Fractional series operators on discrete Hardy spaces

TL;DR

This paper investigates the boundedness of a fractional series operator on discrete Hardy spaces, establishing conditions for boundedness and providing counterexamples for certain mappings.

Contribution

It introduces a fractional series operator on discrete Hardy spaces and analyzes its boundedness, revealing new insights and limitations in these function spaces.

Findings

The operator is bounded from $H^{p}( Z)$ to $\, ext{ell}^{q}( Z)$ under specific conditions.

Counter-example shows the operator is not bounded from $H^{p}( Z)$ to $H^{q}( Z)$ in general.

Conditions for boundedness depend on parameters $\, ext{alpha}$, $\, extbeta$, and $\, extgamma$.

Abstract

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 $H^{q}(\mathbb{Z})$.