Sparse bounds on variational norms along monomial curves

TL;DR

This paper investigates whether the $r$-variation of truncated Hilbert transforms along monomial curves can be pointwise dominated by sparse operators, extending sparse bounds to variational norms in a geometric setting.

Contribution

It establishes sparse bounds for the $r$-variation of Hilbert transforms along monomial curves, a novel extension of sparse domination techniques to variational operators in this context.

Findings

Proves sparse domination for the $r$-variation of Hilbert transforms along monomial curves.

Extends sparse bounds to variational norms in geometric harmonic analysis.

Provides tools for pointwise control of variational operators along polynomial curves.

Abstract

Consider a monomial curve $\gamma:\mathbb{R}\to\mathbb{R}^{d}$ and a family of truncated Hilbert transforms along $\gamma$, $\mathcal{H}^{\gamma}$. This paper addresses the possibility of the pointwise sparse domination of the $r$-variation of $\mathcal{H}^{\gamma}$ - namely, whether the following is true: \begin{equation*}V^{r}\circ\mathcal{H}^{\gamma}f(x)\lesssim \mathcal{S}f(x)\end{equation*} where $f$ is a nonnegative measurable function, $r>2$ and $\mathcal{S}f(x) = \sum_{Q\in\mathcal{Q}}\langle f\rangle_{Q,p}\chi_{Q}(x)$ for some $p$ and some sparse collection $\mathcal{Q}$ depending on $f,p$.