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There exists a finite constant $C({\\Omega})>0$ such that \\[ \\int_{\\Omega}e^{\\frac{4\\pi u^{2}}{H_{d}(u)}}dxdy\\le C(\\Omega),\\;\\;\\forall u\\in C^{\\infty}_{0}(\\Omega), \\] where $H_{d}=\\int_{\\Omega}|\\nabla u|^{2}dxdy-\\frac{1}{4}\\int_{\\Omega}\\frac{u^{2}}{d(z,\\partial\\Omega)^{2}}dxdy$ and $d(z,\\partial\\Omega)=\\min\\limits_{z_{1}\\in\\partial\\Omega}|z-z_{1}|$.}\n  The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in $\\mathbb{R}^{2}$ via "},"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":"1512.07163","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-12-22T17:20:47Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"db01b500ab8a107e6a17f73f30e8f4d5dfcffaaf8b26df9c1fa632a078f61fa8","abstract_canon_sha256":"2dea6758cf5702ba084141c5e91e2dd1475e959d614bfb27bf399713d962ab7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:52.746158Z","signature_b64":"D/VZ8DR1J6Rf7pdAdoUIVYH9bEIWZtjzXzjQ9beyAxmNyncYNbifXx5gRI4AHA0kpPp0NN/Ak1/o4lIhHLvAAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7674800dc9fd554ce9ae579acb2db184d2db980d598ca525d63b4c3d917bc451","last_reissued_at":"2026-05-18T01:23:52.745395Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:52.745395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp Trudinger-Moser inequality on any bounded and convex planar domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Guozhen Lu, Qiaohua Yang","submitted_at":"2015-12-22T17:20:47Z","abstract_excerpt":"Wang and Ye conjectured in [22]:\n  Let $\\Omega$ be a regular, bounded and convex domain in $\\mathbb{R}^{2}$. 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