Pith Number

pith:Y43DL2V3

pith:2026:Y43DL2V35SULWLAJ4ST35SEFSB

not attested not anchored not stored refs resolved

On singular integrals with non-negative kernels in the Heisenberg group

Lingxiao Zhang, Sean Li, Vasileios Chousionis

L^2 boundedness of the K_4 singular integral on a 1-Ahlfors regular set in the Heisenberg group implies it lies in a 1-Ahlfors regular curve.

arxiv:2605.17680 v1 · 2026-05-17 · math.CA · math.MG

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{Y43DL2V35SULWLAJ4ST35SEFSB}

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.

Claims

C1strongest claim

If E subset He is a 1-Ahlfors regular set and the singular integral operator associated with the kernel K_4 is L^2(E)-bounded, then E is contained in a 1-Ahlfors regular curve.

C2weakest assumption

The 1-Ahlfors regularity of the set E together with the non-negativity and precise homogeneity of the kernel K_4 are assumed to control the maximal function and cancellation properties needed for the implication to hold.

C3one line summary

L2-boundedness of the SIO with kernel K_4 on 1-Ahlfors regular sets in the Heisenberg group characterizes containment in 1-Ahlfors regular curves, with negative results for alpha in (0,2) and a bounded operator on a purely 1-unrectifiable set.

References

22 extracted · 22 resolved · 0 Pith anchors

[1] V . Chousionis, J. Mateu, L. Prat, and X. Tolsa. Calderón-Zygmund kernels and rectifiability in the plane.Adv. Math., 231(1):535–568, 2012 2012

[2] Nonnegative kernels and 1-rectifiability in the Heisenberg group.Anal 2017

[3] Singular integrals onC 1,α intrinsic graphs in step 2 Carnot groups.J 2025

[4] Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990 1990

[5] A new family of singular integral operators whoseL 2-boundedness implies rectifiability.J 2017

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:52.427713Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c73635eabbeca8bb2c09e4a7bec885907ae1f73860d413c29ab9e17523165f37

Aliases

arxiv: 2605.17680 · arxiv_version: 2605.17680v1 · doi: 10.48550/arxiv.2605.17680 · pith_short_12: Y43DL2V35SUL · pith_short_16: Y43DL2V35SULWLAJ · pith_short_8: Y43DL2V3

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/Y43DL2V35SULWLAJ4ST35SEFSB \
  | 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: c73635eabbeca8bb2c09e4a7bec885907ae1f73860d413c29ab9e17523165f37

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f20c870ef6d46c6293f68e03f6c40050090fcb08fa56aca1d3142d19765cf305",
    "cross_cats_sorted": [
      "math.MG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CA",
    "submitted_at": "2026-05-17T22:41:12Z",
    "title_canon_sha256": "030dd550635b01938a8a6333c504a8839bf2da42b813b623a42fc7d42bd1b714"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17680",
    "kind": "arxiv",
    "version": 1
  }
}