Fractional series operators on discrete Hardy spaces

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

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