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 Λ_{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.

arXiv comment16 pages. To appear in Analysis & PDE

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