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I see this term coming up quite a lot, but I have not figured if there's a general definition. For example wikipedia https://en.wikipedia.org/wiki/Maximal_function shows several examples but does not provide a general definition (if there's any).

In my understanding a maximal function is any function involving a $ \sup $. For example in the proof of Caratheodory theorem, derivatives of measures or Riesz representation theorem they all involve supremum of something.

Is there a more general definition rather than per example based?

mathlander
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user8469759
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    The original one is due to Hardy and Littlewood, but there are several variants. It is an evolving topic, hence the definitions are not 100% established yet. This should not make you uncomfortable. – Giuseppe Negro Oct 05 '22 at 10:25
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    @GiuseppeNegro Out of curiosity, if this can be answered in a comment. Why are they so useful? It seems to me they kinda of occur naturally. Legendre transform to me is a maximal function as well, the one you mentioned is another one, duality problems in optimization also resemble it. It seems like there's something special about using functions defined by composing with sup and inf. – user8469759 Oct 05 '22 at 13:04
  • @user8469759 A general strategy for proving a convergence result is to prove the result on a dense subspace, and then prove a bound on a maximal function that allows you to get the result by approximation. Thuis works to prove the pointwise Ergodic theorem and the Lebesgue differentiation theorem. – Mason Oct 06 '22 at 01:53
  • This is a typical application of the maximal function: https://math.stackexchange.com/q/3639820/8157. It is the proof of the Hardy-Littlewood-Sobolev inequality, which is a Lp bound for the fractional integral operator. The maximal function appears towards the very end. Note how the maximal function is used "to control the function at all scales". The exact meaning of this informal phrase is left to you to ponder. – Giuseppe Negro Oct 12 '22 at 17:39

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A good definition of a maximal function would be a function using a supremum. Since maximal and maximize have the same root word, I think this is what people mean by "maximal function."

mathlander
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    I think I figured that much. But I was looking for a more established definition in the community if there's any. – user8469759 Oct 12 '22 at 13:03