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Using this result we then prove that the above density holds if (i) $p_-\\geq n$ or if (ii) $2\\leq p_-< n$ and $p_+<\\frac{np_-}{n-p_-}$. Moreover our result allows us to give an alternative proof, for the case $p_-\\geq 2$, that the local boundedness of the maximal operato"},"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":"1406.5385","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-20T13:45:20Z","cross_cats_sorted":[],"title_canon_sha256":"a71dbdd79b37295c111bdbd00132324f8196e77212d1089d22bad6d6e6e007a1","abstract_canon_sha256":"4de5ac1d05e7bff542742d1ca6fe1bb9cdb578b550fab0d9875eceed64df0da8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:02.272166Z","signature_b64":"NO8/yNyR/kkI2TcVgMKhTwni9PUFN1k4dLQfV5cqMiobCN1/J+qErWARm3AcWISNpyNtySpM0e6CVy2Qt19BCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce3e784b6964d298dac68b81bcb723cf765ab22904ca14f5d4c0866a03e73964","last_reissued_at":"2026-05-18T01:37:02.271608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:02.271608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Density of smooth functions in variable exponent Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Nikos Yannakakis, Thanasis Kostopoulos","submitted_at":"2014-06-20T13:45:20Z","abstract_excerpt":"We show that if $p_-\\geq 2$, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space $W^{1,p(\\cdot)}(\\mathbb R^n)$, is that the Riesz potentials of compactly supported functions of $L^{p(\\cdot)}(\\mathbb R^n)$, are also elements of $L^{p(\\cdot)}(\\mathbb R^n)$. Using this result we then prove that the above density holds if (i) $p_-\\geq n$ or if (ii) $2\\leq p_-< n$ and $p_+<\\frac{np_-}{n-p_-}$. 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