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 $f,p$.
Categories: math.CA