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Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := \sup_{h>o} \frac{1}{h}\int_{[x,x+h]} f(t) dt.$$ Establish the rising sun inequality $$\lambda \mu (\{f^* > \lambda\})\leq \int_{\{f^*>\lambda\}} f(t)dt,$$ furthermore, the above is in fact equal when $\lambda > 0$.

I have been stuck on this for the past couple of days. I am still trying to show the case when $\lambda = 0$.

Here is the hint from the problem: when $f$ is compactly supported on the compact interval $[a,b]$, then $F(x) = \int_a^x f(t)dt-(x-a)\lambda$ is continuous on the compact interval; apply the Rising Sun lemma to $F(x)$.

Side note: for $\lambda >0$ $$\mu (\{x\in \mathbb{R} : \sup_{h>o} \frac{1}{h}\int_{[x,x+h]} |f(t)| dt > \lambda\})\leq \frac{1}{\lambda}\int_\mathbb{R}|f(t)|dt$$ is called One-sided Hardy-Littlewood maximal inequality. This can be proven with the hint from above instead of the standard method using Vitali cover lemma.

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migrated from Apr 13 '14 at 15:45

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It is rather rare that posting problems from textbooks here is acceptable. I don't think this is an instance of that. – Mariano Suárez-Alvarez Apr 13 '14 at 8:50
@Xiao: this question is likely to be closed soon. Instead you may have a look e.g. to Wheeden-Zygmund's Measure and Integral: An Introduction to Real Analysis for an elementary and clear proof (to be arranged for your one-side version) – Pietro Majer Apr 13 '14 at 15:35
But the proof from Zygmund's book is not related to my problem; I know the standard method of proving Hardy-Littlewood inequality in $\mathbb{R}^N$. The main problem is that I have this set $\{f^*>\lambda\}$ under the integral. And I could work out the case when $f>0$ or replace with $|f|$, but it does not help me for this general case. – Xiao Apr 13 '14 at 19:51
@MarianoSuárez-Alvarez' comment (and its votes, probably) referred to textbook problems on MathOverflow, where the question was first posted, not to – zyx Apr 14 '14 at 23:55

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