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Here $Q=2n+2$ is the homogeneous dimension of $\\mathbb H^n$. Assume that $\\{e^{-s\\mathcal L}\\}_{s>0}$ is the heat semigroup generated by $\\mathcal L$. The Littlewood-Paley function $\\mathfrak{g}_{\\mathcal L}$ and the Lusin area integral $\\mathcal{S}_{\\mathcal L}$ associated with the Schr\\\"odinger operator $\\mathcal L$ are defin"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.09398","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-16T05:26:02Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"198c9abdff9dd0d1b3b9737477b6268cc867e700b8cbb2d916c8af56d1adedd6","abstract_canon_sha256":"d1158ba0904145301d392478580f180610563fd323cc15e36c3ba791da9cce7c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:59.314217Z","signature_b64":"DRR3UaUyiJj4Ho9+ulem1KuK08BR9gFwYmMwwl6+zyxwO4ewaznkwA6WoGg5YkbBrMlVxs4OefF+z63G9JdQDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c9e41a1ca615c8c3f6d9307aa1389599ee71c80dd4308afc9df2e4a25120184","last_reissued_at":"2026-05-17T23:39:59.313751Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:59.313751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Morrey spaces for Schr\\\"odinger operators with certain nonnegative potentials, Littlewood-Paley and Lusin functions on the Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Hua Wang","submitted_at":"2019-07-16T05:26:02Z","abstract_excerpt":"Let $\\mathcal L=-\\Delta_{\\mathbb H^n}+V$ be a Schr\\\"odinger operator on the Heisenberg group $\\mathbb H^n$, where $\\Delta_{\\mathbb H^n}$ is the sublaplacian on $\\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\\\"older class $RH_q$ with $q\\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\\mathbb H^n$. Assume that $\\{e^{-s\\mathcal L}\\}_{s>0}$ is the heat semigroup generated by $\\mathcal L$. 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