Let $f$ be in $L^{2}(\mathbb{R}^n)$ then $f^{*}\in L^{2}(\mathbb{R}^n)$?
Is there any weaker result like $f^* \in L^2_{loc}(\mathbb{R}^n)$ or $ff^*\in L^{1}((\mathbb{R}^n)$?
Notation:
$f^*(x)=\sup_{x\in B}\int_{B}|f(y)|\;dy$
where $B$ are ball containing $x$.
I want a proof of the weaker statements without the main result( which I don't know if is true). A perfect answer is: proof the main result or give it a counter example and proof the weaker statements or give counter examples to all of them.
Whatever I am happy if you can proof the weaker thesis.