Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces
Abstract
In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator Mγ, defined on ℤn, n∈ℕ, γ∈ [0,n), between the classical Lorentz sequence spaces λp(v) and λq(w), where 0<p,q<∞, {v(m)} and {w(m)} are non-negative sequences defined on ℕ. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator Mγx, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence {x(k)}, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_γ$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.
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BibTeX
@misc{gogatishvili2026characterization,
title = {{Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces}},
author = {Amiran Gogatishvili and Nurzhan A. Bokayev and Nurgul K. Kuzeubayeva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06124},
eprint = {2610.06124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06124},
}
RIS
TY - UNPB
TI - Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces
AU - Amiran Gogatishvili
AU - Nurzhan A. Bokayev
AU - Nurgul K. Kuzeubayeva
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06124
UR - https://arxiv.org/abs/2610.06124
AB - In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator M_γ, defined on ℤ^n, n∈ℕ, γ∈ [0,n), between the classical Lorentz sequence spaces λ_p(v) and λ_q(w), where 0<p,q<∞, {v(m)} and {w(m)} are non-negative sequences defined on ℕ. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator M_γx, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence {x(k)}, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_γ$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.
ER -