{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "hardy-littlewood-one-dimensional-maximal-inequalities-1930", "title": "Hardy and Littlewood: controlling the largest local averages", "summary": "An average can hide a short burst. What if one looks around every point and chooses the interval with the largest average? Hardy and Littlewood show how to control these maximal averages in one dimension. Their rearrangement comparisons lead to an Lp norm bound p/(p−1) for the one-sided average, when p is greater than one. For their two-direction endpoint-interval maximal function, the corresponding integral bound is twice that factor raised to p. A finite-interval endpoint result also explains why an extra logarithmic integrability condition can control the maximal average when ordinary integrability alone cannot.", "boundary": "These are the original one-dimensional interval averages, not arbitrary-dimensional or weighted maximal theorems. The sharp factor two in the rearrangement comparison is not asserted to be the optimal strong Lp constant. The L log L endpoint requires a finite interval; there is no general strong L1 bound here. Toeplitz-style rearrangement and prior Hardy, Elliott and Young inputs are distinct. The paper credits T. W. Chaundy for the substance of Lemma6 proof without making him an article coauthor.", "area": "Harmonic analysis / maximal operators", "topic": "Mathematics", "year": 1930, "lane": "HUMAN", "result": "THEOREM", "package_id": "hardy-littlewood-one-dimensional-maximal-inequalities-1930", "target": { "record_id": "urn:uuid:8d3b306f-2014-409d-aed1-f80977d7f2a1", "version": 1, "content_hash": "sha256:4b2d469ec9595fecb15a3a1e4c82f4e59b9231a4bc679e1ac143189c9d28ec8c" }, "humans": [ "G. H. Hardy", "J. E. Littlewood" ], "additional_contributors": [ { "name": "T. W. Chaundy", "role": "The original p88footnote credits the substance of the proof of Lemma6. This supporting mathematical contribution is distinct from the two article authors; no individual share is measured.", "sources": [ { "title": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf" } ] } ], "authors_complete": true, "human_role_note": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. 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What if one looks around every point and chooses the interval with the largest average? Hardy and Littlewood show how to control these maximal averages in one dimension. Their rearrangement comparisons lead to an Lp norm bound p/(p−1) for the one-sided average, when p is greater than one. For their two-direction endpoint-interval maximal function, the corresponding integral bound is twice that factor raised to p. A finite-interval endpoint result also explains why an extra logarithmic integrability condition can control the maximal average when ordinary integrability alone cannot.", "scope": "An average can hide a short burst. What if one looks around every point and chooses the interval with the largest average? Hardy and Littlewood show how to control these maximal averages in one dimension. Their rearrangement comparisons lead to an Lp norm bound p/(p−1) for the one-sided average, when p is greater than one. For their two-direction endpoint-interval maximal function, the corresponding integral bound is twice that factor raised to p. A finite-interval endpoint result also explains why an extra logarithmic integrability condition can control the maximal average when ordinary integrability alone cannot. These are the original one-dimensional interval averages, not arbitrary-dimensional or weighted maximal theorems. The sharp factor two in the rearrangement comparison is not asserted to be the optimal strong Lp constant. The L log L endpoint requires a finite interval; there is no general strong L1 bound here. Toeplitz-style rearrangement and prior Hardy, Elliott and Young inputs are distinct. The paper credits T. W. 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Hardy", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "J. E. Littlewood", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "These are the original one-dimensional interval averages, not arbitrary-dimensional or weighted maximal theorems. The sharp factor two in the rearrangement comparison is not asserted to be the optimal strong Lp constant. The L log L endpoint requires a finite interval; there is no general strong L1 bound here. Toeplitz-style rearrangement and prior Hardy, Elliott and Young inputs are distinct. The paper credits T. W. Chaundy for the substance of Lemma6 proof without making him an article coauthor.", "domain": "Harmonic analysis / maximal operators" }, "sources": [ { "citation": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf" } ], "statement": "# Hardy and Littlewood: controlling the largest local averages\n\n## Averages can hunt for the biggest burst\n\nImagine measuring an irregular signal. A long average may conceal a short burst, so you inspect many intervals around a point and keep the largest average. This maximal average is useful because it detects concentration, but the choice of interval can amplify the signal. Hardy and Littlewood ask how large that amplification can become over an entire one-dimensional domain. Their original discussion even connects maximal averages to familiar questions about averages in cricket.