Pith Number

pith:V6MR4WEP

pith:2021:V6MR4WEPFAUP7LHT7SUMMJHN5G

not attested not anchored not stored refs pending

Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces

Houkun Zhang, Jiang Zhou

arxiv:2106.04926 v1 · 2021-06-09 · math.FA · math.AP

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{V6MR4WEPFAUP7LHT7SUMMJHN5G}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-07-05T02:47:50.790345Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

af991e588f2828ffacf3fca8c624ede9bc8a96d188f038e74cd4e27c56c60b30

Aliases

arxiv: 2106.04926 · arxiv_version: 2106.04926v1 · doi: 10.48550/arxiv.2106.04926 · pith_short_12: V6MR4WEPFAUP · pith_short_16: V6MR4WEPFAUP7LHT · pith_short_8: V6MR4WEP

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/V6MR4WEPFAUP7LHT7SUMMJHN5G \
  | 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: af991e588f2828ffacf3fca8c624ede9bc8a96d188f038e74cd4e27c56c60b30

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2674c1668bc24a2be7a525de705b287afeaba79644f2165839fcdbb3a70cdb87",
    "cross_cats_sorted": [
      "math.AP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2021-06-09T09:16:17Z",
    "title_canon_sha256": "c94e0b744d37823431c336e9caac90380cc5cfd4f6ff716691e9c47e256e9e4d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2106.04926",
    "kind": "arxiv",
    "version": 1
  }
}