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First, we obtain an atomic decomposition for functions in $H^1_{L_1,L_2}(\\mathbb{R}^{n_1}\\times\\mathbb{R}^{n_2})$ where the Hardy space $H^1_{L_1,L_2}(\\mathbb{R}^{n_1}\\times\\mathbb{R}^{n_2})$ associated with $L_1$ and $L_2$ is defined by square function norms, then prove an interpolation property f"},"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":"1509.07548","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-24T22:02:49Z","cross_cats_sorted":[],"title_canon_sha256":"957cfa35cbcefd54caed01fe9152fe40dcb6e57c977a1d5f717000b003a3b649","abstract_canon_sha256":"fcdc1fb9ba08e66afac131eaf4ace585a4f2417fb36582b298e96b8e2ff71ce5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:02.495311Z","signature_b64":"s3vIWHwCIDlZrmUYexuL7LSSgkbKiymKsnHZvl9WpKnLDSDtpQERtAwU4S2LMDRu7ZLZGJhrs915wRB/4XZHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61c593fa64be918508c81090e043600b1c72fc68dcc93dc552d7b75a76d86911","last_reissued_at":"2026-05-18T01:32:02.494796Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:02.494796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"End-point estimates for singular integrals with non-smooth kernels on product spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ji Li, Lixin Yan, Xuan Thinh Duong","submitted_at":"2015-09-24T22:02:49Z","abstract_excerpt":"The main aim of this article is to establish boundedness of singular integrals with non-smooth kernels on product spaces. 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