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Absolute Marchaud integrability is not necessary for membership in Riemann--Liouville Lp images

  • Salaheddine Lamzib
  • Aadil Lahrouz
  • Ahmed Zeghal
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Abstract

For every fractional order strictly between zero and one and every finite interval, there is a bounded representing function whose Riemann--Liouville fractional integral satisfies the weighted endpoint condition proposed by Vainikko but fails the absolute Marchaud condition almost everywhere. One bounded example therefore disproves the necessity implication in Vainikko's conjectured characterization simultaneously for every exponent strictly larger than one, including the essentially bounded endpoint. A different sine-series construction gives, at each exponent strictly between one and infinity, failure at every interior point for one representative. For the bounded example, signed inversion remains valid for every finite exponent greater than or equal to one. The converse implication in Vainikko's conjecture is not decided here.

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Cite

BibTeXRIS
BibTeX
@misc{lamzib2026absolute,
  title = {{Absolute Marchaud integrability is not necessary for membership in Riemann--Liouville $L^p$ images}},
  author = {Salaheddine Lamzib and Aadil Lahrouz and Ahmed Zeghal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06867},
  eprint = {2610.06867},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06867},
}
RIS
TY  - UNPB
TI  - Absolute Marchaud integrability is not necessary for membership in Riemann--Liouville L^p images
AU  - Salaheddine Lamzib
AU  - Aadil Lahrouz
AU  - Ahmed Zeghal
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.06867
UR  - https://arxiv.org/abs/2610.06867
AB  - For every fractional order strictly between zero and one and every finite interval, there is a bounded representing function whose Riemann--Liouville fractional integral satisfies the weighted endpoint condition proposed by Vainikko but fails the absolute Marchaud condition almost everywhere. One bounded example therefore disproves the necessity implication in Vainikko's conjectured characterization simultaneously for every exponent strictly larger than one, including the essentially bounded endpoint. A different sine-series construction gives, at each exponent strictly between one and infinity, failure at every interior point for one representative. For the bounded example, signed inversion remains valid for every finite exponent greater than or equal to one. The converse implication in Vainikko's conjecture is not decided here.
ER  -