Some questions on the Hardy Littlewood Maximal Function

Question

I'm working on the following problem. $H$ is the Hardy Littlewood Maximal function:

If $f \in L^1(\mathbb{R}^n), f \neq 0$, there exists $C, R>0$ s.t. $Hf(x) \geq C|x|^{-n}$ for $|x| > R$. Hence $m(\{x:Hf(x) > \alpha \}) \geq C'/\alpha$ when $\alpha$ is small, so the estimate in the maximal function is essentially sharp.

I have some questions:

1. I can see there should be a constant $C$ by just writing out the defintion of $H$ and doing some manipulations. I can't see why this should be strictly positive though. $C$ will involve an integral of $f$ over a ball of radius $R$. For this to be zero, $f$ can be zero a.e. The hypothesis doesn't exclude this though - only $f=0$. Am I missing something?

2. What does "essentially sharp" mean?

3. In general, I'm having trouble interpreting what this problem is trying to point out. The maximal theorem gives an upper bound $m(\{x:Hf(x) > \alpha \})$ - is this problem having us investigate the lower bound?

Answer 1

The inequality is trivially true with $C=0$. What we need to prove is that there is a $C>0$ for with the inequality holds. For simplicity I will assume the non-centered maximal function. Let $R>0$ be such that $$ \int_{|x|<R'}|f(x)|\,dx\ge\frac{\|f\|_1}{2}\quad\forall R'\ge R. $$ If $a\in\Bbb R^n$ with $|a|>R/2$, then $$ Hf(a)\ge\frac{1}{m(B_{2|a|}(0))}\int_{B_{2|a|}(0)}|f(x)|\,dx\ge C\,\|f\|_1\,|a|^{-n}, $$ where $C>0$ depends only on $n$.

From here you get a lower bond on $m(\{x:Hf(x)>\alpha\})$ which is of the same order of the upper bound given by the maximal theorem: $$ \frac{c}{\alpha}\,\|f\|_1\le m(\{x:Hf(x)>\alpha\})\le\frac{C}{\alpha}\,\|f\|_1. $$ This is what is meant by essentially sharp.