SWhat 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.
Scientific 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
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.
Contributors
- The named authors in the research record · SCIENCE: 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.
- T. W. Chaundy · SCIENCE: 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.
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