Fractional series operators on discrete Hardy spaces

Abstract

For $0 \leq \gamma < 1$ and a sequence $b=\{ b(i) \}_{i \in \mathbb{Z}}$ we consider the fractional operator $T_{\alpha, \beta}$ defined formally by

$$(T_{\alpha, \beta} \, b)(j) = \sum_{i \neq \pm j} \frac{b(i)}{|i-j|^{\alpha} |i+j|^{\beta}} \quad (j \in \mathbb{Z}),$$

where $\alpha, \beta > 0$ and $\alpha + \beta = 1 - \gamma$. The main aim of this note is to prove that the operator $T_{\alpha, \beta}$ is bounded from $H^{p}(\mathbb{Z})$ into $\ell^{q}(\mathbb{Z})$ for $0 < p < \frac{1}{\gamma}$ and $\frac{1}{q} = \frac{1}{p} - \gamma$. For $\alpha = \beta = \frac{1-\gamma}{2}$ we show that there exists $\epsilon \in (0, \frac 13 )$ such that for every ${0 \leq \gamma < \epsilon}$ the operator $T_{\frac{1-\gamma}{2}, \frac{1-\gamma}{2}}$ is not bounded from $H^{p}(\mathbb{Z})$ into $H^{q}(\mathbb{Z})$ for $0 < p \leq \frac{1}{1 + \gamma}$ and $\frac{1}{q} = \frac{1}{p} - \gamma$.