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We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\\Omega$ such that $$ \\mathop{\\int_{\\Omega} \\int_{\\Omega}}_{|u(x) - u(y)| > \\delta} \\frac{1}{|x-y|^{d+p}} \\, dx \\, dy < + \\infty, $$ for some $\\delta > 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincar\\'e inequalities, and Hardy inequalities."},"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":"1908.06179","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-16T21:16:59Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"7511b248105be7ef372def5db73d918cfb561a36e0439ee7293509befc827336","abstract_canon_sha256":"e8f3b1282def5766a2a2dac4e9723eafaa9332e23bb28de5d7ff09560b93a2dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:58:13.330414Z","signature_b64":"IrCZpYjgAKrDRJDZNSyrgrQbCM8zNrA+LQjGpIIXvL5SgzKkDg1dFGE+IXHiiBigOKCU9yV38gAAzCNf2dH7Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f31087e363239e406b8fbf9da676c5500f5be822e6ae1df16d8da05bab42cbd9","last_reissued_at":"2026-07-04T23:58:13.330031Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:58:13.330031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Arka Mallick, Hoai-Minh Nguyen","submitted_at":"2019-08-16T21:16:59Z","abstract_excerpt":"Let $d \\ge 1$, $p \\ge d$, and let $\\Omega$ be a smooth bounded open subset of $\\mathbb{R}^d$. 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