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Mark Kim
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Those are basic yet difficult questions. I don't know much about the uncentered case, but here is some information on the centered case.

A nonempty set $$B \subseteq \mathbb{R}^d$$ is centrally symmetric with respect to $$p \in B$$ if $$B$$ is invariant under the affine transform $$x \mapsto 2p - x$$. We say that $$B$$ is a centrally symmetric convex body if $$B$$ is compact, convex, and centrally symmetric with respect to the origin.

Consider the centered Hardy-Littlewood maximal function over a centrally symmetric convex body $$B \subseteq \mathbb{R}^d$$ $$\mathcal{M}_Bf(x) = \sup_{r > 0} \frac{1}{m(rB)} \int_{rB} |f(x+y)| \, dy,$$ which is defined for all $$f \in L^1_{\mathrm{loc}}(\mathbb{R}^d)$$. As Willie said in his comment, the classical result is:

Theorem 1 (Stein-Stromberg, 1983). There exists a constant $$c > 0$$ such that $$\|\mathcal{M}_B\|_{L^1(\mathbb{R}^d) \to L^{1,\infty}(\mathbb{R}^d)} \leq c \, d \log d$$ for each $$d > 1$$ and every centrally symmetric convex body $$B \subseteq \mathbb{R}^d$$. Over the standard Euclidean ball $$B_2$$, we have the improved bound $$\|\mathcal{M}_{B_2}\|_{L^1(\mathbb{R}^d) \to L^{1,\infty}(\mathbb{R}^d)} \leq c' \, d$$ for all $$d \geq 1$$.

In the same paper, Stein and Strömberg conjectured the following, which is unresolved as of today (May 19, 2014):

Conjecture 2 (Stein-Strömberg maximal function conjecture, 1983). For each fixed centrally symmetric convex body $$B \subseteq \mathbb{R}^d$$, $$\sup_{d \geq 1} \|\mathcal{M}_B\|_{L^1(\mathbb{R}^d) \to L^{1,\infty}(\mathbb{R}^d)} < \infty.$$

Stein and Strömberg also conjectured a stronger statement, which asserts a uniform control over all dimensions $$d$$ and centrally symmetric convex bodies $$B$$. J. M. Aldaz disproved the conjecture in 2011:

Theorem 3 (Aldaz, 2011). If $$B = B_\infty$$, the $$l_\infty$$ ball, then $$\lim_{d \to \infty} \|\mathcal{M}_B\|_{L^1(\mathbb{R}^d) \to L^{1,\infty}(\mathbb{R}^d)} = \infty.$$

No significant improvement on the Euclidean-ball Stein-Strömberg maximal function conjecture is known. The best constant is known only in dimension 1, as Shaoming mentioned:

Theorem 4 (Melas, 2003). $$\|\mathcal{M}_{B_2}\|_{L^1(\mathbb{R}) \to L^{1,\infty}(\mathbb{R})} = \frac{11 + \sqrt{61}}{21}.$$

In fact, A. Naor and T. Tao showed that the convex-body Stein-Strömberg bound $$d \log d$$ is essentially sharp in a large class of metric measure spaces. For details, see Tao's blog post on the paper. Perhaps there is no uniform bound, after all:

These results suggest (at least to us) that uniform bounds...may fail to exist if one uses euclidean balls (the original question of Stein and Strömberg) since there seems to be no reason to believe that the maximal operator associated to euclidean balls is substantially smaller than the maximal operator associated to cubes. (Aldaz-Lázaro, 2013; p.228)

As for the $$L^p \to L^p$$ bounds, the classical result is due to E. M. Stein:

Theorem 5 (Stein, 1982). For each $$1 < p \leq \infty$$ and every centerally symmetric convex body $$B \subseteq \mathbb{R}^d$$, $$\sup_{d \geq 1} \|\mathcal{M}_B\|_{L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d)} < \infty.$$

Improved results in this direction are mostly due to J. Bourgain. The classical one is the following:

Theorem 6 (Bourgain, 1986: paper 1, paper 2). For each $$p > 3/2$$, $$\sup_{d,B} \|\mathcal{M}_B\|_{L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d)} < \infty,$$ where the supremum is taken over all $$d \geq 1$$ and centrally symmetric convex bodies $$B \subseteq \mathbb{R}^d$$.

It is then natural to conjecture the following:

Conjecture 7 (Bourgain's maximal function conjecture, 1986). For each $$p > 1$$, $$\sup_{d,B} \|\mathcal{M}_B\|_{L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d)} < \infty,$$ where the supremum is taken over all $$d \geq 1$$ and centrally symmetric convex bodies $$B \subseteq \mathbb{R}^d$$.

Once again, Bourgain's maximal function conjecture is unresolved as of today (May 19, 2014). Bourgain proved his own conjecture for the $$l^\infty$$-ball $$B_\infty$$ in December 2012. Another partial result of note is that of D. Müller from 1990, which provides an estimate based on various geometric conditions on the body $$B$$. See Müller's paper for details.

Mark Kim
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