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We show that $|G|^{p(\\cdot)} \\in L^q(\\mathbb{R}^n)$ implies $|D u|^{p(\\cdot)} \\in L^q(\\mathbb{R}^n)$ for any $q \\geq 1$. We also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis."},"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":"1312.5570","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-19T14:52:16Z","cross_cats_sorted":[],"title_canon_sha256":"35b03e1592b42e60b81b8fab616e17bb99ff872410cf138cde38f2a8c4bc0872","abstract_canon_sha256":"e056392af8c2fd29fb47ab06f421070180cbf7bc167a8f5262b86f9fb52b3bda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:14.515623Z","signature_b64":"EwnVgGkPh2NVreCj5jAmaVHGi0W865w/EsILn81yyuPrLcFXXfghytOA2T3ryRqW4uSWMC0eNU3Fm2FbNLE3CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59954e532d2af7ee9a5a42c6f15dcb364aa18358de34c3240b3b201bc15a71c1","last_reissued_at":"2026-05-18T03:04:14.515160Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:14.515160Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global gradient estimates for the $p(\\cdot)$-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lars Diening, Sebastian Schwarzacher","submitted_at":"2013-12-19T14:52:16Z","abstract_excerpt":"We consider Calder\\'on-Zygmund type estimates for the non-homogeneous $p(\\cdot)$-Laplacian system $ -\\text{div}(|D u|^{p(\\cdot)-2} Du) = -\\text{div}(|G|^{p(\\cdot)-2} G),$ where $p$ is a variable exponent. We show that $|G|^{p(\\cdot)} \\in L^q(\\mathbb{R}^n)$ implies $|D u|^{p(\\cdot)} \\in L^q(\\mathbb{R}^n)$ for any $q \\geq 1$. 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