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We give an alternative proof of that fact that also shows the $W^{1,1}$ norm of $\\Phi$ can be bounded by $5\\pi n+1$. Answering a question raised by Lazarev and Lieb, we show that if $p>1$ then there is no bound for the $W^{1,p}$ norm of any such multiplier in terms of the norms of $f_{1},...,f_{n}$."},"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":"1212.5759","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-23T03:44:09Z","cross_cats_sorted":[],"title_canon_sha256":"b02ae67af5d6234c859550d94db55917462b6ea5ccd932a519976650a72c5682","abstract_canon_sha256":"eaf06237191ac57334d3fc8f81d44febd36f9d23ccc733c7d522c18b9ce4982e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:51.467127Z","signature_b64":"zMwfqyIBloT3iAKY8yJmvG8u3KON75q33HqVAg9npOcggdsXIcj2yS29EFg/VF8wskEzct0as1MeHZrdLiKzBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59e8cafb16605065e5e25d87f298d3b21f703a572cb73fb1d76ffffcdff4c60d","last_reissued_at":"2026-05-18T03:37:51.466371Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:51.466371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Lazarev-Lieb Extension of the Hobby-Rice Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Vermont Rutherfoord","submitted_at":"2012-12-23T03:44:09Z","abstract_excerpt":"O. 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