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We present the state-of-the-art on the conjecture that $\\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \\leq q \\leq \\infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \\[\\mathfrak{p}_d(q) = 2+\\cfrac{2}{d+\\cfrac{2}{q-2}}\\] for $d\\geq1$ and $\\frac{2d}{d+1} \\leq q \\leq \\infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that t"},"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":"2402.09787","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-02-15T08:35:35Z","cross_cats_sorted":[],"title_canon_sha256":"729d5499a53c6a9fd5d1900bd3e055ce1fce7cffc9427894e90e859dfc91276a","abstract_canon_sha256":"c471477196586166e6af28159a19b142de841c703bebccd4dc2a1edba5bc2f1d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:04:50.480213Z","signature_b64":"fknzK/P+TNWs+BmBZf5qCaYZX7h3Dr3CEOvK+c+KsJ+FeQac5ldLmLjQY+On/ob/oFqtIxSZKhJ92Ze++JqFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47954c7a137a7bea4ce00187dda793d7a9be547d65e57e8c6533702a460e072e","last_reissued_at":"2026-05-29T01:04:50.479504Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:04:50.479504Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical exponents of the Riesz projection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Adri\\'an Llinares, Kristian Seip, Ole Fredrik Brevig","submitted_at":"2024-02-15T08:35:35Z","abstract_excerpt":"Let $\\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\\mathbb{T}^d)$ to the Hardy space $H^p(\\mathbb{T}^d)$, where $\\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that $\\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \\leq q \\leq \\infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \\[\\mathfrak{p}_d(q) = 2+\\cfrac{2}{d+\\cfrac{2}{q-2}}\\] for $d\\geq1$ and $\\frac{2d}{d+1} \\leq q \\leq \\infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. 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