Classical discrete operators on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces

TL;DR

This paper proves the boundedness of classical discrete operators like the Hilbert transform and Riesz potential on variable exponent sequence spaces, using weighted inequalities and the Rubio de Francia algorithm, under certain maximal function conditions.

Contribution

It establishes boundedness results for discrete classical operators on variable $ ext{ell}^{p(ullet)}( ext{Z})$ spaces, extending known theory with new weighted inequality techniques.

Findings

Discrete Hilbert transform is bounded on variable $ ext{ell}^{p(ullet)}( ext{Z})$ spaces.

Discrete Riesz potential is bounded on these spaces.

Vector-valued inequalities for the fractional maximal operator are obtained.

Abstract

We show, by applying discrete weighted norm inequalities and the Rubio de Francia algorithm, that the discrete Hilbert transform and discrete Riesz potential are bounded on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces whenever the discrete Hardy-Littlewood maximal is bounded on $\ell^{p'(\cdot)}(\mathbb{Z})$. We also obtain vector-valued inequalities for the discrete fractional maximal operator.