\n\n## Changing the order reveals a comparison\n\nA decreasing rearrangement lists the same values from large to small while preserving how much space they occupy. The paper compares maximal averages with averages of this reordered function, first for finite sequences and then for integrals. This creates a controlled route from local choices to a global inequality. The comparison does not claim that spatial order is irrelevant to every property. It isolates a particular bound that survives reordering and the passage from sums to integrals.\n\n## What the exponent buys\n\nAn Lp norm raises the signal magnitude to a power p, integrates and then takes the pth root. Larger p penalizes peaks more strongly. For p greater than one, the original one-sided maximal average has norm bound p divided by p minus one. At p equal to two this factor is two. The two-direction integral inequality has factor two times this ratio raised to p, giving eight when p is two. Eight is an integral bound here, not a measured amplification.\n\n## The endpoint cannot be obtained by wishful substitution\n\nAs p approaches one, the displayed norm factor grows without bound. The paper does not replace that divergence with a universal strong L1 estimate. On a finite interval, an extra logarithmic integrability condition supplies a different endpoint statement. Also, the sharp factor two in a rearrangement comparison is not automatically the best strong Lp constant. The authors explicitly keep these claims separate. Later multidimensional and weighted theorems are outside the original record.\n\n## An agent can compare before trying to generalize\n\nGive your agent a short nonnegative list and ask it to compute several left-looking interval averages at each position. It should state its finite averaging convention and compare the largest averages with the original list. Then ask which result changes when intervals on both sides are allowed. This small exercise explains why taking the best local average requires a bound. It does not establish the continuous theorem, determine an optimal constant or certify a modern maximal-operator implementation.\n\n## Scientific scope\n\nThese are the original one-dimensional interval averages, not arbitrary-dimensional or weighted maximal theorems. The sharp factor two in the rearrangement comparison is not asserted to be the optimal strong Lp constant. The L log L endpoint requires a finite interval; there is no general strong L1 bound here. Toeplitz-style rearrangement and prior Hardy, Elliott and Young inputs are distinct. The paper credits T. W. Chaundy for the substance of Lemma6 proof without making him an article coauthor.\n\n## Sources for this explanation\n\n- [A maximal theorem with function-theoretic applications](https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf): Original sectionsII–IV,theorems2,5–12;original constants,finite-interval endpoint and Chaundy lemma credit.\n", "summary": "An average can hide a short burst. What if one looks around every point and chooses the interval with the largest average? Hardy and Littlewood show how to control these maximal averages in one dimension. Their rearrangement comparisons lead to an Lp norm bound p/(p−1) for the one-sided average, when p is greater than one. For their two-direction endpoint-interval maximal function, the corresponding integral bound is twice that factor raised to p. A finite-interval endpoint result also explains why an extra logarithmic integrability condition can control the maximal average when ordinary integrability alone cannot.", "title": "Hardy and Littlewood: controlling the largest local averages" }, "record_id": "urn:uuid:8d3b306f-2014-409d-aed1-f80977d7f2a1", "version": 1, "source_ref": { "record_id": "urn:uuid:8d3b306f-2014-409d-aed1-f80977d7f2a1", "version": 1, "content_hash": "sha256:4b2d469ec9595fecb15a3a1e4c82f4e59b9231a4bc679e1ac143189c9d28ec8c" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:7842c0e96a515aa770c88873de4f12d492da1caaca6ec732eaff75f45784e0b6" }, "target": { "record_id": "urn:uuid:8d3b306f-2014-409d-aed1-f80977d7f2a1", "version": 1, "content_hash": "sha256:4b2d469ec9595fecb15a3a1e4c82f4e59b9231a4bc679e1ac143189c9d28ec8c" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "hardy-littlewood-one-dimensional-maximal-inequalities-1930", "entry_id": "hardy-littlewood-one-dimensional-maximal-inequalities-1930", "revision": 1, "title": "Hardy and Littlewood: controlling the largest local averages", "context": "Harmonic analysis / maximal operators", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "An average can hide a short burst. What if one looks around every point and chooses the interval with the largest average? Hardy and Littlewood show how to control these maximal averages in one dimension. Their rearrangement comparisons lead to an Lp norm bound p/(p−1) for the one-sided average, when p is greater than one. For their two-direction endpoint-interval maximal function, the corresponding integral bound is twice that factor raised to p. A finite-interval endpoint result also explains why an extra logarithmic integrability condition can control the maximal average when ordinary integrability alone cannot.", "scope": "An average can hide a short burst. What if one looks around every point and chooses the interval with the largest average? Hardy and Littlewood show how to control these maximal averages in one dimension. Their rearrangement comparisons lead to an Lp norm bound