Pith Number

pith:ROGJZZ3O

pith:2026:ROGJZZ3O2LHVZG422ZXP7O3JX7

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Superharmonically Weighted Dirichlet Spaces

A. Hanine, H. Bahajji-El Idrissi, O. El-Fallah, Y. Elmadani

Invariant subspaces in superharmonically weighted Dirichlet spaces reduce to outer functions when the Laplacian measure is finite or its boundary support is countable.

arxiv:2605.13787 v1 · 2026-05-13 · math.FA · math.CA

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Claims

C1strongest claim

We provide a description of invariant subspaces when the measure Δω is finite measure or if the supp(Δω)∩T is countable. Finally, we prove that a smooth outer function f∈Dα such that Z(f) is regular is cyclic in Dα if and only if cα(Z(f))=0.

C2weakest assumption

The weight ω is a positive superharmonic function on the unit disk, and for the final cyclicity statement the outer function is smooth with a regular zero set whose capacity is well-defined in the space Dα.

C3one line summary

Superharmonically weighted Dirichlet spaces admit explicit descriptions of invariant subspaces when the Laplacian measure is finite or countably supported on the circle, and smooth outer functions with regular zero sets are cyclic in the standard case precisely when the associated capacity vanishes.

References

52 extracted · 52 resolved · 0 Pith anchors

[1] D. R. Adams, On the existence of capacitary strong type estimates inRn, Ark. Mat.14(1976), 125–140 1976

[2] J. Agler, J. E. McCarthy, Pick interpolation and Hilbert function spaces, volume 44 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002 2002

[3] Aleman, The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, Hagen, (1993) 1993

[4] Aleman, Hilbert spaces of analytic functions between the Hardy and the Dirichlet space, Proc 1992

[5] A. Aleman, M. Hartz, J. McCarthy, S. Richter, Free outer functions in complete Pick spaces. Trans. Amer. Math. Soc. 376 (2023), no. 3, 1929-1978 2023

Receipt and verification

First computed	2026-05-18T02:44:15.667952Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8b8c9ce76ed2cf5c9b9ad66effbb69bfc9ef51846b88887bba9ce6c51d9d2da6

Aliases

arxiv: 2605.13787 · arxiv_version: 2605.13787v1 · doi: 10.48550/arxiv.2605.13787 · pith_short_12: ROGJZZ3O2LHV · pith_short_16: ROGJZZ3O2LHVZG42 · pith_short_8: ROGJZZ3O

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ROGJZZ3O2LHVZG422ZXP7O3JX7 \
  | 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: 8b8c9ce76ed2cf5c9b9ad66effbb69bfc9ef51846b88887bba9ce6c51d9d2da6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "da1b3071844e3782b5b6bdfbc50d29a0fd3341e6cb6d16445801d1cd64a8930c",
    "cross_cats_sorted": [
      "math.CA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-05-13T17:07:42Z",
    "title_canon_sha256": "ab8b2b53c3e500fb7460791a6b5015473a697a0ffd6b0c69efb0ef9579c02c34"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13787",
    "kind": "arxiv",
    "version": 1
  }
}