Fractional series operators on discrete Hardy spaces

Abstract

We estudy the $H^p(\mathbb{Z})$ - ${\ell }^q(\mathbb{Z})$ boundedness of the fractional series operator $T_{\gamma }$ given by \[ (T_γb)(j) = \sum_{i \neq \pm j} \frac{b(i)}{|i-j|^α|i+j|^β}, \] where $0\le \gamma <1$, $\alpha ,\beta >0$ 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 $H^q(\mathbb{Z})$.

arXiv comment10 pages