Fractional series operators on discrete Hardy spaces

We estudy the $H^{p}(\mathbb{Z})$ - $\ell^{q}(\mathbb{Z})$ boundedness of the fractional series operator $T_{\gamma}$ given by $$(T_{\gamma}b)(j) = \sum_{i \neq \pm j} \frac{b(i)}{|i-j|^{\alpha}|i+j|^{\beta}},$$ where $0 \leq \gamma < 1$, $\alpha, \beta > 0$ and $\alpha + \beta = 1 -\gamma$. By means of a counter-example, we also show that the operator $T_{\gamma}$ is not bounded from $H^{p}(\mathbb{Z})$ into $H^{q}(\mathbb{Z})$.