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.

16 pages. To appear in Analysis & PDE

References in corpus (3)

• Sharp weighted norm estimates beyond Calderón-Zygmund theory
• Sparse bilinear forms for Bochner Riesz multipliers and applications
• Square function estimates for discrete Radon transforms

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• A metric approach to sparse domination
• Averages along the Square Integers: ${\ell }^p$ improving and Sparse Inequalities
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