{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0071",
  "title": "Failure of Uniform Bounds for Fractional Carleson Operators at Exponent Zero",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove that the L2 operator norm of the even-phase fractional Carleson maximal operator, with kernel exp(i lambda |t|^epsilon)/t and supremum over real lambda, grows at least as sqrt(log(1/epsilon)) as epsilon decreases to zero. Explicitly, the norm is at least sqrt(3N/(2048*pi)) for every N>=1 whenever 0<epsilon<=2^(-16-12N). This disproves the universal estimate in Guo's AIM Question 12 (equation (0.17), Carleson theorems and multilinear operators), also for compactly supported smooth inputs. One bounded, finite-valued modulation parameter aligns packets centered at exp(16^j); an explicit three-region kernel estimate transfers the logarithmic construction to positive exponents. The inherited logarithmic reduction and multiplier in the frozen UnsolvedMath report are credited. The odd-phase question near exponent one and optimal growth are not addressed. This six-page English preprint was developed with AI assistance and self-audited; it has not undergone independent peer review or formal verification. Supporting exact and high-precision checks supplement the written analytic proof. No absolute priority is certified.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "keywords": [
    "AIM-ANALYSIS-0071",
    "fractional Carleson operator",
    "uniform estimates",
    "counterexample",
    "wave packets"
  ],
  "manuscript_version_date": "2026-10-28",
  "publication_date": "2026-10-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0071/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0071/paper.pdf?v=1a787c05895b",
  "doi": "10.5281/zenodo.23716806",
  "zenodo_record_url": "https://zenodo.org/records/23716806",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every N>=1 and 0<epsilon<=2^(-16-12N), the even-phase fractional Carleson operator has L2 norm at least sqrt(3N/(2048*pi)). Hence its norm grows at least as sqrt(log(1/epsilon)) near zero and the universal bound in AIM Question 12 is false, also on compactly supported smooth inputs. This answers the even-phase exponent-zero uniform-bound question. The odd-phase exponent-one question and optimal growth remain outside the result. The frozen report's logarithmic reduction is credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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      "sha256": "1a787c05895bafa1548ef810cb084cce9a122dffde73bd971ddab0f421eeb345"
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    "source.zip": {
      "sha256": "17b8af3aef50fb4866d57ab47cddca3885cf612ece1208885b4dde4d35ca5e88"
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    "verification_report.md": {
      "sha256": "f1de5b84e9c4573390f6500ce70670457af4f9893f9b0e1468160578216c37b1"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}