# What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function?

The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup_{\delta>0,\left|y-x\right|<\delta} \text{Avg}_{B(y,\delta)} \left|f\right|$, where $\text{Avg}_{B(y,\delta)} \left|f\right| = \int_{\left|z\right|<\delta} f(y-z) dz$.

The following results regarding the (uncentered) Hardy-Littlewood maximal function are well-known and can be found in many basic texts on analysis (e.g. Loukas Grafakos' "Classical Fourier Analysis", Chapter 2, pages 78-81):

• The Hardy-Littlewood maximal function is a bounded operator from $L^1(\mathbb{R}^n)$ to $L^{1,\infty}(\mathbb{R}^n)$ (i.e., weak $L^1$) of norm at most $3^n$ ($n$ is the dimension of the Euclidean space).

• Since the Hardy-Littlewood maximal function is also a bounded operator from $L^{\infty}(\mathbb{R}^n)$ to itself with norm at most $1$ (this is clear), we can apply the Marcinkiewicz interpolation theorem to conclude that for all $1 < p < \infty$, the operator norm of the Hardy-Littlewood maximal function is at most $2\left(\frac{p}{p-1}\right)^{\frac{1}{p}}3^{\frac{n}{p}}$. In fact, there is a slightly better bound: $\frac{p}{p-1}3^{\frac{n}{p}}$.

• The bound given above grows exponentially with the dimension $n$ (if $p$ is fixed). It is a fact that it cannot be improved to a bound that does not grow exponentially with the dimension $n$.

My questions:

Is an exact value for the norm of the (uncentered) Hardy-Littlewood maximal function, viewed as a bounded operator from $L^p$ to itself ($1<p<\infty$), known? If so, what is it?

Also, what is the norm of the Hardy-Littlewood maximal function when it is viewed as an operator from $L^1$ to weak $L^1$ (if it is known)?

Are the answers to the analogous questions regarding the centered Hardy-Littlewood maximal function known?

I apologize if this question is too basic. It seems like a fairly simple question but it is not clear (at least to me) how to solve it.

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.

• Thank you so much for your answer, Mark! (I'm very sorry I didn't see it until now.) I really, really appreciate that you took the time to write such a detailed answer. Thank you! Mar 29, 2015 at 15:54

I just searched in google, "best constants for uncentered maximal functions", by Grafakos and Smith. "the best constants for the centered H-L maximal inequality", by AD Melas.(1-D, weak type (1,1)). maybe you can find more results following this two.

• A few comments (a) the second author of the first paper you refer to is Montgomery-Smith, not just Smith. (b) The sharp constants computed by Grafakos and Montgomer-Smith for the uncentered operator, and the sharp constant by Melas, are both results in 1 dimension. Less is known for higher dimensions; though there are known improvements over the exponential bounds for the centered maxmimal functions. The classic paper in that direction is the 1983 paper of Eli Stein and J. Stromberg, though I'm sure there must be other references since then. Jul 6, 2011 at 13:50

Uncenteed maximal operator is proven to be unbounded as $L^p \to L^p$ for general norm (see L. Grafakos and J. Kinnunen, "Sharp inequalities for maximal functions associated with general measures") Actually the same result is for $L^1 \to L^{1,\infty}$ (ADAM OSEKOWSKI "BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS"). In those papers one can find some results regarding strong maximal function.