilbert transforms along various curves in multi-dimensional space, leading to new weighted norm inequalities applicable to a broad class of curves.

Contribution

It introduces the first sharp sparse bounds for Hilbert transforms along curves, including monomial and nonvanishing torsion curves, advancing the understanding of their weighted inequalities.

Findings

Sharp sparse bounds for Hilbert transforms along curves

Weighted norm inequalities derived from these bounds

Applicability to monomial and $C^n$ curves with nonvanishing torsion

Abstract

We obtain sharp sparse bounds for Hilbert transforms along curves in $\mathbb{R}^n$, and derive as corollaries weighted norm inequalities for such operators. The curves that we consider include monomial curves and arbitrary $C^n$ curves with nonvanishing torsion.