7-1

Solution 7.1.
Let f be measurable in \mathbb{R}^{n} and different from zero in some set of positive measure. Show that there is a positive constant c such that  f^{*}(x) \geq c|x|^{-n}  for  |x| \geq 1 .
Solution:
Let  E \subset \mathbb{R}^{n} be a bounded subset of positive measure on which |f|>0, let  x \in \mathbb{R}^{n} and |x| \geq 1, and let Q^{\prime} be the smallest cube centered at x containing E . If the side length of Q^{\prime}is k|x|, take  c=\frac{\int_{E}|f|}{k^{n}} . Then we have
f^{*}(x)=\sup _{Q} \frac{1}{|Q|} \int_{Q}|f| \geq \sup _{Q} \frac{1}{|Q|} \int_{Q \cap E}|f| \geq \frac{1}{\left|Q^{\prime}\right|} \int_{Q^{\prime} \cap E}|f|=\frac{\int_{E}|f|}{k^{n}|x|^{n}}=c|x|^{-n},
where the sup is taken over all cubes centered at x.