{"paper":{"title":"The dual of the Hardy space associated to the Dunkl-Schr\\\"odinger operator with reverse H\\\"older class potential","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The dual of the Hardy space H^1 for the Dunkl-Schrödinger operator is the space BMO(L_k)","cross_cats":[],"primary_cat":"math.FA","authors_text":"P. Athulya, S.K. Verma","submitted_at":"2026-05-14T06:51:22Z","abstract_excerpt":"Let $\\mathcal{L}_k = -\\Delta_k + V$ be a Schr\\\"odinger operator associated with the Dunkl Laplacian $\\Delta_k$, where $V$ is the non-negative potential function belonging to the reverse H\\\"older class $RH_k^q(\\mathbb{R}^n)$ with $q> \\max\\{1, \\frac{n+2\\gamma}{2}\\}$. Here, $2\\gamma$ denotes the degree of homogeneity of the weight function $w_k$, which is determined by the normalized root system and the non-negative multiplicity function $k$. In this paper, we investigate the dual space of the Hardy space $H_{\\Tilde{\\mathcal{L}}_k}^1$ associated with the Dunkl-Schr\\\"odinger operator. The dual spa"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The dual space of H_{tilde L_k}^1 is BMO(L_k), a subspace of the Dunkl BMO_k space, characterized via atomic decomposition where the cancellation condition of atoms depends on the critical radius function associated with V.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The potential V belongs to the reverse Hölder class RH_k^q(R^n) with q > max{1, (n+2γ)/2}, which is required for the atomic decomposition and the definition of the critical radius function to work as stated.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The dual of the Hardy space H^1 for the Dunkl-Schrödinger operator L_k is the BMO(L_k) space, characterized via atoms whose cancellation depends on the critical radius function of the reverse Hölder potential V.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The dual of the Hardy space H^1 for the Dunkl-Schrödinger operator is the space BMO(L_k)","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"840a0de40cd2620ab109e7bc7282f86d5cf3ae0fd49d37ba15acc02a5423f9d1"},"source":{"id":"2605.14456","kind":"arxiv","version":1},"verdict":{"id":"4135a445-c222-4848-acbf-c21c618b8ea7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:00:50.786497Z","strongest_claim":"The dual space of H_{tilde L_k}^1 is BMO(L_k), a subspace of the Dunkl BMO_k space, characterized via atomic decomposition where the cancellation condition of atoms depends on the critical radius function associated with V.","one_line_summary":"The dual of the Hardy space H^1 for the Dunkl-Schrödinger operator L_k is the BMO(L_k) space, characterized via atoms whose cancellation depends on the critical radius function of the reverse Hölder potential V.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The potential V belongs to the reverse Hölder class RH_k^q(R^n) with q > max{1, (n+2γ)/2}, which is required for the atomic decomposition and the definition of the critical radius function to work as stated.","pith_extraction_headline":"The dual of the Hardy space H^1 for the Dunkl-Schrödinger operator is the space BMO(L_k)"},"references":{"count":39,"sample":[{"doi":"","year":2011,"title":"D. 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Semi- group Forum.101(3):507-533 (2020)","work_id":"b7b56ebb-585f-4b6e-be11-f21c514c0671","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"J.P. Anker, J. Dziuba´ nski, A. Hejna, Harmonic functions, conjugate harmonic functions and the Hardy spaceH 1 in the rational Dunkl setting. J. Fourier Anal. Appl.25(5):2356-2418 (2019)","work_id":"07e9b4a1-b33c-413a-aa59-7f39612495fd","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":39,"snapshot_sha256":"a909caebeb1cd196c8b2db1d98dedb14e8d2cb4eb0b1c9b0d0950a30d2b615d5","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f1fab4b81dacdd12697277fe6d129236a8b66832886e72e39cb9faf7bd3f5692"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}