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We investigate a new Adams-Adimurthi-Druet type inequality on the whole space $\\mathbb{R}^n$ which is strongly influenced by the vanishing phenomenon. Specifically, we prove\n  \\begin{equation}\\nonumber\n  \\sup_{\\underset{\\|\\nabla^{m} u\\|_{\\frac{n}{m}}^{^{\\frac{n}{m}}}+\\|u\\|_{\\frac{n}{m}}^{\\frac{n}{m}} \\leq 1}{u\\in W^{m,\\frac{n}{m}}(\\mathbb{R}^n)}} \\int_{\\mathbb{R}^n}\\Phi\\left(\\beta \\left(\\frac{1+\\alpha\\|u\\|_{\\frac{n}{m}}^{\\frac{n}{m}}}{1-\\ga"},"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":"2505.24297","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-05-30T07:13:23Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"cd099877fcf6e2a1b87e09e24ff669279c0eea80c46134c2688ffcbaf1f3c36b","abstract_canon_sha256":"90f569d5e20c83e413ac49e75e5048cdcc03789b3da6d1d712c28aed54de1dde"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:12:46.362288Z","signature_b64":"7epR/keG++vaeUXdpgI5/J9O1rHiVBP7c7Vgx3U2hYGBipwHRK68tf5JM9A5LK9XEoCCbWMS9pvG9W8t9SXFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2551794268d6ecc574296428600872480df0b253b487a7988d94de878a805a0","last_reissued_at":"2026-07-05T11:12:46.361762Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:12:46.361762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Abiel Costa Macedo, F\\'abio Sodr\\'e Rocha, Jos\\'e Francisco de Oliveira","submitted_at":"2025-05-30T07:13:23Z","abstract_excerpt":"Let $W^{m,\\frac{n}{m}}(\\mathbb{R}^n)$ with $1\\le m < n$ be the standard higher order derivative Sobolev space in the critical exponential growth threshold. 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