Skip to content
DraconiticResearch
Search Sources API
182,469 studies31 sourcesUpdated 30 min ago
PreprintarXiv licencearXiv

The endpoints p-=1 and p+=∞ for the maximal operator on Lp(·) over spaces of homogeneous type

  • Alina Shalukhina
Open at sourcePDFDOICite

Abstract

We investigate the boundedness of the Hardy-Littlewood maximal operator M on variable Lebesgue spaces Lp(·)(Ω,d,μ) over spaces of homogeneous type, focusing on the endpoints p-=1 and p+=∞. At the lower endpoint, we prove that the boundedness of M forces p->1 for all exponents if and only if the space (Ω,d,μ) is non-atomic. At the upper endpoint, we generalize the Muckenhoupt-Nekvinda sufficient condition of Adamadze, Diening, and Kopaliani. On bounded spaces of homogeneous type, whenever p->1, this yields a boundedness criterion based on the Muckenhoupt condition alone. We also construct an unbounded space where the Muckenhoupt condition suffices for the maximal operator's boundedness.

Related

  1. Weighted boundedness of maximal operator associated with invertible linear transformations on variable Lebesgue spaces
    Gonzalo H. Ibañez Firnkorn, Lucas A. Vallejos
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. Muckenhoupt weights for the Hardy-Littlewood maximal operator on hyperbolic spaces
    Tapendu Rana
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to ℋ∞-calculus to Fractional Domains
    Reinhard Farwig, Kazuyuki Tsuda
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Iterated toric descent obstruction and homogeneous spaces over geometric global fields
    Felipe Gambardella, Nguyen Manh Linh
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Characterization of the Boundedness of a Generalized Fractional Maximal Operator in Weighted Lorentz Spaces for Certain Parameter Values
    Azhar Abek, Amiran Gogatishvili, Nurzhan Bokayev
    MathematicsPeer-reviewedCC BY 4.0General mathematicsCrossref

Cite

BibTeXRIS
BibTeX
@misc{shalukhina2026endpoints,
  title = {{The endpoints $p_-=1$ and $p_+=\infty$ for the maximal operator on $L^p(\cdot)$ over spaces of homogeneous type}},
  author = {Alina Shalukhina},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33792},
  eprint = {2610.33792},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33792},
}
RIS
TY  - UNPB
TI  - The endpoints p_-=1 and p_+=∞ for the maximal operator on L^p(·) over spaces of homogeneous type
AU  - Alina Shalukhina
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33792
UR  - https://arxiv.org/abs/2610.33792
AB  - We investigate the boundedness of the Hardy-Littlewood maximal operator M on variable Lebesgue spaces L^p(·)(Ω,d,μ) over spaces of homogeneous type, focusing on the endpoints p_-=1 and p_+=∞. At the lower endpoint, we prove that the boundedness of M forces p_->1 for all exponents if and only if the space (Ω,d,μ) is non-atomic. At the upper endpoint, we generalize the Muckenhoupt-Nekvinda sufficient condition of Adamadze, Diening, and Kopaliani. On bounded spaces of homogeneous type, whenever p_->1, this yields a boundedness criterion based on the Muckenhoupt condition alone. We also construct an unbounded space where the Muckenhoupt condition suffices for the maximal operator's boundedness.
ER  -