Pith Number

pith:36OKUDMY

pith:2026:36OKUDMYAUM7II5NCCE7XIRMJ2

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A tree-like fractal Dirichlet space lying between strong and weak elliptic Harnack inequalities

Caoxu Huang, Guanhua Liu

A constructed tree-like fractal obeys the weak elliptic Harnack inequality but violates the strong version under a selected self-similar measure.

arxiv:2605.15479 v1 · 2026-05-14 · math.AP · math.PR

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Claims

C1strongest claim

Under a properly selected self-similar measure, the weak elliptic Harnack inequality holds but the strong analogue fails on the constructed tree-like fractal Dirichlet space.

C2weakest assumption

That a specific self-similar measure exists on the infinitely branched tree fractal such that the Dirichlet form remains regular while the strong Harnack inequality is violated (abstract, paragraph 2).

C3one line summary

Construction of a tree-like fractal Dirichlet space separating weak and strong elliptic Harnack inequalities under a chosen self-similar measure.

References

9 extracted · 9 resolved · 0 Pith anchors

[1] M. T. Barlow, M. Murugan. Stability of elliptic Harnack inequality. Ann. of Math. (2) 187 (2018) 777–823.doi.org/10.4007/annals.2018.187.3. 4 2018 · doi:10.4007/annals.2018.187.3

[2] A Simple Structural Analysis Method for DAEs 2002 · doi:10.1023/a:

[3] A. Grigor’yan, J. Hu, K.-S. Lau. Estimates of heat kernels for non-local regular Dirichlet forms. Trans. Amer. Math. Soc. 366 (2014) 6397–6441. doi.org/10.1090/S0002-9947-2014-06034-0 2014 · doi:10.1090/s0002-9947-2014-06034-0

[4] B. M. Hambly, T. Kumagai. Transition density estimates for diffusion pro- cesses on post critically finite self-similar fractals. Proc. London Math. Soc. 78 (1999) 431–458.doi.org/10.1112/S00246115990 1999 · doi:10.1112/s0024611599001744

[5] J. Hu, Z. Yu. The weak elliptic Harnack inequality revisited. Asian J. Math. 27 (2023) 771–828.doi.org/10.4310/AJM.2023.v27.n5.a4 2023 · doi:10.4310/ajm.2023.v27.n5.a4

Formal links

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Receipt and verification

First computed	2026-05-20T00:01:00.671114Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

df9caa0d980519f423ad1089fba22c4e8aee2960f30c782468eb8b05d4467547

Aliases

arxiv: 2605.15479 · arxiv_version: 2605.15479v1 · doi: 10.48550/arxiv.2605.15479 · pith_short_12: 36OKUDMYAUM7 · pith_short_16: 36OKUDMYAUM7II5N · pith_short_8: 36OKUDMY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/36OKUDMYAUM7II5NCCE7XIRMJ2 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: df9caa0d980519f423ad1089fba22c4e8aee2960f30c782468eb8b05d4467547

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9478c80c9d0553a19ea573497e7771302880413dc63fccae36177f077f288cb6",
    "cross_cats_sorted": [
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-14T23:42:14Z",
    "title_canon_sha256": "04a63fc8db1842080a92469eedf4e34599ca21e64efab8489601731e21ac8463"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15479",
    "kind": "arxiv",
    "version": 1
  }
}