Sparse bounds on variational norms along monomial 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}^γf(x)\lesssim \mathcal{S}f(x)\end{equation*} where $f$ is a nonnegative measurable function, $r>2$ and $\mathcal{S}f(x)={\sum }_{Q\in \mathcal{Q}}⟨f{⟩}_{Q,p}{\chi }_Q(x)$ for some $p$ and some sparse collection $\mathcal{Q}$ depending on $f,p$.