I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their connections?
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2$\begingroup$ I think it is meant more as analogy or example than as motivation. It appears to be in H+L, A maximal theorem with function-theoretic applications, Acta Mathematica 1930, Volume 54, Issue 1 -- looking online seems to produce some, erm, copies that can be viewed $\endgroup$– Yemon ChoiCommented Mar 18, 2015 at 22:58
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$\begingroup$ The review by Askey of M. L. Cartwright, Manuscripts of Hardy, Littlewood, Marcel Riesz and Titchmarsh, Bull. London Math. Soc. 14 (1982), no. 6, 472–532, MR0679927 (84c:01042), says (in part), "We know what Hardy wrote as the "gas'' for the maximal function paper (cricket, of course), but it will be very interesting if more can be found about how Hardy and Littlewood discovered this very important result." But maybe this is where you read it. $\endgroup$– Gerry MyersonCommented Mar 18, 2015 at 23:01
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$\begingroup$ Also, Wikipedia en.wikipedia.org/wiki/… says, "In their original paper, G.H. Hardy and J.E. Littlewood explained their maximal inequality in the language of cricket averages." $\endgroup$– Gerry MyersonCommented Mar 18, 2015 at 23:08
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$\begingroup$ See also Alexandra Bellow, Transference principles in ergodic theory, in Christ et al., eds., Harmonic Analysis and Partial Differential Equations, especially pages 27-28. Also, C James Elliott, Rearrangement of maximal functions, math.unm.edu/~crisp/courses/math565/spring08/… $\endgroup$– Gerry MyersonCommented Mar 18, 2015 at 23:12
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One hesitates to explain a joke, but this is quite a nice joke, and I cannot resist answering a cricket question (especially now). Suppose a batsman scores 20, 100, 30, 40, 70 and 0 in his last six innings (0 being the most recent). Being upset at scoring 0 in his last innings, he might say to himself -- at least I am averaging 35 in my last two innings; or going further, that I'm averaging 36.67 in my last three; or averaging 35 in my last four; or 48 in my last five; or 43.33. Probably he would most prefer the fifth statement which gives the largest average, or the most satisfaction.
Suppose now the batsman does this over an entire season of cricket, after each innings computing his satisfaction until then. Call the total satisfaction the sum of the satisfactions after each innings in the season. Then for a given stock of scores in a season, the Hardy-Littlewood maximal theorem gives that the batsman's total satisfaction is a maximum if his scores are in descending order throughout the series -- or in other words, his satisfaction is a maximum precisely when his batting has been in decline throughout the season!
There is a nice discussion of this in Bela Bollobas's problem book "The art of mathematics: coffee time in Memphis" -- see problem 85 on Satisfied Cricketers: the Hardy-Littlewood maximal theorem there.
The English cricket team has clearly taken this Theorem a little too seriously!
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$\begingroup$ I upvoted for the last line of the answer :) $\endgroup$– ajayCommented Mar 19, 2015 at 9:50
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7$\begingroup$ From what I understand the theorem is rather trivial when applied to Bollobas's own cricketing career - run out for 0 off the second ball of his only innings. $\endgroup$ Commented Mar 19, 2015 at 13:27
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1$\begingroup$ @BenGreen: I did think that you might like this answer, except for the last line -- sorry! $\endgroup$– LuciaCommented Mar 19, 2015 at 13:32