#### Vol. 12, No. 5, 2019

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Sparse bounds for the discrete cubic Hilbert transform

### Amalia Culiuc, Robert Kesler and Michael T. Lacey

Vol. 12 (2019), No. 5, 1259–1272
##### Abstract

Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $ℤ$ 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 $ℤ$ there exists an $\left(r,r\right)$-sparse form ${\Lambda }_{r,r}$ for which

$|〈{H}_{3}f,g〉|\le C{\Lambda }_{r,r}\left(f,g\right).$

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.

##### Keywords
Hilbert transform, cubic integers, sparse bound, exponential sum, circle method
##### Mathematical Subject Classification 2010
Primary: 11L03, 42A05