Let $B$ be a centrally symmetric convex body in $\mathbb R^n.$ The maximal function associated to $B$ is defined by
$$
Mf = \sup\_{r>0}(\chi\_{B})\_{r}\*|f|.
$$
Bourgain (<http://www.jstor.org/stable/info/2374532>) proved that this operator is bounded on $\mathbb R^n$ for all $p>3/2$ with constant depending only on $p.$


The question is that: Is this operator bounded for $p>\lambda,$ ($1<\lambda\le 3/2$) with constant independent of the dimension? Or can we find a counterexample for that?
Any references?


Thank you.


Hahn.



Maximal function associated to convex bodies
In the case of the cube, it was shown by Bourgain ([very recently](http://arxiv.org/abs/1212.2661)) that the constants remain independent of $n$ for all $p>1$. The problem appears to be open for the case of more general centrally symmetric convex bodies. On page 3 of his recent preprint, Bourgain writes:


"While it is reasonable to believe that this statement holds in general, our argument is based on a very explicit analysis which does not immediately carry over to other convex symmetric bodies."


For $p=1$, J. Aldaz [has shown](http://arxiv.org/abs/0805.1565) that the (weak $L^1$) constant can't be taken independent of $n$ in the case of a cube. The case of the ball is open (this problem was briefly discussed on Gil Kalai's blog [here](http://gilkalai.wordpress.com/2012/11/17/a-few-mathematical-snapshots-from-india-icm2010/)).


