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I know this is finite for any integrable function. But unable to say anything for an arbitrary measurable function. Can you please give a particular example of a measurable function for which the Hardy-Littlewood maximal function is not finite?

One more help I need about a reference where the converse of the Muckenhoupt's theorem is proved, that is the boundedness of the Maximal function from weighted Leb space into itself implies the weigt belong to Ap.

Thanking you

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  • $\begingroup$ Sorry for the mistake. At the begining, by the word this is finite I mean to say the Maximal function is finite for any integrable function. $\endgroup$ – Mathlover Oct 7 '18 at 21:29
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1) Any measurable function whose integral over some set of finite measure is infinite will have infinite maximal function; for example, $\frac 1 x \chi_{(0, 1)}$. Any other example which is not locally integrable works too.

2) A good reference for $A_p$ weights is Duoandikoetxea's Fourier Analysis (MR1800316), Chapter 7.

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  • $\begingroup$ Thank you very very much. $\endgroup$ – Mathlover Oct 7 '18 at 21:52

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