HL-Maximum function | Mathematics | Page 1

rustic owl May 30, 2023, 3:56 PM

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Hey does someone know how to solve this problem. Mf(x) is the Hardy Littlewood max function.

humble dune May 30, 2023, 4:47 PM

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rustic owl Hey does someone know how to solve this problem. Mf(x) is the Hardy Littlewood m...

by integrable do you mean that M f \in L^1?

rustic owl May 30, 2023, 4:48 PM

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yes

humble dune May 30, 2023, 4:49 PM

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doesn't c/|x|^d not belonging to L^1 directly prove M f doesn't belong to L^1

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please tell me if I'm wrong

rustic owl May 30, 2023, 4:53 PM

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I think so but the main part that I dont understand is showing that the inequation is valid.

humble dune May 30, 2023, 4:53 PM

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the inequation is valid right?

rustic owl May 30, 2023, 4:54 PM

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yeah but I need to show it

humble dune May 30, 2023, 4:54 PM

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oh it's pretty easy

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wait shit

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I'm pretty sure that inequality is supposed to be reversed

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but idk

rustic owl May 30, 2023, 5:00 PM

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could be a mistake I suppose

humble dune May 30, 2023, 5:01 PM

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I found this one stackexchange answer that says otherwise but I don't understand it fully

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step 2 seems weird, everything else is aight

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oh I get it

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pretty cool proof

rustic owl May 30, 2023, 5:04 PM

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why B(x,2|x|))?

humble dune May 30, 2023, 5:04 PM

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so they assume that a is an arbitrary real > 0 and < |x|

humble dune May 30, 2023, 5:05 PM

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rustic owl why B(x,2|x|))?

they just assumed r = 2|x|

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and so they can translate the ball to the origin so that B(0,a) is a subset of B(0,2|x|)

#HL-Maximum function