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.
Categories: math.CA