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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.

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  • $\begingroup$ Is there a typo in this? The operator does not seem to depend on $p$ $\endgroup$ Commented Feb 5, 2013 at 20:07
  • $\begingroup$ I guess, we consider $M:L_p (R^n) \rightarrow L_p (R^n)$. $\endgroup$ Commented Feb 5, 2013 at 21:35

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In the case of the cube, it was shown by Bourgain (very recently) 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 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).

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  • $\begingroup$ Thank you very much for the updating information, Mark Lewko. I knew the result of Aldaz for the case $p=1,$ but not Bourgain's one. The recent Bourgain's paper is really good source for what I am looking for. Hahn. $\endgroup$ Commented Feb 5, 2013 at 22:30

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