p/(p−1) for the one-sided average, when p is greater than one. For their two-direction endpoint-interval maximal function, the corresponding integral bound is twice that factor raised to p. A finite-interval endpoint result also explains why an extra logarithmic integrability condition can control the maximal average when ordinary integrability alone cannot. These are the original one-dimensional interval averages, not arbitrary-dimensional or weighted maximal theorems. The sharp factor two in the rearrangement comparison is not asserted to be the optimal strong Lp constant. The L log L endpoint requires a finite interval; there is no general strong L1 bound here. Toeplitz-style rearrangement and prior Hardy, Elliott and Young inputs are distinct. The paper credits T. W. Chaundy for the substance of Lemma6 proof without making him an article coauthor.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. These are the original one-dimensional interval averages, not arbitrary-dimensional or weighted maximal theorems. The sharp factor two in the rearrangement comparison is not asserted to be the optimal strong Lp constant. The L log L endpoint requires a finite interval; there is no general strong L1 bound here. Toeplitz-style rearrangement and prior Hardy, Elliott and Young inputs are distinct. The paper credits T. W. Chaundy for the substance of Lemma6 proof without making him an article coauthor.", "references": [ { "title": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred." }, { "who": "T. W. Chaundy", "credit": "SCIENCE", "did": "The original p88footnote credits the substance of the proof of Lemma6. This supporting mathematical contribution is distinct from the two article authors; no individual share is measured." } ], "target": { "record_id": "urn:uuid:8d3b306f-2014-409d-aed1-f80977d7f2a1", "version": 1, "content_hash": "sha256:4b2d469ec9595fecb15a3a1e4c82f4e59b9231a4bc679e1ac143189c9d28ec8c" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "380000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "hardy-littlewood-one-dimensional-maximal-inequalities-1930", "target": { "record_id": "urn:uuid:8d3b306f-2014-409d-aed1-f80977d7f2a1", "version": 1, "content_hash": "sha256:4b2d469ec9595fecb15a3a1e4c82f4e59b9231a4bc679e1ac143189c9d28ec8c" }, "sections": [ { "title": "Averages can hunt for the biggest burst", "text": "Imagine measuring an irregular signal. A long average may conceal a short burst, so you inspect many intervals around a point and keep the largest average. This maximal average is useful because it detects concentration, but the choice of interval can amplify the signal. Hardy and Littlewood ask how large that amplification can become over an entire one-dimensional domain. Their original discussion even connects maximal averages to familiar questions about averages in cricket." }, { "title": "Changing the order reveals a comparison", "text": "A decreasing rearrangement lists the same values from large to small while preserving how much space they occupy. The paper compares maximal averages with averages of this reordered function, first for finite sequences and then for integrals. This creates a controlled route from local choices to a global inequality. The comparison does not claim that spatial order is irrelevant to every property. It isolates a particular bound that survives reordering and the passage from sums to integrals." }, { "title": "What the exponent buys", "text": "An Lp norm raises the signal magnitude to a power p, integrates and then takes the pth root. Larger p penalizes peaks more strongly. For p greater than one, the original one-sided maximal average has norm bound p divided by p minus one. At p equal to two this factor is two. The two-direction integral inequality has factor two times this ratio raised to p, giving eight when p is two. Eight is an integral bound here, not a measured amplification." }, { "title": "The endpoint cannot be obtained by wishful substitution", "text": "As p approaches one, the displayed norm factor grows without bound. The paper does not replace that divergence with a universal strong L1 estimate. On a finite interval, an extra logarithmic integrability condition supplies a different endpoint statement. Also, the sharp factor two in a rearrangement comparison is not automatically the best strong Lp constant. The authors explicitly keep these claims separate. Later multidimensional and weighted theorems are outside the original record." }, { "title": "An agent can compare before trying to generalize", "text": "Give your agent a short nonnegative list and ask it to compute several left-looking interval averages at each position. It should state its finite averaging convention and compare the largest averages with the original list. Then ask which result changes when intervals on both sides are allowed. This small exercise explains why taking the best local average requires a bound. It does not establish the continuous theorem, determine an optimal constant or certify a modern maximal-operator implementation." } ], "references": [ { "title": "A maximal theorem with function-theoretic applications", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5446-11511_2006_Article_BF02547518.pdf", "locator": "Original sectionsII–IV,theorems2,5–12;original constants,finite-interval endpoint and Chaundy lemma credit." } ] }, "reading_status": "EXPLANATORY_GUIDE" }