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 $$\frac{1}{q} = \frac{1}{p} - \gamma$$ .