Sparse Bounds for the Discrete Cubic Hilbert Transform

Abstract

Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $\mathbb{Z}$ by \begin{eqnarray*} H_3f(n) = \sum_{m \not = 0} \frac{f(n- m^3)}{m}. \end{eqnarray*} We prove that there exists $r <2$ and universal constant $C$ such that for all finitely supported $f,g$ on $\mathbb{Z}$ there exists an $(r,r)$-sparse form ${\Lambda}_{r,r}$ for which \begin{eqnarray*} \left| \langle H_3f, g \rangle \right| \leq C {\Lambda}_{r,r} (f,g). \end{eqnarray*} This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.