{"paper":{"title":"Superharmonically Weighted Dirichlet Spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Invariant subspaces in superharmonically weighted Dirichlet spaces reduce to outer functions when the Laplacian measure is finite or its boundary support is countable.","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"A. Hanine, H. Bahajji-El Idrissi, O. El-Fallah, Y. Elmadani","submitted_at":"2026-05-13T17:07:42Z","abstract_excerpt":"In this paper, we consider weighted Dirichlet spaces $\\cD_\\omega$, where $\\omega$ is a positive superharmonic weight on the unit disc $\\DD$. These spaces include the standard weighted Dirichlet spaces $\\cD_\\alpha$ and appear in the description of their invariant subspaces. Our goal is to study the spaces $\\cD_\\omega$. We show that an explicit description of invariant subspaces reduces to the description of those generated by a bounded outer function, and then to the problem of describing cyclic functions, known as the Brown--Shields conjecture. We develop tools, analogous to those used in the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Invariant subspaces in superharmonically weighted Dirichlet spaces reduce to outer functions when the Laplacian measure is finite or its boundary support is countable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5f1fa5ffca894c4f19e23ad090ed4361e44ef3623c7a1cb3b39fcaa03d5cc3d7"},"source":{"id":"2605.13787","kind":"arxiv","version":1},"verdict":{"id":"92596a60-a883-4edd-bd0c-47a86f7159a5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:32:26.022854Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"Invariant subspaces in superharmonically weighted Dirichlet spaces reduce to outer functions when the Laplacian measure is finite or its boundary support is countable."},"references":{"count":52,"sample":[{"doi":"","year":1976,"title":"D. R. Adams, On the existence of capacitary strong type estimates inRn, Ark. Mat.14(1976), 125–140","work_id":"13ea7ecb-f755-41a8-94aa-9f79160af7a9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"J. Agler, J. E. McCarthy, Pick interpolation and Hilbert function spaces, volume 44 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002","work_id":"aefe37b1-3833-49d0-8dba-d1ddf4102111","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"Aleman, The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, Hagen, (1993)","work_id":"aa8fd154-f5ef-4296-9a64-6e5afe5182b4","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"Aleman, Hilbert spaces of analytic functions between the Hardy and the Dirichlet space, Proc","work_id":"573c6daf-b6a3-4a72-9c3d-338d364428e8","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"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","work_id":"b8781c6d-d9a0-4bb2-9dae-039ce9375093","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":52,"snapshot_sha256":"16aadfc1de7ee0131ba017fe40ba3f8f6c14e57c4a92a44d3023d42f47d28fd0","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}