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

$$H_{3}f\left(n\right)=\sum _{m\ne 0}\frac{f\left(n-m^3\right)}{m}.$$

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 $\left(r,r\right)$-sparse form ${\mathrm{\Lambda }}_{r,r}$ for which

$$\left|\langle H_{3}f,g\rangle \right|\le C{\mathrm{\Lambda }}_{r,r}\left(f,g\right).$$

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.